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Њ = # p .7Բ # SORITES # X
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An International Digital Journal of Analytical Philosophy
j d Indexed and Abstracted in THE PHILOSOPHER'S INDEX
. ISSN 1135-1349
! G
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,H P# Editor: Lorenzo Pe9a
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Associate Editor: Txetxu Aus1n
+
[ ! ; (Spanish Institute for Advanced Studies)
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t JeanYves B)ziau, Enrique Alonso, GuillermoHurtado, ManuelLiz,
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P n) ) t Issue #17 " October 2006 n) =p-p-p-
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SORITES ($2"($)
. y ISSN 1135-1349
) Issue #17 " October 2006
" Copyright by SORITES and the authors
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Main InterNet Access:
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b 8 M X` hp x (#%'0*,.8135@8: 5 SORITES
1 ISSN 1135-1349
H 0 Roll of Referees
M b Rainer Bornp9!1 JohannesKepler Universitaet Linz (Austria)
Amedeo Contep9!A University of Pavia (Italy)
Newton C.A. da Costap9!< University of SMo Paulo (Brazil)
Marcelo Dascalp9!= University of Tel Aviv (Israel)
Dorothy Edgingtonp|9!@ Birbeck College (London, UK)
Graeme Forbesp9!- Tulane University (New Orleans, Louisiana, USA)
Manuel Garc1a-Carpinterop 9!= University of Barcelona (Spain)
Laurence Goldsteinp!9!9 University of Hong Kong (Hong Kong)
Jorge Graciap_9!1 State University of New York, Buffalo (USA)
Nicholas Griffinp9!- McMaster University (Hamilton, Ontario, Canada)
Rudolf Hallerp'9!3 KarlFranzensUniversitaet Graz (Austria)
Terence Horganp 9!6 University of Memphis (Tennessee, USA)
Victoria Iturraldep9!* Univ. of the Basque Country (San Sebastian, Spain)
Tomis E. Kapitanp9!: Northern Illinois University (USA)
Manuel Lizp(9!- University of La Laguna (Canary Islands, Spain)
Peter MenziespT9!( Australian National University (Canberra, Australia)
Carlos Moyap9!> University of Valencia (Spain)
Kevin Mulliganp9!: University of Geneva (Switzerland)
JesCs PadillaGlvezp9!1 JohannesKepler Universitaet Linz (Austria)
Philip PettitpT9!( Australian National University (Canberra, Australia)
Graham Priestp9!. University of Queensland (Brisbane, Australia)
Eduardo Rabossip9!6 University of Buenos Aires (Argentina)
DavidHillel Rubenp9! School of Oriental and African Studies, University of London
Mark Sainsburyp9!A King's College (London, UK)
Daniel Schulthesspn9!7 University of Neuch tel (Switzerland)
Peter Simonsp9!= University of Leeds (Leeds, UK)
Ernest Sosap_9!, Brown University (Providence, Rhode Island, USA)
Friedrich Stadlerp9!5 Institut Wien Kreis
(Vienna, Austria) & = p-p-p- `h
p x( !0$& b/
jcM"z *!#%2(*, b # -t \ P C 'qP# =
/ g 5 SORITES
1 ISSN 11351349
, Issue #17 " October 2006
v m 0 Table of Contents
a b
o bb " Abstracts of the Papers9!pi 9!Z 03
o bb " About Properties of LInconsistent Theories
by Vyacheslav Moiseyev9!pi 9!Z 07
o bb " Paraconsistent logic! (A reply to Slater)
by Jean-Yves B)ziau9!pi 9!Z 17
o bb " The Logic of Lying
by Moses @k/9!pi 9!Z 27
o bb " Sparse Parts
by Kristie Miller9!pi 9!Z 31
o bb " Are Functional Properties Causally Potent?
by Peter Alward9!pi 9!Z 49
o bb " Subcontraries and the Meaning of `If8Then'
by Ronald A. Cordero9!pi 9!Z 56
o bb " Does Frege's Definition of Existence Invalidate the Ontological
Argument?
by Piotr Labenz9!pi 9!Z 68
o bb " Why Prisoners' Dilemma Is Not A Newcomb Problem
by P. A.
Woodward9!pi 9!Z 81
o bb " A Paradox Concerning Science and Knowledge
by Margaret Cuonzo9!pi 9!Z 85
o bb " Between Platonism and Pragmatism: An alternative reading of Plato's
Z | Theaetetus
by Paul F. Johnson9!pi 9!Z 95
o bb " Blob Theory: Nadic Properties Do Not Exist
by Jeffrey Grupp9!p 9!Y 104
Z L o bb " The SORITES Charter9!p 9!Y 132
o bb " Release Notice9!p 9!Y 137 = p-p-p- b # 'V 2 P kC 4P# X01 Í Í. X01 Í Í. 4 X 1 Í . X* Í . 4 # 4o \ P C +XP# = 5 a v 1 )
W B a " SORITES Issue #17 " October 2006issnĠ1135-1349`!%"] ă
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W B a " SORITES ĠIssue #17 " October 2006. issnĠ1135-1349`!%"] ă
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A d d x @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
( Issue #17 " October 2006. Pp. 36
- W Abstracts of the Papers
% Copyright by SORITES and the authors -- , a 5 a v
s * ] Abstracts of the Papers
+
X B
About Properties of LInconsistent Theories
X * A by Vyacheslav Moiseyev
, a In the paper a new type of the formal theory, Linconsistent theory
, is constructed and
some properties of such theories are investigated. First a theory T* is defined as a set of
limiting sequences of formulas from a theory T with a language L. A limiting sequence
X { An }n=1 of the formulas from T is said to be a theorem of the theory T* if there exists an
X m0 such that for any nm the formula An of the language L is a theorem of the theory T.
X T is embeded into T*. Then, a theorem of T* is called an Lcontradiction if the limit of this
X theorem equals B U B , where B is a formula of the language L. Finally, the theory T* is said
X o to be an Linconsistent theory if there exists an Lcontradiction in T*. It is proved that the
theory T* is consistent, complete, etc., iff the theory T is consistent, complete, etc. However,
T* contains more theorems and inferences than T (see Theorems 9 and 10). Linconsistent
theory T* can be presented as a new approach to the Philosophical Logic, dealing with an
extension of Method of Limits to thinking. Namely some philosophical antinomies, for
example Kantian ones, could be presented as Lcontradictions in an Linconsistent theory.
a+ ك
X a! Paraconsistent logic! (A reply to Slater)
X a, by Jean-Yves B)ziau ă
We answer Slater's argument according to which paraconsistent logic is a result of a
verbal confusion between contradictories
and subcontraries
. We show that if such notions
are understood within classical logic, the argument is invalid, due to the fact that most
paraconsistent logics cannot be translated into classical logic. However we prove that if such
notions are understood from the point of view of a particular logic, a contradictory forming
function in this logic is necessarily a classical negation. In view of this result, Slater's
argument sounds rather tautological.
a+ ك
X % , The Logic of Lying
X
' / by Moses @k/
) a By definition, a lie is a dishonestly made statement. It is a wilful misrepresentation, in
one's statement, of one's beliefs. Both a truthful person and a liar could hold false beliefs. We
should not uncritically regard an untruthfully made statement as an untrue statement, or a
truthfully made statement as a true statement. The only instance when a lie is necessarily false '+ = p-p-p- is when the liar's corresponding belief that was distorted was true. In other instances, the lie
could be either true or false. We conclude that a lie is not necessarily a false statement.
a+ ك
X ;a/ Sparse Parts
X a- by Kristie Miller ă
Four dimensionalism, the thesis that persisting objects are four dimensional and thus
extended in time as well as space, has become a serious contender as an account of
persistence. While many four dimensionalists are mereological universalists, there are those
who find mereological universalism both counterintuitive and ontologically profligate. It would
be nice then, if there was a coherent and plausible version of four dimensionalism that was
nonuniversalist in nature. I argue that unfortunately there is not. By its very nature four
dimensionalism embraces theses about the nature of objects and their borders that make any
version of nonuniversalist four dimensionalism either incoherent or at least highly
implausible.
a+ ك
X k a Are Functional Properties Causally Potent?
X a. by Peter Alward ă
Kim has defended a solution to the exclusion problem which deploys the causal
inheritance principle
and the identification of instantiations of mental properties with
instantiations of their realizing physical properties. I wish to argue that Kim's putative solution
to the exclusion problem rests on an equivocation between instantiations of properties as
X bearers of properties and instantiations as property instances. On the former understanding,
the causal inheritance principle is too weak to confer causal efficacy upon mental properties.
And on the latter understanding, the identification of mental and physical instantiations is
simply untenable.
a+ ك
X H Subcontraries and the Meaning of If8Then
X + by Ronald A. Cordero
) a In this paper I maintain that useful, assertable conditional statements with subcontrary
antecedents and consequents do actually occur. I consider the paradoxical results of applying
rules of inference like Transposition in such cases and argue that paradox can be avoided
through an interpretation of conditionals as claims that the truth of one statement would permit
a sound inference to the truth of another.
a+ ك
X h% a Does Frege's Definition of Existence Invalidate the Ontological Argument?
X & a. by Piotr Labenz ă
It is a well-known remark of Frege's that his definition of existence invalidated the
ontological argument for the existence of God. That has subsequently often been taken for
granted. This paper attempts to investigate, whether rightly so. For this purpose, both Frege's
ontological doctrine and the ontological argument are outlined. * p-++!! Ԍ Arguments in favour and against both are considered, and reduced to five specific
questions. It is argued that whether Frege's remark was right depends on what the answers to
these questions are, and that for the seemingly most plausible ones " it was not.
a+ ك
X &
a Why Prisoners' Dilemma Is Not A Newcomb Problem
X a- by P. A. Woodward ă
David Lewis has argued that we can gain helpful insight to the (all too common)
Prisoners' Dilemmas that we face from the fact that Newcomb's Problems are easy to solve,
and the fact that Prisoners' Dilemmas are nothing other than two Newcomb Problems side by
side. The present paper shows that the (all too common) Prisoners' Dilemmas that we face are
significantly different from Newcomb Problems in that the former are iterated while the latter
are not. Thus Lewis's hope that we can get insight into the former from the latter is illusory.
a+ ك
X Y a A Paradox Concerning Science and Knowledge
X a, by Margaret Cuonzo ă
Quine's and Duhem's problem regarding the laying of blame
that occurs when an
experimental result conflicts with a scientific hypothesis can be put in the form of a standard
X ) philosophical paradox. According to one definition, a philosophical paradox is an argument
with seemingly true premises, employing apparently correct reasoning, with an obviously false
or contradictory conclusion. The Quine/Duhem problem, put in the form of a paradox, is a
special case of the skeptical paradox. I argue that both the Quine/Duhem paradox and the
skeptical paradox enjoy the same type of solution. Both paradoxes have the kind of restricted
solution that Stephen Schiffer calls mildly unhappyface
solutions. Although there can be
no solution to these two paradoxes that gives an accurate account of the relevant notions (e.g.,
knowledge), replacement notions are given for the ones that lead to the paradoxes.
a+ ك
X H ~a Between Platonism and Pragmatism: An alternative reading of Plato's Theaetetusă
X a, by Paul F. Johnson ă
In a letter to his friend Drury, Wittgenstein claims to have been working on the same
X problems that Plato was working on in the Theaetetus. In this paper I try to say what that
problem might have been. In the alternative reading of the dialogue that I construct here,
attention is drawn to Socrates' frequent appeal in the course of discussion to the ordinary ways
of speaking that he, and Theaetetus, and everyone else in Athens at the time engaged in. The
more abstruse theories of Heraclitus and Protagoras which Socrates and Theaetetus are
discussing are found to do violence to these ordinary ways of talking, and found seriously
wanting as a result. A case is made that the conventions and presuppositions of ordinary
conversational speech are inherently normative, and constitute a valid standard against which
philosophical theories may be measured. Lines of affinity are drawn between these claims
advanced by Plato and the recent work of contemporary neopragmatists, and Robert
Brandom's work in particular.
a+ ك k* p-++!! Ԍ X a Blob Theory: Nadic Properties Do Not Exist
X _ ga- by Jeffrey Grupp ă
I argue for blob theory: the philosophic position that nadic properties do not exist. I
discuss hitherto unnoticed problems to do with the theories of property possession in the
ontological theories of ordinary objects: the bundle theory of objects and substance theories
of objects. Specifically, I argue that theories of property possession involved with the bundle
theory and substance theories of objects are contradictory, and the best theory we have been
given by metaphysical realists is a theory that reality is propertyless. K p-++!! 5 a v 7 * H2 v W B a " About Properties of LInconsistent Theories
by Vyacheslav Moiseyev`!%"] ă
y I d d d y 7 =
a a @
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a d d x @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 716
# About Properties of LInconsistent Theories
!
Copyright by Vyacheslav Moiseyev and SORITES -- , a 5 a v
s o About Properties of LInconsistent Theories
X , Vyacheslav Moiseyev
+
a Apparently, there have been two traditions in the history of logic, these are Line of
Parmenide and Line of Heraclitus. Former is originated from the ideas of ParmenideAristotle
and is based on the Law of Identity. This line constitutes formal logic. Latter is originated
from the ideas of HeraclitusPlato and has been expressed itself in the ideas of dialectics, or
dialectical logic. Contemporary mathematical logic is the worthy result of the development of
the first line. Possibility of good precision and clear procedures of justification is the most
strong side of this line. On the other hand, dialectics always have been trying to deny the
meaning of Law of Identity. Dialectical ideal have been expressed itself in the idea of
contradiction. But a very big problem have been subsisted here. This is the problem which we
X shall call Problem of Logical Demarcation (PLD). Breafly speaking, essence of the problem
is in the following idea. Mistakes are contradictions too and if dialectics does not want to be
simply mistaken reasoning, then it must show a criterion with the help of which we could to
separate contradictionsmistakes from dialectical contradictions (antinomies). We shall call
X x such criterion as Criterion of Logical Demarcation (CLD). Although dialectics has not been
able to show CLD but there have been many interesting attempts to find the Criterion. One
can refer here to Plato, Nicholas from Cusa, Russian Philosophy of AllUnity, etc.
It seems to us that one of the interesting ideas here is the idea of some connection
between CLD and concept of limit. For example, Nicholas from Cusa tried to express idea of
God in the image of a straight line which is limit for the infinite sequence of tangent
circumferences. Our paper is an attempt to extend this trend and to formulate a version of
CLD, where dialectical contradictions (antinomies) can be expressed as limits of ifinite
sequences of formulas in a formal language. Main new idea is here in the technique of work
X ! with the limiting sequences of formulas, not terms. This idea is fully correlated with the
method of extension of rational numbers by irrational ones in mathematical analysis. As is
well known, every irrational number can be represented by a limiting sequence of rational
numbers. Then we can represent rational numbers itselves as a particular case of limiting
sequences, i.e., as stationary sequences. Thus we are passing to a new type of objects and we
can define operations with these objects generalizing of operations on rational numbers. The
same approach is demonstrated below but in the logical sphere.
Basic task here is to define limiting sequences of formulas. Separate formulas in a formal
language can be considered as analogues of rational numbers in analysis. Stationary sequences
of formulas must be a particular case of the definition of limiting sequence. We shall carry
out the task of limiting sequence of formulas definition by use of limiting sequences of terms.
Let us see the following simple example. Let 1/n = 1/n and 1/n c 1/n+1 be formulas in a
theory T generalizing theory of real numbers. Then for every n = 1, 2, 3, we can prove that
the corresponding formulas are theorems. Let us propose that we can prove also formula z, = p-p-p- X lim(1/n) = 0 in T (I shall mean the limit of sequence of terms an as n under the symbol
X lim(an)), i.e., limit of sequence of numbers 1/n is zero. (1/n = 1/n U 1/n c 1/n+1) is also
formula in T and we can consider the following infinite sequence of formulas
(1/1 = 1/1 U 1/1 c 1/2), (1/2 = 1/2 U 1/2 c 1/3), (1/3 = 1/3 U 1/3 c 1/4),
Every element of the sequence is formed as the result of substitution of constants 1/1, 1/2,
1/3, etc., for the places of variables in formula (x=x U x c y). For example, first formula can
X z be represented as (x=x U x c y) x,y [1/1, 1/2], i.e., as the result of substitution of constants 1/1
and 1/2 for variables x and y respectively. Hence we can rewrite the sequence of formulas in
the following form
X (x=x U x c y) x,y [1/1, 1/2], (x=x U x c y) x,y [1/2, 1/3], (x=x U x c y) x,y [1/3, 1/4],
It permits to us to use the following designation for the expression of this sequence
X u {(x=x U x c y) x,y [1/n, 1/n+1]}
Let us define the limit of this sequence as the result of substitution of limits of sequences
of terms for the variables. In our case we receive
X " lim((x=x U x c y) x,y [1/n, 1/n+1]) =Df (x=x U x c y) x,y [lim(1/n), lim(1/n+1)]
Since lim(1/n) = lim(1/n+1) = 0, we finally receive
X lim((x=x U x c y) x,y [1/n, 1/n+1]) = ((x=x U x c y) x,y [0, 0]) = (0 = 0 U0 c 0),
i.e., contradiction.
However, though limit of sequence of formulas is contradiction, every formula from the
sequence is theorem of T. Such consequence of formulas plays a role similar to role of
consequence of rational numbers which limit is absent between rational numbers, i.e., is an
X e irrational number. We shall call consequences of theorems which limit is contradiction as L X P contradiction, i.e., limit contradiction. Instead of working with contradiction we can work with
limiting consequence of formulas which limit is the contradiction. Logic of limiting
consequences of formulas is not poorer than logic of formulas since the last is generalized by
the former on the basis of stationary sequences.
Finally, we can formulate CLD with the help of the idea of Lcontradiction. Namely,
X U contradiction AUA is called an antinomy (dialectical contradiction) relatively consistent
X D theory T if AUA is formula of the language of T and there exists an extension of the theory
T to a theory T* of limiting consequences of formulas from T such that there exists an Lcontradiction from T* which limit equals AUA. Therefore it is clear that satisfactory decision
of CLD and PLD is the consequence of satisfactory formulation of the theory T* and its
properties. Below we shall investigate namely this problem.
Suppose T is a formal theory with a language L such that there exist formulas in L, which
X <% can be represented in the metalanguage as expressions of the form lim ( an ) = a
, where an
,
X %& a
are names for terms from L, n
is name for natural number n, and these formulas can
X ' be interpreted in a model M of the theory T as the equality of the limit of a sequence {an}
with individuals from M to an individual a from M. We shall say that such theory T is called
X ( tlimiting theory ( t
is used from term
). A theory of sets and theory of real numbers are
examples of tlimiting theories. )
p-++!! Ԍ X Let T be a tlimiting theory with a language L, where An is a formula from L such that
X _ (*) An = A x1, x2, , xm [ a1n p1, a2n p2, , amn pm ],
X where pj N ,
X ) N is set of natural numbers,
and j=1,,m.
X In other words, the formula An is the result of the substitution of terms a1n p1, a2n p2, ,
X amn pm for free entrances of the variables x1, x2, , xm into a formula A , where each of the
X terms ajn pj is an element of an infinite sequence { ajk } (k is variable of sequence here), and
theorems of the form
X limajn = aj
are deduced in the theory T for every j.
X
The sequence { An }n=1 is defined for the formulas An of the sort (*). By definition, put
X 0 A = lim An = A x1, x2, , xm [ lim ( a1n ), lim ( a2n ) ,, lim ( amn )].
X This definition allows us to reduce a concept of formula limit to limits of terms, which
are included into a formula.
X o DEFINITION 1. Sequences { An }n=1 of the formulas An of the sort (*) and also stationary
X sequences of the formulas from L are called limiting sequences of the formulas from L.%"
Let a language L* be the set of all the terms from L and also the set of all limiting
sequences of the formulas from L. The language L can be embeded into the language L* with
X the help of the injective map E*: L L* such that if ! is a term from L, then E*( ! )= ! , if A
X is a formula from L, then E*( A ) is the stationary sequence of the formulas A .
X I o DEFINITION 2. Limiting sequences of the formulas { An }n=1 are called formulas of the
X 4 language L*.%"
Thus the languages L and L* do not differ between themselves by the alphabets and sets
of terms but only sets of the formulas.
X o DEFINITION 3. We say that two formulas { An }n=1 and { Bn }n=1 from L* are called equal
X and this is denoted by { An }n=1 = { Bn }n=1
if the formula lim An (i.e. formula, which
X is limit of sequence { An }n=1) can be obtained from the formula lim Bn by right renaming
of bound variables.%"
X ! o DEFINITION 4. A formula { An }n=1 of the language L* is said to be a metatheorem of the
X " theory T if there exists an m0 such that for any nm the formula An of the language L
is a theorem of the theory T.%"
X 2% o DEFINITION 5. A limiting sequence of the formulas { An }n=1, which is a metatheorem of the
X & theory T, is called an Lcontradiction ( L
from limit
) if the limit of this sequence,
X ' lim An , equals B U B , where B is a formula of the language L.%"
X k( o DEFINITION 6. The set of metatheorems of the theory T is called a theory T*.%"
X ) In this case, the metatheorems of the theory T can be called also theorems of the theory
X * T*. The language L* is the language of the theory T*. We shall say that T* is called a ft * p-++!! ԫ X limiting theory ( f
from formula
). The approach, circumscribed above, can be considered
as a methodology of buildingup of ftlimiting theories on the basis of tlimiting theories. The
X theory T* is said to be an Linconsistent theory if there exists an Lcontradiction in T*.
X 5 o DEFINITION 7. The theory T* is called consistent if not all formulas from L* are theorems
of the theory T* (see also Theorem 20).%"
o THEOREM 1. If the theory T is consistent, then the theory T* is consistent.%"
X o PROOF. Suppose the theory T is consistent; then there exists a formula A from the language
X L such that A is not a theorem of the theory T. Let { An }n=1 be the stationary sequence,
X where for any n An is A . The sequence { An }n=1 is a formula from L* but it is not a
theorem of the theory T*. Therefore the theory T* is consistent.%"
o THEOREM 2. If the theory T* is consistent, then the theory T is consistent.%"
o PROOF. Assume the converse. Then the theory T* is consistent and the theory T is not. If T
is nonconsistent, then any formula of the theory T is the theorem of this theory. If T* is
X ) consistent, then there exists a formula { An }n=1 from L* such that { An }n=1 is not a
theorem of the theory T*. Hence for any m0 there exists an nm such that the formula
X An is not a theorem of the theory T. This contradiction proves the theorem.%"
o THEOREM 3. The theory T is consistent iff the theory T* is consistent.%"
o PROOF. See Theorems 1 and 2.%"
o DEFINITION 8. Let M be a structure for the language L. We shall say that a formula
X { An }n=1 from the language L* is valid in M if there exists an m0 such that for any nm
X the formula An is valid in M.%"
X U o DEFINITION 9. A structure M for the language L is called a model of the theory T* if any
theorem of the theory T* is valid in M.%"
X We shall say that a structure M for the language L is called a structure for the language
X L*.
o THEOREM 4. Let M be a model of the theory T; then M is a model of the theory T*.%"
X J o PROOF. Let a formula { An }n=1 from the language L* be a theorem of the theory T*. Then
X 3 there exists an m0 such that for any nm the formula An is a theorem of the theory T,
X " i.e., An is valid in the model M of the theory T. Therefore the formula { An }n=1 is valid
in M. Hence M is a model of the theory T*.%"
o THEOREM 5. Let M be a model of the theory T*; then M is a model of the theory T.%"
X " o PROOF. Let M be a model of the theory T* and A be a theorem of the theory T. Suppose
X # { An }n=1 is the stationary sequence, where An = A for any n; then { An }n=1 is a theorem
X $ of the theory T* and { An }n=1 is valid in M, i.e., there exists an m0 such that for any
X % nm the formula An is valid in M. Since An is A , we see that the formula A is valid in
M. Therefore M is a model of the theory T.%"
o THEOREM 6. M is model of the theory T iff M is model of the theory T*.%"
o PROOF. See Theorems 4 and 5.%" 7) p-++!! Ԍ X o DEFINITION 10. A formula { An }n=1 from the language L* is said to be an axiom of the
X theory T* if there exists an m0 such that for any nm the formula An is an axiom of the
theory T.%"
X 9 o DEFINITION 11. Suppose n is a set of formulas from L and there exists an m0 such that
X ( for any nm n An is an inference of an formula An from n in the theory T. Let the
X sequence of the sets {n}n=1 have the limit and the sequence of the formulas { An }n=1
X also have the limit. Then the object {n An }n=1 (i.e. sequence of inferences n An ,
X where n=1,2,3,) is called an inference in the theory T*. Denote by {n }n=1 T*
X { An }n=1
, or {n}n=1 { An }n=1
, any inference {n An }n=1.%"
X I o DEFINITION 12. An inference {n An }n=1 in the theory T* is said to be a proof in the
X 8
theory T* if there exists an m0 such that for any nm n is a set of axioms of the theory
X ' T or n is empty.%"
X o DEFINITION 13. A sequence of the sets {n}n=1 is called regular if {n}n=1 = {Bk=1N{ Akn }}
X {
(here {Bk=1N{ Akn }} is sequence, where n=1,2,3,, of unions Bk=1N{ Akn } of oneelement
X j sets { Akn }), while N is a finite natural number or infinity, and { Akn }n=1 (here { Akn } is
sequence of formulas of L, where n=1,2,3,) is a formula from L*. In this case, denote
X B by {{ Akn }n=1} k=1N
any {n }n=1 and denote by {{ Akn }n=1} k=1N { An }n=1
any
X 1 inference {n}n=1 { An }n=1. We shall say that the formula { An }n=1 is deduced from the
X set {{ Akn }n=1} k=1N of the formulas { Akn }n=1 in the theory T*.%"
X o THEOREM 7. If { An }n=1 is a theorem of the theory T*, then { An }n=1 is deduced in the
theory T* from axioms of the theory T*.%"
X o PROOF. Let { An }n=1 be a theorem and not be an axiom of the theory T*; then there exists
X an m0 such that for any nm the formula An is a theorem and is not an axiom of the
X theory T, i.e., there exists an inference Bn1 , Bn2 , , Bnk ( n ) An in the theory T, where Bn1 ,
X Bn2 , , Bnk ( n ) are axioms of the theory T. Here n = { Bn1 , Bn2 , , Bnk ( n )}. Further, if n
X } An , then n An , where n = B k=1n k. The sequence {Bk=1n k}n=1 has the limit, this
X l limit equals = Bk=1 k, and, for any nm, we have An . can be represented
X [ as { B1 , B2 , , BN }, where N is a finite number or infinity, and B1 , B2 , , BN are axioms
X J of the theory T. Let {*n}n=1 be the new sequence, where, for any n, *n = . Further,
X 9 for any nm, we have *n An and the sequences {*n}n=1, { An }n=1 have the limits.
X ( Hence the inference {*n}n=1 { An }n=1 is defined in the theory T*. Besides, the
X sequence {*n}n=1 is regular. Indeed, {*n}n=1 = {Bk=1N{ Bkn }}n=1, where Bkn = Bk for
X any k. It follows that {*n}n=1 = {{ Bkn }n=1} k=1N, where the stationary sequences of the
X axioms from T { Bkn }n=1 are the axioms of the theory T*. In other words, if { An }n=1 is
X ! a theorem and not an axiom of the theory T*, then there exists the inference {{ Bkn }n=1}
X " k=1N { An }n=1 of the theorem { An }n=1 of the theory T* from axioms of the theory T*.
X # If { An }n=1 is an axiom of the theory T*, then there exists the inference { An }n=1 of the
X $ theorem { An }n=1 of the theory T* from the empty set of axioms of the theory T*.%"
Let Thm* be the set of all the theorems of the theory T*, Thm be the set of all the
theorems of the theory T.
o THEOREM 9. If the theory T is consistent and the theory T* contains an Lcontradiction, then
there does not exist a map h: Thm* Thm such that%"
1) h is a bijection, *
p-++!! Ԍ X 2) h({ An }n=1) is a theorem of the theory T iff { An }n=1 is a theorem of the theory T *,
X _ 3) if { An }n=1 is a stationary sequence, then h({ An }n=1) = An .
o PROOF. Assume the converse, i.e., there exists a map h with properties 1, 2 and 3. It follows
X that if { An }n=1 is a theorem of the theory T*, then there exists the theorem A from T
X such that h({ An }n=1) = A . Let { Bn }n=1 be the stationary sequence such that Bn is A for
X y any n. Therefore, we have h({ An }n=1) = A = h({ Bn }n=1). Since h is bijection, we obtain
X b { An }n=1 = { Bn }n=1, i.e., any theorem of the theory T* equals some stationary sequence
X K of theorems from the T. On the other hand, let { Cn }n=1 be an Lcontradiction, i.e.,
X 4 Cn Ġ= C , where { Cn }n=1 is a theorem from T* and C is a contradiction. By assumption,
X { Cn }n=1= { Dn }n=1, where { Dn }n=1 is a stationary sequence of the theorems from T, i.e.,
X
for any n, Dn is D and D is a theorem from T. This implies that Cn is Dn but Cn is C , and
X
C is a contradiction, Dn is D , and D is a theorem from T. Since the theory T is consistent,
X we see that C can not be a theorem of T, i.e., C can not be equal to D . This contradiction
proves the theorem.%"
X o DEFINITION 15. Let the map take each formula { An }n=1 from L* to lim An and take each
X term ! from L* to ! . The map : L* L is called a natural embedding of the language
X L* into the language L. On the other hand, let the map * take each formula A from L
X to { An }n=1, where An is A , and take each term ! from L to ! . The map *: L L* is
X called a natural embedding of the language L into the language L*.%"
X = Obviously, if A is a theorem of the theory T, then *( A ) is also a theorem of the theory
T*. The return relation, as follows from Theorem 9, is not correct.
Maps and * can be extended to the set of inferences in the theories T* and T. Namely
X z if an inference {n An }n=1 is given in the theory T*, then we can define the object ({n
X i An }n=1) = lim{n An } = limn lim An (in accordance with Theorem 10, the object limn
X X lim An is not always an inference of the theory T. In this case, the sign
is used as a
X A formal character). On the other hand, if an inference A is given in the theory T, then by
X 0 definition, put *( A ) ={n An }n=1, where n is and An is A for any n. Obviously, if
X A is an inference in the theory T, then *( A ) is an inference in the theory T*. The
return relation is not always correct (see below).
o THEOREM 10. If the theory T is a consistent theory and the theory T* contains an L X V contradiction, then there exists an inference {n An }n=1 in the theory T* such that
X E ({n An }n=1) is not an inference of the theory T.%"
X o PROOF. Let {n An }n=1 be an inference in the theory T*, where for any n n = H, i.e., the
X ! inference is a proof, and { An }n=1 is an Lcontradiction. In this case, ({n An }n=1) =
X " lim{n An } = limn lim An = lim An , where lim An is a contradiction. Since the
X w# theory T is consistent, we see that the object lim An can not be an inference in the theory
T.%"
X % o DEFINITION 16. Let the theory T be a theory with axiom schemes and { An }n=1 be an axiom
X & of the theory T*. If there exists an m0 such that for any nm An belongs to one axiom
X ' scheme A, then we say that { An }n=1 belongs to the axiom scheme A.%"
X ( The theory T* is called a theory with axiom schemes if the theory T is a theory with
X ) axiom schemes and for any axiom { An }n=1 in T* there exists an axiom scheme A such that
X * { An }n=1 belongs to A. * p-++!! Ԍ o THEOREM 11. Let T and T* be theories with axiom schemes and axioms of different
schemes be mutually independent in the theory T; then axioms of different schemes in
the theory T* are mutually independent.%"
X 1 o PROOF. Assume the converse. Therefore if A , B are axioms of different schemes in the
X theory T, then there does not exist an inference A B in the theory T. Besides, there
X exist axioms of different schemes { An }n=1 and { Bn }n=1 in the theory T* such that
X { An }n=1 and { Bn }n=1 are not mutually independent, i.e., there exists an inference { Bn }n=1
X { An }n=1 in the theory T*. It follows that there exists an m0 such that for any nm
X there exists an inference Bn An in the theory T, where Bn and An are axioms of
different schemes in the theory T. This contradiction proves the theorem.%"
o THEOREM 12. Let the theories T and T* be theories with axiom schemes. If any two axioms
of different schemes in the theory T* are mutually independent, then any two axioms of
different schemes in the theory T are mutually independent.%"
o PROOF. Assume the converse, i.e., any two axioms of different axiom schemes of the theory
X & T* are mutually independent and there exist axioms A and B of different schemes in the
X theory T such that there exists an inference A B in the theory T. Then there exists the
X inference { Bn }n=1 { An }n=1 in the theory T*, where An is A and Bn is B for any n.
X Besides, the stationary sequences { An }n=1 and { Bn }n=1 are axioms of different schemes
in the theory T*. This contradiction proves the theorem.%"
o THEOREM 13. Let the theories T and T* be theories with axiom schemes. Then axioms of
different schemes in the theory T are mutually independent iff axioms of different
schemes in the theory T* are mutually independent.%"
o PROOF. This follows from Theorems 11 and 12.%"
o THEOREM 14. Let the theories T and T* be theories with axiom schemes. If there does not
exist an axiom of the theory T, which can be deduced from any finite set of axioms of
another schemes of the theory T, then this is correct for axioms of the theory T*.%"
o PROOF. Assume the converse, i.e., the condition of the theorem is true and there exist an
X axiom { An }n=1 and a set of axioms of another schemes {{ Bkn }n=1}k=1N in the theory T*
X such that there exists an inference {{ Bkn }n=1}k=1N { An }n=1 in the theory T* and N is
X a finite number. The expression {{ Bkn }n=1}k=1N
guesses that the set of the axioms
X {{ Bkn }n=1}k=1N is regular, i.e., {{ Bkn }n=1}k=1N = {{ Bkn }k=1N}n=1 and there exists an m0
X } such that for any nm there exists an inference { Bkn } k=1N An in the theory T, where the
X l formulas Bkn and An are axioms of different schemes in the theory T. This contradiction
proves the theorem.%"
o THEOREM 15. Let the theories T and T* be theories with axiom schemes. If there does not
exist an axiom of the theory T*, which can be deduced from any finite set of axioms of
another schemes of the theory T*, then this is correct for axioms of the theory T.%"
o PROOF. Assume the converse, i.e., the condition of the theorem is true and there exist an
X & axiom A and axioms B1 , B2 , , Bn , which are of another schemes than A , such that there
X ' exists an inference B1 , B2 , , Bn A in the theory T. Let {{ Bki }i=1}k=1 { Ai }i=1 be the
X ( inference of the theory T*, where Ai is A and Bki is Bk for any i. This inference will be
X ) an inference of the axiom { Ai }i=1 from the axioms { Bki }i=1 of another schemes. This
contradiction proves the theorem.%" ~* p-++!! Ԍ o THEOREM 16. Let the theories T and T* be theories with axiom schemes. There does not
exist an axiom of the theory T*, which can be deduced from any finite set of axioms of
another schemes of the theory T* iff this is correct for axioms of the theory T.%"
o PROOF. See Theorems 14 and 15.%"
X o DEFINITION 17. The theory T (T*) is called semantically complete if any formula from the
language L (L*), which is valid in any model M of the theory T (T*), is a theorem of the
theory T (T*).%"
o THEOREM 17. If the theory T is semantically complete, then the theory T* is semantically
complete.%"
o PROOF. Assume the converse, i.e. the theory T is semantically complete and the theory T*
X
is not. It follows that there exists a formula { An }n=1 of the language L* such that
X { An }n=1 is valid in any model M of the theory T* and at the same time { An }n=1 is not
X a theorem of the theory T*. If { An }n=1 is valid in a model M of the theory T*, then there
X
exists an m0 such that for any nm the formula An is valid in the model M. Since,
according to Theorem 6, the model M of the theory T* is simultaneously a model of the
X theory T, we see that for any nm the formula An is valid in the model M of the theory
T. Since M is any model of the theory T and the theory T is semantically complete, we
X _ see that for any nm the formula An is a theorem of the theory T, i.e., the formula
X N { An }n=1 from the language L* is a theorem of the theory T *. This contradiction proves
the theorem.%"
o THEOREM 18. If the theory T* is semantically complete, then the theory T is semantically
complete.%"
o PROOF. Assume the converse, i.e., the theory T* is semantically complete and the theory T
X is not. It follows that there exists a formula A of the language L such that A is valid in
X any model M of the theory T and at the same time A is not a theorem of the theory T.
X Let { An }n=1 be the formula of the language L* such that An is A for any n. The model
M, according to Theorem 6, is simultaneously a model of the theory T*. Then the
X k formula { An }n=1 is valid in any model M of the theory T*. Since the theory T* is
X T semantically complete, we see that the formula { An }n=1 is a theorem of the theory T*,
X = i.e., the formula A is a theorem of the theory T. This contradiction proves the theorem.%"
o THEOREM 19. The theory T* is semantically complete iff the theory T is semantically
complete.%"
o PROOF. See Theorems 17 and 18.%"
X C" o THEOREM 20. Let a formula { An }n=1 of the language L* be a formula, for which there
X ,# exists an m0 such that for any nm An is B U B , where B is a formula of the language
X $ L. Then for any formula { Cn }n=1 from the language L* there exists an inference in the
X % theory T* { An }n=1 { Cn }n=1.%"
X c& o PROOF. According to the definition, an inference { An }n=1 { Cn }n=1 is an object { An
X L' Cn }n=1, where there exists an p0 such that for any np An Cn is an inference of the
X ;( theory T. If there exists an m0 such that for any nm An is B U B , then an inference
X *) An Cn is always well defined in the theory T for any nm, i.e., p=m.%" *) p-++!! Ԍ After the technical side of the theory we would like to return to more philosophical
aspects of the problem. Below we try to show an example of representation of concrete
philosophical antinomy as an Lcontradiction.
Let S be a consistent formal set theory with the posibility to prove in S a theorem of
X existence of set theoretical universes U0, U1, U2, , where Ui E Ui+1, Ui c Ui+1 for any i, and
X if X E Ui and X Ui, then XUi+1.
X n Let then Ri+1 = {XUi: XX} be an (i+1)Russell set. We can prove that Ri+1 Ui, Ri+1
X ] Ri+1, Ri+1E Ri+2 and Ri+1 Ri+2 are theorems in S for any i. Let S be tlimiting theory, i.e., limit
of infinite sequence of sets is defined in ordinary sense. In particular, limit of infinite sequence
X 5 of sets {Xn}, where XnEXn+1 for any n, equals infinite unification B{Xn}. From here we
X $
receive that infinite sequence {Ri+1}i=0 of iRussell Sets has limit. I shall denote this limit as
X
R. Therefore there exists Linconsistent theory S*, where the infinite consequence of
X formulas {Ri+1 Ri+1 U Ri+1 Ri+2}i=0 is Lcontradiction. Really limit of this consequence is
X contradiction R R U R R. If this contradiction is considered as analogue of Russell
paradox, then we receive the proof that this paradox is antinomy, not mistake.
X 3 On the basis of limiting consequence {Ri+1}i=0 and limiting consequences of formulas we
could to try to interprete some philosophical antinomies, for example, first antinomy of Kant'
Critique. This antinomy asserts that Universe is and at the same time is not limited in space X time. Let XX be axiom of the theory S. Then Ri+1 = Ui+1. Hence iRussell set equals i X Universe, where i=1,2,3,. If X Ri, then we can say that X is ilimited. Thus assertion that
X X is not ilimited is expressed itself in the formula XRi. The first Kantian antinomy can be
interpreted not so much separate proposition as limiting consequence of propositions, i.e., as
X Lcontradiction {Ri+1 Ri+1 U Ri+1 Ri+2}i=0, or {Ui+1 Ui+1 U Ui+1 Ui+2}i=0. Therefore Russell
paradox has obvious connection with the dialectical tradition of philosophical logic and his
formulation today is a certain manifestation of outgrowing of Line of Parmenides in
contemporary logical thinking.
By the same way we can express another philosophical antinomies, for example Hegel
X antinomy of being which is also nonbeing. Formula X Ui can be expressed the idea that X
X is ibeing. Let {Xi}i=0 be a sequence of sets, where Xi+1Ui and Xi+1Ui+1 for any i=0,1,, and
X there exists limit X of sequence {Xi}i=0. For example, Xi+1 = Ri+1. Then sequence {Xi}i=0
can be an expression of principle, which is being and also nonbeing. Really we have L X ] contradiction {Xi+1Ui U Xi+1Ui+1}i=0. Let U be an operation, where {Xi+1Ui}i=0 U
X L {Xi+1Ui+1}i=0 equals by definition {Xi+1Ui U Xi+1Ui+1}i=0. We can understand U
as
metaconjunction such that metaconjunction of two sequences of formulas is sequence of
conjunctions of formulas (by the similar way, we can define another logical operations in logic
X
" of limiting sequences of formulas). Then left member {Xi+1Ui}i=0 of metaconjunction can
X " be read as X is nonbeing
, in accordance with limit lim(Xi+1Ui) = XU. Accordingly,
X # right member {Xi+1Ui+1}i=0 of metaconjunction can be read as X is being
, in accordance
X $ with the limit lim(Xi+1Ui+1) = XU. Finally we can interprete Lcontradiction {Xi+1Ui U
X % Xi+1Ui+1}i=0 as Hegel antinomy there exists a principle which is being and nonbeing
.
In our opinion, by the similar way another philosophical and religious antinomies may
be interpreted in suitable Linconsistent theories. Taking into account the analogy between
method of construction of mathematical continuum and method of Linconsistent theories
formation, one may conclude that numerous antinomies, constantly have been reproduced in
the history of human thinking, are examples of logical irrationalities
. These are antinomies * p-++!! of all the limiting concepts of philosophy, for example, World
, Being
, Consciousness
,
Will
, Freedom
, Personality
, etc. And just as there exists common method of
mathematical irrationalities expression, there could be a common method of logical
irrationalities representation. Author hopes that ideas of this paper could to help us to come
nearer to this method.
b L v Vyacheslav Moiseyev
7 Moscow Medical Stomatological University
Q p-++!! N %" 5 a 5 H2 W B
a " Paraconsistent logic! (A reply to Slater)
by JeanYves B)ziau`!%"] ă
y I d d d y 5 =
a a @
a d d x
d d x @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 1725
$ 7 Paraconsistent logic! (A reply to Slater)
#
Copyright by JeanYves B)ziau and SORITES -- , a 5 a v
s ! = Paraconsistent logic! (A reply to Slater)
X - V Jean-Yves B)ziau # V 2 P kC 4P# X01 Í Í. 4 X1* Í 4 V * Work supported by a grant of the Swiss National Science FoundationV X1* Í 4 X01 Í Í 4 # 4o \ P C +XP#
+
t M
a `%" Je ne discute jamais du nom pourvu qu'on m'avertisse quel sens on lui donne.
v M `2%"@ Blaise Pascal, Les Provincialesă
t M Ea1 Contents
v 0. Paraconsistent logic?
1. Contradictories, subcontraries and contraries in the tradition
2. Contradictories, subcontraries and contraries in classical logic
3. Contradictories, subcontraries and contraries in paraconsistent logic
31. Da Costa's logic C1
32. Priest's logic LP
4. A general result about contradictories and paraconsistent logic
5. Conclusion
t M P 6. Bibliography
v a+
X p a* 0. Paraconsistent logic ă
Paraconsistent logic is the study of logics in which there are some theories embodying
X contradictions but which are not trivial, in particular in a paraconsistent logic, the ex
X contradictione sequitur quod libet, which can be formalized as Cn(T, a,a)= F is not valid.
Since nearly half a century various systems of paraconsistent logic have been proposed and
X { studied. This field of research is classified under a special section (B53) in the Mathematical
X f Reviews and watching this section, it is possible to see that the number of papers devoted to
paraconsistent logic is each time greater and has recently increased due in particular to its
applications to computer sciences (see e.g. Blair and Subrahmanian, 1989).
However in a recent paper entitled Paraconsistent logics?
, a philosopher from Perth,
B.H.Slater, pretends to show in less than ten lines that paraconsistent logic doesn't exist. Here
is his laconic argument:
t M # o If we called what is now red
, blue
, and vice versa, would that show that pillar boxes are blue, and
the sea is red? Surely the facts wouldn't change, only the mode of expression of them. Likewise, if we
called subcontraries
, contradictories
, would that show that it's not red
and it's not blue
were
contradictories? Surely the same point holds. And that point shows that there is no paraconsistent logic
.
(Slater 1995, p.451)ƶ
X ( Are these few lines, the death sentence of paraconsistent logic? ( G = p-p-p- Ԍ Slater' argumentation is based on the traditional notions of contradictories
and
subcontraries
. Unfortunately the Perthian doesn't give precise definitions of them. After
giving such definitions and proving a general result about them, we will show that Slater's
argument is not valid or, in the best case, is tautological.
X a 1. Contradictories, subcontraries and contraries in the tradition ă
Such notions as subcontraries
and contradictories
belong to traditional logic, i.e.
logic in the tradition of Aristotle. The first point is to precise what is their meaning in this
tradition and the second point is to see how they can be understood in the light of modern
mathematical logic.
One of the sad defect of Slater's argument is that both of these points are eluded and that
therefore his argument is viciated by fuzziness. The farther precision Slater is getting at is
X e when he says that contradictories cannot be true together - by definition
(Slater 1995, p.453).
Even this precision is quite ambiguous because, due to the fact the Perthian doesn't give any
definition of contradictories, one may imagine that the definition of contradictories is that two
sentences are contradictories iff they cannot be true together, which is not the correct
definition according to the tradition as we shall see very soon.
Of course one can imagine that it is not necessary to precise what is the exact meaning
of notions such as contradictories and subcontraries, that everybody knows what their meaning
is, and that this meaning is clear. But it is not so obvious, due to the fact that these notions
belong to traditional logic, and that most concepts of traditional logic appear as confuse in the
light of modern logic, and that at least their interpretations is not straightforward.
We will not enter into philological details to explain what is the meaning of
contradictories
, and subcontraries
. The following excerpt from p.56 of (Kneale and
Kneale 1962) will provide all the necessary information for our discussion including the
standard definitions of contradictories, subcontraries and contraries (the concept of subalterns
is not relevant for us here):
t M p o jj 8 the square of opposition, is also not to be found in Aristotle's text, but it provides a useful
v M 1 summary of his doctrine. According to his explanations, statements are opposed as contradictories when
v M they cannot both be true and cannot both be false, but as contraries only when they cannot both be true
v M but may both be false [De Interpretatione 7 (17b 16-25)] 8 Although he does not use these expressions
v M z subaltern and sub-contrary), Aristotle (8) assumes that subcontraries cannot be false though they may both
t M = be true. This is shown by his description of them as contradictories of contraries.ƶ
For more details about the square of opposition, the reader may consult e.g. (Parsons
1997).
X ! \a 3. Contradictories, subcontraries and contraries in classical logic ă
X # Let F be the set of propositional formulas built with the connectives , U, V, .
X
$ Formulas will be denoted by a, b, etc., sets of formulas by T, U, etc. The set C of classical
X $ valuations is defined as usual: it is a set of functions from F to {0,1} and its members obey
X % the standard conditions, in particular we have: for any v in C and for any a in F , v(a)=1 iff
X & v(a)=0.
With this framework we are now able to define precisely the discussed notions in the
context of the semantics of classical logic.
X v* Given two formulas a and b, we say that they are: v* p-++!! Ԍ X - contradictories iff for any v in C , v(a)=0 iff v(b)=1;
X a o - contraries iff for any v in C , v(a)=0 or v(b)=0 and there exists v in C , v(a)=0 and v(b)=0;%"
X o - subcontraries iff for any v in C , v(a)=1 or v(b)=1, and there exists v in C , v(a)=1 and
X v(b)=1;%"
Let us note that if we remove the second part of the definition of subcontraries
X m there exists v in C , v(a)=1 and v(b)=1
, which translates may both be true
, then all
contradictories are subcontraries. In this case confusing subcontraries with contradictories
would not be the same as switching red with blue, or cats with dogs, but rather would amount
X * of confusing dogs with canines. Let us call global confusion this kind of error by contrast to
X
the first one that we can call switching confusion. As Slater claims through his red and blue
example that paraconsistent logicians are making a switching confusion rather than a global
one, it seems implicit that he doesn't consider that all contradictories are subcontraries, neither
do we here.
X 1 It is clear that for any formula a, a and a are contradictories. The connective is said
X to be a contradictory forming relation.
X Which examples of subcontraries can we find? For any two atomic formulas a and b, a
X l and aVb are subcontaries, as the reader can easily check. This can be illustrated by Plato
is a cat
and Plato is not a cat or snow is blue
, which cannot both be false but can both
be true.
X Can we define the relation which associates to any formula a the set of formulas {aVb;
X b F } as a subcontrary forming relation? That sounds reasonable but we must be aware that
X in this case this relation includes pairs of formulas like a and aV(aUa) which are
contradictories.
It is clear that inside classical logic, there are a lot of subcontrary forming relations;
however the question is: are paraconsistent negations part of these subcontrary forming
relations? And the answer is: no. Because these negations are not definable in classical logic.
For example da Costa's paraconsistent negation of the logic C1 is not definable in
classical logic because it is not self-extensional (i.e. the replacement theorem does not hold
for it).
A paraconsistent negation is not in general a subcontrary forming relation inside classical
logic, maybe be it is a subcontrary forming relation from another point of view - this question
will be examined later on - but anyway we must remember that in general paraconsistent
negations are not definable in classical logic and that for example the logic C1 of da Costa
X " is strictly stronger than classical logic in the sense that classical logic is definable in C1 but
not the converse. The same happens with intuitionistic logic, and that is why from this point
of view, intuitionistic negation is not a contrary forming relation, erroneous conclusion that
someone may reach applying an argument similar to Slater's one.
Thus paraconsistent logic is not merely the result of changing the names of concepts of
classical logic already existing, but the appearance of a new phenomenon. This is a first point
against Slater.
Even if someone thinks that notions such as negation and contradictory cannot be used
in another way that the way they are used in classical logic, he must admit that there are + p-++!! notions of non classical logic that cannot be defined in classical logic (and that therefore,
however they are named, these notions cannot be named by names naming some notions
definable in classical logic).
X 1 As I have pointed out in my review of Slater's paper for Mathematical Reviews
(96e03035), paraconsistent logic is not a result of a verbal confusion similar to the one
according to which in Euclidean geometry point
will be exchanged with line
, but rather
the shift of meaning of negation
in paraconsistent logic is comparable to the shift of
meaning of line
in non-Euclidean geometry.
X 6 La 3. Contradictories, subcontraries and contraries in paraconsistent logic ă
X ea* 31. Da Costa's logic C1 ă
The set of formulas of the logic C1 is the same set of formulas of classical logic. This
logic was presented syntactically in (Costa 1963) and its semantics presented in (Costa 1976).
X <
The semantics for C1 is a non truth-functional semantics. Its set D of bivaluations can be
X % defined like this: v D iff v is a function from F into {0, 1} obeying the following conditions:
X - if v(a)=0 then v(a)=1
X - if v(aUa)=1, the v( (aUa))=0
X T - if v(a)=0, then v(? a)=0
X - if v(a#b)=1 and v(a) c v(a) and v(b) c v(b),
X then v((a#b))=0, where #{U, V, }.
These are the conditions for negation. We will not recall the conditions for the other
connectives which are similar to the classical case (note however that the semantics for C1
cannot be generated by distributions on atomic formulas as it is the case in classical logic or
other truth-functional semantics).
X It is clear that if we redefine the notion of contradictories, subcontraries, and contraries
X inside C1 (i.e. using D instead of C in the definition of SECTION 2), then the paraconsistent
negation of C1 is not a contradictory forming relation but is a subcontrary forming relation.
It is worth mentioning that da Costa has also developed a logic in which there is a
paraconsistent negation which is neither a contradictory forming relation, nor a sucontrary
forming relation, nor a contrary forming relation, from the point of view of the set of
valuations of this logic (Loparic and Costa 1984).
X ! a+ 32. Priest's logic LP ă
Priest has proposed a rival system to da Costa's one called LP (logic of paradox),
presented for the first time in (Priest 1979). Priest claims that his logic is better than da
Costa's, in particular because, according to him and Routley, da Costa's paraconsistent
negation is not a negation but a subcontrary forming relation.
The argumentation of Priest and Routley appears in (Priest and Routley 1989). In the
same paper the two pseudo-Australian claim that their argumentation against C1 cannot be
applied to LP: ;) p-++!! Ԍ t M o Someone might try to make out that the negation of this system is not really a negation. But in virtue of
X all the above points, they would have little ground to stand on. (Priest and Routley 1989, p.169)ƶ
However Slater in his paper attacks also Priest's logic and says that the paraconsistent
negation of Priest is also only a subcontrary forming relation. Although the argumentation of
the Perthian is quite imprecise, and in particular is false in the sense that Priest's negation is
not a subcontrary forming relation inside classical logic, it contains a valid remark that we will
try to make clear.
Priest's semantics for his logic LP can be presented in different manners. It can be seen
X as a three-valued (truth-functional) semantics. The set of valuations P is a set of functions
X from F to {0, , 1}, obeying the following conditions for negation: for any v in P , and any
X a in F ,
X A - v(a)=0 iff v(a)=1
X - v(a)= iff v(a)= .
Now if we want to interpret the discussed traditional notions in this context (more
generally in the context of a logic with more than two values), we must fix what truth
is
and what falsity
is. It is clear that if we interpret truth by 1 and falsity by 0, then is a
contradictory forming relation. And that is apparently why Priest thinks that his paraconsistent
negation is really a negation. But his argumentation is viciated as Slater himself confusedly
perceived.
The reason why Priest's argumentation is wrong is the following: he considers as
designated elements (in the sense of matrix theory) not only 1 but also , as we can see when
he defines the notions of logical truth and semantic consequence. The last one is defined by:
X . a aCn(T) iff for every v P , v(b)=0 for one bT, or, v(a)=1 or v(a)= .
X This definition allows to have aCn(b,b), for any atomic formulas a and b, and
therefore to say that LP is paraconsistent. Had 1 been taken as the only designated value, LP
would have not been paraconsistent.
Priest's conjuring trick is the following: on the one hand he takes truth to be only 1 in
X order to say that his negation is a contradictory forming relation, and on the other hand he
takes truth to be and 1 to define LP as a paraconsistent logic. However it is reasonable to
demand to someone to keep his notion of truth constant, whatever it is. Therefore we have
only the two following possibilities, which show that Priest cannot run away: in one case LP
is paraconsistent and its negation is only a subcontrary forming relation from the point of view
X B! of P , in the other case LP's negation is a contradictory forming relation but LP is not
paraconsistent.
We cannot have the penny and the bun, that is what we will show explicitly in the next
section.
X % >a 4. A general result about contradictories and paraconsistent logic ă
It seems to us that the real question is to know whether a paraconsistent negation can be
a contradictory forming relation from the point of view of its own semantics. We have seen
that it is neither the case of da Costa's negation, nor of Priest's negation. In this section we
will show that in general it is not possible for a paraconsistent negation to be a contradictory
forming relation from the point of view of its own semantics. * p-++!! Ԍ For proving this result we will have to discuss and present succintly some general remarks
on logic and semantics. This will permit us by the way to precise some points made about
Priest's logic.
The notion of contradictories depends on the notions of truth and falsity. One may think
that in the case of many-valued logics, the notion of contradictories would therefore be
seriously challenged. But following the traditional matrix approach to many-valued logic, it
is not really challenged because fundamentally a bivalent division is kept, as stressed by
G.Malinowski:
t M 4 o ^jj The matrix method inspired by truth-tables embodies a distinct shadow of two-valuedness in the
t M division of the matrix universe into two subsets of designated and undesignated elements.
(Malinowski 1993, p.72)X j
What happens is that matrices are used to define logical truth and also consequence
relation in a way that there is no doubt that designated values should be taken as truth and
undesignated values as falsity. Of course it would be possible to use many-valued matrices
in a more radical way, breaking the bivalent paradigm, as proposed in (Malinowski 1994), but
this is not what is done generally and in particular this is not what Priest is doing, as we have
seen.
a# GENERAL DEFINITION OF CONTRADICTORIES
X ` The notion of contradictories can be defined for any set of bivaluations B on a given set
X I L , i.e. when B is a set of functions from L to {0,1}:
X Given two objects x and y of L , we say that x and y are contradictories iff for every vB,
X v(x)=0 iff v(y)=1.
8a, DEFINITION OF LOGIC
X W We call a logic L any structure L = < L ;Cn> where L is any set and Cn any function from
the power set of L into itself.
X Remark We therefore do not presuppose that Cn obeys any axiom, or that L is a structure
of a particular kind. Our reasoning can thus be applied to any logical language.
$ a% DEFINITION OF CLASSICAL NEGATION
X J Given a logic L =< L ;Cn>, a unary function on L is said to be a classical negation iff
X 9 for every x L and T E L ,
X ea* xCn(T) iff Cn(T,x)= L ă
This definition is equivalent to other standard definitions of classical negation (see B)ziau
1994).
We can ask: is classical negation a contradictory forming relation (i.e. a relation such that
X 4% for every x, x and x are contradictories)?,,5 But contradictories in which sense?
Contradictories from the point of view of any set of bivaluations which can define the
logic of this negation, i.e. any adequate bivalent semantics for this logic. Before turning to the
definition of adequate bivalent semantics, let us note that therefore the notion of contradictory
here makes sense only if the logic can be defined by a set of bivaluations. This is the case of T) p-++!! a wide class of logics, including most of many-valued logics (on this topic see Costa and
B)ziau 1994).
Note also that the theorem we will prove below makes sense only if we are in the case
of logics which can be defined by a set of bivaluations, but that the proof of the theorem does
X not depend on any specific axioms for Cn.
a! DEFINITION OF ADEQUATE BIVALENT SEMANTICS
X Given a logic L =< L ;Cn>, a set of functions B from L to {0, 1} is called an adequate
X bivalent semantics iff for every x L and T E L :
X * a xCn(T) iff for every v B , if v(y)=1 for every yT then v(x)=1.
a2 THEOREM
X is a classical negation (in a given logic L ) if for every x, x and are contradictories
X (from the point of view of any adequate bivalent semantics for L ).
X = o Proof. Suppose that for every x, x and x are contradictories and that is not a classical
negation.%"
X 1) There exists x, T and y, such that xCn(T) and yCn(T,x). If yCn(T,x), then there
X z exists v, such that v(T)=1, v(x)=1, v(y)=0. But if xCn(T) and v(T)=1, then v(x)=1. Therefore
X i x and x are not contradictories, because they can both be true.
X 2) There exists x and T such that xCn(T) and Cn(T,x)= L . If xCn(T), then there exists
X v such that v(T)=1 and v(x)=0. Now suppose that v(x)=1, then xCn(T,x), which is absurd
X due to the fact that Cn(T,x)= L . Therefore v(x)=0. Therefore x and x are not
contradictories, because they can both be false.
X Remark The converse of this theorem is false. It can be proved (with some few additional
X negligible hypotheses) that if is a classical negation, then for every x, x and x are
X contraries, but it cannot be proved that x and x are subcontraries. One counter example is
the following: as a corollary of a general result, the set of characteristic functions of
deductively closed sets of formulas is an adequate bivalent semantics for classical logic. But
X it is clear that given two atomic formulas a and b, aCn(b) and aCn(b).
a1 HCOROLLARY
X Y Given a paraconsistent negationH (in a logic L ), x and x cannot be contradictories for
X H every x (from the point of view of any adequate bivalent semantics for L )}.
In another words: a paraconsistent negation cannot be a contradictory forming relation
from the point of view of its own semantics (and the same holds of course for intuitionistic
negation, Curry's negation, Johansson's negation, etc.).
We have not given a precise definition of paraconsistent negation, and in fact there is no
uniform definition, but to infer the COROLLARY from the THEOREM, we just need to
suppose that a paraconsistent negation is different from classical negation. So if we consider
X ( the rejection of the ex contradictione sequitur quod libet, Cn(T, a, a) = F , as a necessary
condition for a negation to be paraconsistent, it is enough to get the COROLLARY. ) p-++!! Ԍ X a/ 5. Conclusion ă
In view of the above result, to say that a negation is not a negation because it is not a
contradictory forming relation, is just to say that a negation is not a negation because it is not
a classical negation, because only classical negation is a contradictory forming relation.
To state, without argumentation, that only classical negation is a negation and to claim
that paraconsistent negations are therefore not negations, is just to make a tautological
affirmation without any philosophical value.
But the real discussion does not reduce to such a trivial point. The question is to know
what are the properties of classical negation which are compatible with the rejection of the
X ex contradictione sequitur quodlibet, rejection which is the basis of paraconsistent negation
(on this topic see B)ziau 2000).
Paraconsistent logic has shown in fact that a paraconsistent negation
can have some
strong properties, that for example it does not reduce to a mere modal operator and that it can
make sense to use the word negation
in the context of paraconsistency, in a similar way
that it can make sense to speak of intuitionistic negation
or of Johansson's negation
.
Moreover, obviously the meaning of the word negation
in natural language does not
reduce to the meaning of classical negation of classical logic and nobody has yet tried to
prohibit the use of this word in natural language.
Finally, a possible way to consider that a paraconsistent negation (or another non classical
negation) is a contradictory forming relation, despite of our negative result of SECTION 4,
X is to change the definition of contradictory forming relation and to say that two formulas a
X and b are contradictories iff one is the negation
of the other.
Of course this can lead to nonsense if we are dealing with something which has nothing
to do with negation. But if we reasonably change the meaning of negation
, it makes sense
to accordingly change the meaning of contradictories
.
It seems that this is the option Priest has now taken after we present to him our present
criticisms to his paper with Routley.
It is worth emphasized that from this point of view Priest's negation LP does not present
any superiority to da Costa's negation C1 or other paraconsistent negations.
X a1 Postface ă
This paper was originally written in 1996, just after I wrote the review of Slater's paper,
X O! Paraconsistent logics?
for Mathematical Reviews; a Romanian translation of it was
X :" published in 2004 in I.Lucica et al. (eds), Ex falso qodlibet, Tehnica, Bucarest. In particular
this paper was written before the publication of Greg Restall's paper, Paraconsistent logics!
,
X $ Bulletin of the Section of Logic 26/3 (1997), with a title which is quite the same. However the
contents of the papers are completely different. After writing this paper I wrote several papers
which are a continuation of it:
X A' o J.-Y.B)ziau, Paraconsistent logic from a modal viewpoint
, Journal of Applied Logic, 3
(2005), pp.7-14. []%"
X ) o J.-Y. B)ziau, New light on the square of oppositions and its nameless corner
, Logical
X v* Investigations, 10, (2003), pp.218-232. []%" v* p-++!! Ԍ X o J.-Y.B)ziau, Are paraconsistent negations negations?
, in Paraconsistency: the logical way
X to the inconsistent, W.Carnielli et al. (eds), Marcel Dekker, New-York, 2002, pp.465-486.%"
X L a. 6. Bibliography ă
X o [Bez94] J.-Y.B)ziau, 1994, Th)orie l)gislative de la n)gation pure
, Logique et Analyse,
X 147-148 , pp.209-225.%"
X o [Bez96] J.-Y.B)ziau, 1996, review of [Sla], Mathematical Reviews, 96e03035.%"
X V o [Bez97] J.-Y.B)ziau, 1997, What is many-valued logic?
, in Proceedings of the 27th
X A International Symposium on Multiple-Valued logic, IEEE Computer Society, Los
Alamitos, pp.117-121.%"
X
o [Bez00] J.-Y.B)ziau, 2000, What is paraconsistent logic?
, in Frontiers in paraconsistent
X v logic, D.Batens et al. (eds), Research Studies Press, Baldock, pp.95-111.%"
o [BlS] H.A.Blair and V.S.Subrahmanian, 1989, Paraconsistent logic programming
,
X
Theoretical Computer Science, 68 , pp.135-153.%"
o [Cos] N.C.A.da Costa, 1963, Calculs propositionnels pour les syst/mes formels
X
inconsistants
, Comptes Rendus de l'Acad)mie des Sciences de Paris, 257 , pp.3790-3793.%"
X k o [CoA] N.C.A.da Costa and E.H.Alves, 1976, Une s)mantique pour le calcul C1
, Comptes
X V Rendus de l'Acad)mie des Sciences de Paris, 238A , pp.729-731.%"
X o [CoB] N.C.A.da Costa and J.-Y.B)ziau, 1994, Th)orie de la valuation
, Logique et Analyse,
X 146 , pp.95-117.%"
X o [KnK] W.Kneale and M.Kneale, 1962, The development of logic, Clarendon Press, Oxford.%"
o [LoC] A.Loparic and N.C.A.da Costa, 1984, Paraconsistency, paracompleteness and
X K valuations
, Logique et Analyse, 106 , pp.119-131.%"
X o [Mal] G.Malinowski, 1993, Many-valued logics, Clarendon Press, Oxford.%"
X
o [Mali] G.Malinowski, 1994, Inferential many-valuedness
, in Philosophical logic in Poland,
J.Wolenski (ed), Kluwer, Dordrecht, pp.75-84.%"
X W o [Par] T.Parsons, 1997, The traditional square of opposition
, Stanford Encyclopedia of
X B Philosophy, http://plato.stanford.edu.%"
X o [Pri] G.Priest, 1979, Logic of paradox
, Journal of Philosophical Logic, 8 , pp.219-241.%"
X ! o [PrR] G.Priest and R.Routley, 1989, Systems of paraconsistent logic
, in Paraconsistent
X ! logic: Essays on the inconsistent, Philosophia, Munich, pp.151-186.%" ! p-++!! Ԍ X o [Sla] B.H.Slater, 1995, Paraconsistent logics?
, Journal of Philosophical logic, 24 ,
pp.451-454. %"
X J Ra. Acknowledgments ă
I would like to thank N.C.A. da Costa, G.Priest and B.H.Slater for discussions and
comments.
b O Q v Jean-Yves B)ziau
, P Institute of Logic and Semiological Research Center
0 c University of Neuch tel, Espace Louis Agassiz 1
? CH - 2000 Neuch tel, Switzerland
D p-++!! N %" 5 a L \ W B
a " The Logic of Lying
by Moses @k/`!%"] ă
y I d d d y =
a a @
d d x
d d x @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 2730
/ The Logic of Lying
& Copyright by Moses @k/ and SORITES -- , a 5 a v
X , The Logic of Lying
X 1 Moses @k/
+
X = / @ Introduction
) a Usually, people normally expect that others should take whatever claims they make as
true. Also, whenever it is said that a claim is a lie, most people generally tend to presume that
the claim is false. It is also generally believed that a truthfully made statement is true.
In this discussion, I wish to call due attention to the nature of a lie to show that it is not
the case that a lie is necessarily a false statement, just as a truthfully made statement is not
necessarily a true statement.
The statements or claims we make are expressions of our beliefs and thoughts; our
statements are reports of the information at our disposal. The information could have come
to us from any sources whatsoever. The epistemic or any other value that we place on each
source and kind of information is a different matter altogether, and not the concern of the
present discussion.
The context for this discussion can be conveniently set in Chisholm's treatment of the
Epimenides
. Accordingly to Chisholm (1977:91),
o If a man says, `I am now lying' is he saying something that is true or something that
is false? Either possibility seems to lead to a contradiction8 we bring the problem
into sharper focus if we consider a man who says more baldly, `What I am saying
is false'.ƶ
By definition, a lie is a dishonestly made statement. It is a statement which deviates from
what its author actually knows, believes or holds to be true. To lie is, therefore, to say is false
what one believes is true, or to say is true what one believes is false. We may say that a lie
is an intentional and deliberate distortion by someone of what he or she believes or takes to
be true. It is a wilful misrepresentation, in one's statement, of one's beliefs. In this regard, it
is important to note that the opposite concept of lying is not truth, but truthfulness. It should
also be noted that whereas truth and falsity are epistemic terms, truthfulness and lying (i.e.
untruthfulness) are moral concepts. It is in this context that we can see how a truthfully made
statement might be untrue, while an untruthfully made statement (i.e. lie) might not be untrue.
The truthful person is one who honestly says what he or she believes or thinks as he or
she believes or thinks it to be. There is agreement between a belief and its corresponding
expression, by a truthful person. However, a person's belief that P is true, or that P is false,
does not imply that P is true, or that P is false. Wittgenstein indicates this point clearly when
he says: * = p-p-p- Ԍ v M o From its seeming to me ! or to anyone ! to be so, it doesn't follow that it is so8 For it is not as though
the proposition `It is so' could be inferred from someone else's utterance: `I know it is so'. Nor from the
X utterance together with its not being a lie (Wittgenstein 1974:2, 3).ƶ
The truthvalue of a statement is therefore independent of the manner of its utterance as well
as the moral status of its author. An honest person and a dishonest person could equally say
what is true, as well as what is false, on the same issue.
Statements, being the expression of their author's beliefs, thoughts, ideas, feelings, etc.,
could, for one reason or the other, be wrong or inaccurate with reference to what they seek
to express. A person might in his statement express his belief, etc. incorrectly owing to
ignorance, mistake, or illness. In each of these cases, the error involved is epistemic, without
any intention to deceive or misinform anyone. On the other hand, when a statement is not an
accurate expression of its author's belief owing to his or her intention to do mischief,
deliberately to misinform and deceive those to whom the statement is communicated, we have
a case of dishonesty that falls within the purview of morality rather than that of epistemology.
Hallen (1998:187204, and 2000:1335) hints at the tendency of people to mistake
truthfulness for truth when they evaluate one another's statements. For instance, people
generally tend not to believe or hold as true whatever a person known to be a liar says. On
the other hand, people generally feel inclined to accept as true whatever anyone adjudged to
be truthful or honest says. Wiredu (1996:106) also remarks how the connection between
truthfulness and truth makes the word `truth' ambiguous and confusing.
However, people generally expect that other persons would accept their claims as true.
Thus, even the person who says `What I am saying is false', or the one who says `I am lying'
would want and expect to be taken as saying the truth. The point to note here is that the
person who declares his or her own statement false might, in his or her declaration, be making
a false statement, such that the allegedly false statement may in fact be true. On the other
hand, the selfacclaimed liar might be saying the truth about himself or herself, but the
statement might also be true. That is, a liar's lie might be a true statement. This points to the
fact that a liar does not necessarily say what is false whenever he lies.
A lie might be a true statement if the belief which the liar held to be true, and which he
or she sought to distort, was in fact false. This follows from the fact that owing to a number
of epistemic defects, a person may sincerely hold a false belief to be true, or a true belief to
be false, and say honestly that it is true, or that it is false, respectively. From this, it is to be
noted that a person's truthfulness does not imply the truth of his or her statements. A truthful
person is not a person who is filled with truths and nothing but truths. In the same vein, a
person's untruthfulness (or habit of lying) does not imply the falsity of his or her statements.
An untruthful person (a liar) is not a person full of nothing but untruths. In other words, a
truthfully made statement could be either true or false, just as a lie, too, could be either true
or false independently of the motive or character of its author.
Whether a statement is true or false is, therefore, not a function of the moral character
of the statement's author, but rather of the situation to which the statement pertains. Hence,
we may have (i) statements that are truthfully made but which are false, (ii) statements that
are truthfully made and are true, and (iii) lies that are true statements. We may thus say very
rightly that truthfulness and lying are to persons as truth and falsity are to statements. The
ability to lie is thus an essential characteristic of persons, just as falsitypossibility is an
essential feature of statements, thoughts and beliefs. * p-++!! Ԍ Both a truthful person and a liar could hold false beliefs. However, whereas the truthful
person expresses and communicates his or her belief without any deliberate or intentional
distortion, the liar deliberately and intentionally communicates the opposite, the negation, the
caricature or the counterfeit of his or her belief. It has to be reemphasised, however, that a
truthfully made statement is not necessarily a true statement, or a statement of truth. This is
so in the same way that Hanson (1952:424) has shown that a factual statement is not
necessarily a statement of fact. In a related reference, Wittgenstein (1953: Part II, 192e)
cautions that we should not mistake a hesitant assertion for an assertion of hesitancy.
Similarly, we should not uncritically regard an untruthfully made statement as an untrue
statement, or a truthfully made statement as a true statement.
The liar might hold as true a belief that is false. That is, a person who lies about his or
her belief could have unintentionally said the truth. This comes to saying that the lie (i.e. the
negation or distortion of the belief held to be true) was false. On the other hand, a lie could,
unintended though, be true if the author of the lie originally held as false a true belief. Either
way, there is no contradiction involved in the statement or the assertion of it. What we have
is a disagreement between a person's belief and his or her statement that purports to express
that belief. Hence, whenever a lie is true, the logical implication is that the liar was mistaken
about the truthvalue of the belief that he or she sought to misrepresent or distortedly
communicate. In the case of a truthful person, his or her statement will be false only when
his belief is false and true whenever his belief is true.
It is important to note that in the case of a liar, his or her lie could be false both when
the corresponding belief is true and when it is false. This is because the lie may sometimes
not be a logical negation of the liar's belief; it could be a different false statement that neither
truly expresses the liar's mistaken false belief, nor falsely expresses his or her true belief.
We can illustrate the possibilities with a simple example. Let us assume that today is
Monday. A liar (or any other person, for that matter) could believe that today is Monday or
that today is any other day of the week, Friday for example. That is, the liar's belief could be
either true or false. If a liar believes falsely that today is Friday, but untruthfully says that
today is Tuesday, for instance, the lie would be a false statement. Also, if the liar believes
truly that today is Monday but, lying, says that today is Tuesday, for instance, the lie would
also be a false statement. The only instance, therefore, when a lie is necessarily false is when
the liar's corresponding belief that was distorted was true. In other instances, the lie could be
either true or false.
We conclude, therefore, that a lie is not necessarily a false statement. This shows, as
Wittgenstein (1953:182) had noted, that the logical relations between the words `lie', `true',
and `false' are more involved8 than we are tempted to think.
X # a0 References ă
X q$ o Chisholm, R.M. (1977), Theory of Knowledge, Englewood Cliffs, New Jersey: PrenticeHall,
Inc.%"
o Hallen, B. (1998), Moral Epistemology ! When Propositions Come out of Mouths: Reply
X ' to Oke
, International Philosophical Quarterly, vol. xxxviii, No. 2, Issue 50.%"
X ) o """ (2000), The Good, The Bad and The Beautiful: Discourse About Values in Yoruba
X ) Culture, Bloomington and Indianapolis: Indiana University Press.%" ) p-++!! Ԍ X o Hanson, N.R. (195254), Fact and Factual Statements
, Analysis, vol. xiiixiv.%"
X a o Wiredu, K. (1996), Cultural Universals and Particulars: African Perspective; Bloomington
and Indianapolis: Indiana University Press.%"
X o Wittgenstein, L. (1953), Philosophical Investigations, trans. G. E .M. Anscombe, New York:
The Macmillan Company.%"
X o """ (1974), On Certainty, ed. G.E.M. Anscombe and G.H. von. Wright; trans. Denis Paul and
G.E.M., Anscombe, Oxford: Basil Blackwell.%"
b ? V v Moses @k/
G Department of Philosophy
E Obafemi Awolowo University
O a IleIfe, Nigeria
F p-++!! N %" 5 a L D R W B
a " Sparse Parts
by Kristie MillerG `!%"] ă
y I d d d y =
a a @
d d x
d d x ! @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 3148
2 Sparse Parts
$ Copyright by Kristie Miller and SORITES -- , a 5 a v
s / Sparse Parts
X . Kristie Miller # V 2 P kC 4P# X01 Í Í 4 X1* Í Í. 4 * = q F * ԍWith thanks to David BraddonMitchell, Mark Colyvan and Dominic Hyde for helpful discussion of these issues. X1* Í Í. 4 X01 Í Í. 4 # 4o \ P C +XP#
+
X
a a. 1 Introduction ă
It is sometimes said that four dimensionalists are guilty of ontological profligacy. They
admit into their ontology crazy objects such as that composed of your cat on Tuesday and my
dog on Wednesday. It is thus at least implied that four dimensionalism is incompatible with
the view that ontology is sparse, that is, the view that only some arrangements of basic
X particulars compose composite concrete objects. G q F ԍHenceforth I use object
to refer to any nonbasic particular that is composed of particulars arranged in some
manner. Moreover, I do not use object
as a term of art to refer to some subset of the composite things that exist.
I do not use object, for instance, to refer just to those composite things that are recognised by our conceptual
apparatus, or which have some sort of natural border. Rather, all composite things that exist are objects. This incompatibilty, however, has as yet not
definitively been shown to be the case. I argue that four dimensionalism in its most common
variety, perdurantism, is indeed incompatible with the view that ontology is sparse, but that
this need not provide reason to reject four dimensionalism. For this incompatibility is merely
a specific instance of a more general problem that a sparse ontology view faces with respect
to parthood, namely the problem that we frequently quantify over nonexistent parts.
X Suppose we agree about the distribution of basic particulars, q F t ԍHenceforth I use particulars
to refer to the most basic things that exist, whatever these things are. and we ask what, if
anything, is composed of those particulars: that is, we ask which objects exist. One natural
response to this question is to hold that only some ways of arranging particulars composes an
X K object, and some other ways do not. Call this the view that composition is restricted. j K q F " ԍProponents of this view include Van Inwagen (1987) and Wiggins(1980). j There
X 4 are various different accounts of just which ways of arranging particulars will result in some
X object being composed. v U q F I& ԍDefenders of restricted composition include Van Inwagen (1990) and Wiggins (1980).v In general however, these accounts attempt to preserve our intuitions
that there is, for instance, no object composed of myself and George Bush, or of a tennis
racket and your left foot. The view that composition is restricted therefore results in what we
might call a sparse ontology, for it countenances the existence of far fewer objects than, for
instance, the view of mereological universalism according to which every way of arranging !
= p-p-p- X particulars composes some object. _ q F y ԍDefenders of unrestricted mereological composition include Lewis (1991) and Heller (1990). I will thus refer to the view that ontology is sparse in this
X way, as sparsism, and to proponents of this view as sparsists.
Sparsism, then, is not simply the view that some subset of all the objects that exist are
special in some way. It is not the view that there are two distinct kinds of objects, natural
objects and gerrymandered objects, and that while gerrymandered objects are mere
mereological fusions, there is some special composition relation that holds between and only
between particulars and the natural objects they compose. Sparsists are not those who see their
project as providing an account of the necessary and sufficient conditions for something to
count as a natural object. That view, whatever we call it, is perfectly consistent with both
mereological universalism and with four dimensionalism.
Four dimensionalism is a theory of persistence. It holds that persisting objects are
temporally extended: they have not only spatial dimensions, but also a temporal dimension.
The most common version of four dimensionalism is perdurantism, according to which objects
X persist by perduring, that is, by being the mereological sum of temporal parts. ~ J G _ q F ԍPerdurantism is to be distinguished from other versions of four dimensionalism such as Ted Sider's stage view
according to which our everyday linguistic terms do not refer to the mereological fusion of temporal parts, but rather
to temporal stages. Claims about the past and future of these persisting objects are then made true by the existence
of the relevant temporal counterparts. While the stage view is not technically a perdurantist view, it will be open to
the same sorts of criticisms that I level against a sparsist version of perdurantism.
For a defence of the stage view see Sider (2001). ~ It is arguably
X
the case that most perdurantists are mereological universalists, P
} _ q F ԍSee for example Lewis (1991); Heller (1990).P and prima facie there seems
be a tension between perdurantism and sparsism. Although it seems coherent to hold that only
some ways of arranging particulars over time composes persisting objects, and that those
objects persist by perduring, this appears to be an odd combination of views. For at least
intuitively, temporal parts do not seem to be the sorts of objects that the sparsist is likely to
countenance in her ontology.
Why so? Well there is much debate about whether there are any nonarbitrary,
X informative criteria that determine when composition occurs and when it does not. _ q F v ԍCf. van Inwagen (1987); Heller (1990); Lewis (1986). pp. 212213; Wiggins (1980); Markosian (1998). Let us
X j call whatever these criteria are, the composition criteria. In this paper I will assume that there
are composition criteria, that is, I will assume that composition is not a brute relation. Further,
I will assume that whatever these criteria are, they will preserve most of our core intuitions
about which objects exist and which do not. For surely the central motivating force behind the
X sparsist position is that we are in general right about what exists and what does not, and about
X which sorts of things exist and which do not.
Sparsist accounts are founded on the idea that an arrangement of particulars composes
some object just if that arrangement exemplifies some property, where this property
supervenes on the complex causal relations of the particulars. So for instance the sparsist
might hold that particulars compose an object just if they are continuous, or if they form a "
p-++!! X functional unit, or if they form a unit that is a member of a certain sortal or natural kind = _ q F y ԍWiggins (1980). pp. 5770= or
X if their collective activity constitutes a life. I
G _ q F ԍVan Inwagen (1990). chapter 10 pg. 98I The idea is that the behaviour of particulars is
such that there exists an integrated, functional unit with global properties that supervene on
those particulars.
Whatever the composition criteria are, they explain why the borders of objects are where
they are, and thus explain how it is that we are able confidently to pick out objects. Indeed,
I take it that the sparsist takes as primary datum the fact that objects have discernible, non X arbitrary borders. n _ q F ԍWhere to say that an object has some border is not to say that it has some determinate border, only that the
border, if it is vague, is discernible. For presumably the idea is not that we begin with the intuition that there
are certain functional or complex causal properties in the world, and then go about marking
X out the borders of the things that exemplify those properties, finally to exclaim ah! so that's
an object.
Rather, we begin with intuitions about where the borders of objects lie, we develop
some general ideas about the features of these borders, and from there we attempt to construct
an account that explains why objects lie within, and only within, those borders. So any sparsist
X M account should respect this core idea that objects have what I will call natural borders.
This is not so say, of course, that all such borders are natural in the sense that they are
carved out by nature. Some borders, such as those of natural kinds, will be natural in the sense
that they are borders recognised by the physical sciences. But if particulars ever compose
artefacts, then the borders of these objects are not carved by nature. Still, the borders of such
X R objects are not merely arbitrary: there is some genuine difference between that which lies to
one side of the border, and that which lies to the other. Broadly speaking then, we will say
that a border is natural to the extent that either it is carved by nature, or it is nonarbitrary.
Now of course, it is sometimes the case that the concept of nonarbitrariness is analysed in
terms of the notion of being carved out by nature. It might be that for some, what it is for
there to exist a nonarbitrary border, is for there to exist a border carved by nature. Clearly
this is not the analysis of nonarbitrary that I wish to embrace. Nor is it the analysis that most
sparsists would want to adopt, since it would mean that a great number of arrangements of
particulars that we take to have natural borders and thus to compose objects, in fact fail to do
so. It would turn out that only natural kinds exist. So a more robust sparsism requires that
there be some account of a nonarbitrary border that does not make recourse to the idea of
being carved by nature. I cannot provide such an account here. Indeed, providing such an
account lies at the very heart of the sparsist project, and the difficulty of this task is one of
the major stumbling blocks for sparsism. Here I assume that there is some such account, and
that at least the general idea of a nonarbitrary border is sufficiently intuitively clear.
The idea that objects have natural borders lies, then, at the very core of sparsism, and thus
is a notion that the sparsist should take seriously. It is a notion, however, that seems to be
inconsistent with perdurantism. For perdurantists are typically committed to the idea that for
any perduring object O and arbitrary temporal interval T during which O exists, there is some
temporal part of O that exists during and only during T. Though the temporal extent of any
temporal part is held to be an essential property of that part, the temporal borders of temporal ## D p-++!! parts are purely arbitrary. Temporal parts do not have natural borders, and are thus precisely
the sorts of objects that the sparsist refuses to admit into her ontology.
Indeed, one of the reasons some three dimensionalists reject four dimensionalism is
because of the apparent arbitrariness of the borders of temporal parts: they do not see how it
is that at the moment one object ceases to exist, another comes into existence that is
qualitatively identical to the previous object at the moment of its cessation. All they see is a
X unitary persisting object. < _ q F e ԍCf. Van Inwagen (2000). < And this is precisely because they see only one natural temporal
border, not a series of such borders that mark out the borders of the various temporal parts.
In the next section I begin by outlining the difficulties for a perdurantist version of
sparsism, and then move on to consider a number of ways these difficulties might be met. I
consider Storrs McCall's sparsist perdurantism, and argue that he is faced with a dilemma. If
temporal parts are mere abstractions then they cannot do the metaphysical work proposed for
them. If they are not abstractions then they appear to have nonnatural borders and thus are
inconsistent with sparsism. Then I consider a revised perdurantism which is consistent with
sparsism, but which faces almost insurmountable metaphysical difficulties. Finally I examine
sparsism itself, and conclude that it faces some difficulties of its own which may provide
reason to prefer perdurantism.
X a/ 2 The Problem ă
Perdurantism is the thesis that objects persist by perduring: by being composed of
temporal parts. Roughly speaking, temporal parts as they are widely construed, are objects that
exist during and only during a particular temporal instant or interval, and which during that
X instant or interval wholly overlap the perduring object of which they are a part. \
G _ q F ԍSider (2001) pg. 60; Zimmerman (1996); Markosian (1994).\ So if I
perdure, then a temporal part of me is some object that exists during and only during interval
T, which during T has my spatial dimensions, and which is part of me simpliciter. More
formally, following Ted Sider we will define both an instantaneous and an extended temporal
part as follows:
t M o x is an instantaneous temporal part of y at instant t=df 1) x is part of y. 2) x exists at, but only at t. 3) x
overlaps every part of y that exists at t.ƶ
o An extended temporal part of x during T is an object that exists at all and only times in T, is part of x at
X every time during T and at every moment in T overlaps everything that is part of x at that moment. 8 _ q F p" ԍSider (2001) pg. 60.8 ƶ
So consider some persisting object O. Perdurantists hold that for every temporal instant
t at which O exists, there is some instantaneous object that exists at that time, which overlaps
O at that time and which is part of O. So too for any arbitrary temporal duration T during
which O exists, they hold that there is some object that exists only during T, which overlaps
O during T and which is part of O. That is, the perdurantist subscribes to what van Inwagen
calls the doctrine of arbitrary temporal parts: "$ p-++!! Ԍ t M o DATP: for every persisting object P, if I is the interval of time occupied by P and subI is any occupiable
subinterval of I whatever, there exists a persisting object that occupies the interval subI and which, for
X every moment t that falls within subI, has at t exactly the same momentary properties that P has. @ _ q F ԍvan Inwagen (1981). pg. 203.@ ƶ
I think that most perdurantists are committed to this doctrine, though it is prima facie
plausible that they need not be. They will almost certainly, however, want to be committed
to a related doctrine. For the heart of the perdurantist thesis is that objects persist by being
composed of parts at times. If an object O persists through interval T, then at each time at
which O exists, some part of O must exist at that time. This does not imply that for any
interval of time T during which O exists, there is some object that exists only during T and
overlaps O during T. But it does imply what we will call the doctrine of instantaneous
temporal parts (DITP):
o DITP: for every persisting object P, if I is the interval of time occupied by P and t
is any occupiable instant of I whatever, there exists an instantaneous object that
occupies t and that at t has the same momentary properties that P has at t.ƶ
Perdurantists will want to adopt DITP. For suppose the perdurantist held that there exist
no instantaneous temporal parts. Rather, persisting objects are composed of extended temporal
parts with natural borders. Consider the example of a member of the genus Lepidoptera. On
this view such an organism is composed of four extended temporal parts: an egg temporal
part, a caterpillar temporal part, a pupa temporal part and a butterfly temporal part. The
Lepidoptera thus perdures, but it is composed only of parts with natural borders. There are two
obvious worries about this proposal. First, in the case of many persisting objects, there do not
appear to be any candidates to be temporal parts with natural borders. Given our definition
of temporal part then, these objects then cannot perdure. Second, in many cases the temporal
borders of such temporal parts would be excessively vague: consider the example of a person
with stages of childhood, adolescence and so forth. While sparsists may embrace ontological
vagueness, it is difficult to see how an object could be composed of parts where the
indeterminacy of the borders ranges over a number of years. Even putting these worries aside,
however, few perdurantists would embrace such an account. For a rejection of DITP is a
rejection of pure perdurantism in favour of some perdurantistendurantist hybrid.
To see this, suppose we grant that a member of the Lepidoptera genus perdures by being
composed of extended temporal parts with natural borders. Then how do these extended
temporal parts persist? These temporal parts persist by perduring just if they are composed of
temporal parts, and so too for the temporal parts of their temporal parts and so forth down the
line. We get perdurantism all the way down
so to speak, just if persisting objects are
ultimately composed of instantaneous objects. If we reject DITP then we are forced to hold
that persisting object perdure in virtue of being composed of objects that do not themselves
X L" perdure. JL"G _ q F h& ԍOf course, someone might resist the idea that we have perdurance all the way down, on the grounds that there
are smallest units of spacetime such as the Planck length and Planck time. In that case DITP should be altered to
be the doctrine of the shortest temporal parts (DSTP). While it is true that these shortest temporal parts would not
perdure (just as instantaneous temporal parts do not perdure), and this technically an adoption of DSTP might be
seen as a hybrid view, the general point remains the same: it cannot be that the onlytemporal parts that exist are
extended temporal parts with natural borders. While it might be argued that such a hybrid view is plausible, it is so only if the L"% } p-++!! temporal parts that endure are not themselves composite persisting objects: if Lepidoptera
perdure then surely so do caterpillars and butterflies!
Even if perdurantists were willing to accept this peculiar hybrid view as a tradeoff for
retaining their sparsist intuitions, they would surely baulk at the loss of virtually all of the
theoretical elegance of perdurantism. Perdurantists hold that if all properties are disguised
relations to times as endurantists maintain, then there are no truly intrinsic properties, for no
object ever exemplifies any property simpliciter. The perdurantist account allows that
persisting objects exemplify properties at times in virtue of being composed of temporal parts
that exemplify those properties simpliciter, and this is the sense in which properties are
intrinsic.
But a temporal part exemplifies a property simpliciter only if the entire temporal duration
of that part exemplifies the property. Temporal part P is red simpliciter only if P is red at all
times at which it exists. Suppose persisting object O is rapidly changing colour from being
all red to all blue to all red again. If O's having the property of being red is to be an intrinsic
property as understood by the perdurantist, there must be some part of O that exists only
during the short period in which O is red. Thus for every momentary intrinsic property that
O exemplifies, there must be some instantaneous temporal part of O that exemplifies that
property simpliciter. If O were composed only of extended temporal parts, then there would
be properties that O exemplified which were not properties of any of O's parts simpliciter and
which therefore would not be intrinsic in the relevant sense. Thus if the perdurantist is to
retain the apparatus with which to explain how persisting objects exemplify intrinsic properties
at times, she must at least subscribe to DITP.
Prima facie though, both DATP and DITP are problematic doctrines for the sparsist, since
X Q they seem to entail that arrangements of particulars can compose objects with wholly arbitrary
temporal borders. So it seems that the very essence of perdurantism is in conflict with the core
of sparsism. Is there any way to resolve this conflict? In the next section I consider a putative
reconciliation of sparsism and perdurantism proposed by Storrs McCall, and argue that in fact
it is no reconciliation at all. I then move on to consider two other proposals. In the first we
take instantaneous objects to be basic and then ask ourselves how those objects need to be
arranged in order to compose some persisting object. In the second I broaden the definition
of temporal part, and argue that a reconfigured perdurantism can accommodate sparsist
intuitions. Unfortunately this version of perdurantism is unsuccessful in its own terms, as a
metaphysics of persistence.
X xa+ 3 Sparse Perdurantism
X B! a+ 3.1 McCalls' Solution ă
One defender of a sparse ontology who embraces perdurantism is Storrs McCall. He
suggests that rather than regarding persisting objects as being composed of more basic stages
united by some unity relation, we should instead think of the four dimensional object as basic,
X \% and the stages as derivative abstractions
. > \% _ q F ' ԍMcCall (1994). pp. 211214> On this view, four dimensional volumes have
a natural shape associated with a sortal or natural kind: they are not made of arbitrary portions
of spacetime. These natural four dimensional volumes can then be divided into temporal stages
or parts, just as the earth can be divided into spatial parts by meridian lines. (& G p-++!! Ԍ It is not clear in exactly what sense McCall means to count temporal parts as abstractions.
The comparison to meridian lines suggests that he means to take a sort of antirealist view of
temporal parts. Just as we can imagine dividing up the earth in many different ways,
corresponding to the different places we might draw meridian lines, so too we can divide up
four dimensional objects in many different ways, each corresponding to one way of drawing
the temporal border of a temporal part. If this is what is mean by abstraction
however, it
simply will not do. Perdurantism is the view that persisting objects have the properties they
do in virtue of having temporal parts that exemplify those properties. Those temporal parts
have to be real parts, not mere abstractions: no abstraction is red, only objects are red.
The other possibility is that McCall simply means by abstraction
, taking a topdown
view of composition, that is, abstracting away from the whole four dimensional object to
determine which temporal parts that object has. This too is problematic. For McCall's sparsism
tells us that only those four dimensional volumes that have natural borders contain objects.
Then four dimensional volumes contain temporal parts just if those volumes have natural
borders. Since most of the temporal parts countenanced by perdurantists do not have natural
X temporal borders, by McCall's own composition criteria, those temporal parts do not exist. 7 _ q F ԍMcCall attempts to solve this problem by arguing that three and four dimensionalism are equivalent theories.
I am sympathetic to this view, but I do not see how it helps in this matter. Either there is some object that exists
within a certain temporal border or there is not. I grant that the three dimensionalism can agree that such an object
exists, but can argue that the object is not a part of the persisting object. The two theories might then come out as
equivalent if it were construed as a debate about what it is to be a part of an object. In this case, however, the issue
is about whether there is some object that exists during a period of time T and overlaps a persisting object at that
time, not whether or not that object is a temporal part. See McCall (1994) pp. 215216.7
It is not sufficient simply to say that we can use our intuitions about which particulars
compose objects to determine which four dimensional objects exist, and then use a topdown
process to maintain that in addition, all of the volumes contained within that four dimensional
volume contain some object: a temporal part. Whatever our composition criteria are, all and
only those arrangements of particulars that meet these criteria compose an object; there is no
distinction to be drawn between four dimensional objects and temporal parts such that the
former but not the latter need meet these criteria.
This does not necessarily mean, however, that there is no way to alter McCall's proposal
so that it allows a reconciliation of sparsism with perdurantism. It is to that possibility that
we next turn.
X a- 3.2 DATP and DITP ă
Suppose as sparsists we had the following intuition: if objects are in constant flux at the
microlevel, then an arrangement of particulars that composes an object at some time t, can
never be identical to an arrangement of particulars that composes an object at t*. This is a
fairly standard perdurantist intuition according to which the only sense in which something
that exists now is identical with something that exists at some other time, is the sense in
which both of those things are parts of the same perduring object. If we accept this intuition,
then we are faced with two questions: which arrangements of particulars at a time compose
some instantaneous object, and which combinations of instantaneous objects compose some
persisting object.
It's easy to see how this second question might be answered if we first grant that every
arrangement of particulars at a time composes some instantaneous object. We could maintain #' p-++!! that instantaneous objects compose some persisting object just if they are related in a
particular way, namely, if they are causally connected in a smooth and continuous manner
such that the existence of an instantaneous object at one time causes the existence of an
instantaneous object at the next time. That is, a series of instantaneous objects compose some
persisting object O just if they form a nice smooth four dimensional volume. This would rule
out punctuate objects and other odd gerrymandered objects, for there would be no
instantaneous objects causally connected in the requisite way. Since not every arrangement of
instantaneous objects would compose some persisting object, we would preserve the sparsist
intuition that ontology is sparse, yet it would still be true that objects persist by perduring.
Further, since the perdurantist need not be committed to DATP, she can hold that there
exist only instantaneous objects and the persisting objects that they compose. Just as some
object that exists at a time is composed of certain basic particulars at that time, so too an
object that exists over time, is composed at each of those times of basic instantaneous objects.
There are no extended temporal parts whose temporal borders are oddly arbitrary: there is no
object that wholly overlaps my dog and exists for only ten minutes on Tuesday. So the
appearance of a plethora of objects with arbitrary temporal borders is removed. For just as the
spatial borders of a mereological simple are natural, so too the temporal borders of an
instantaneous object are natural.
All well and good. The difficulty lies in conceding, as I did, that every arrangement of
particulars at a time composes some instantaneous object. I do not think that many sparsists
will be happy with this concession. Sparsists will not, I think, want to allow that there is some
object that exists only at t, and which is composed of my dog at t, Jupiter at t, and your
pillow case at t. While this object might have a natural temporal border, it certainly does not
have a natural spatial border. This is not to say that this position is a hopeless one. Perhaps
there are sparsists who hold that there is something special about persisting objects, such that
sparsismovertime is a more important doctrine than sparsismatatime. Perhaps such a
person would be willing to concede that there exist odd instantaneous objects, but no odd
persisting objects. But I am not entirely sure what would motivate such a position. Why
should we think that the glue
that holds objects together over time is fundamentally
different from the glue
that holds them together at a time?
This latter question is particularly pertinent given that we are talking about a sparist
version of perdurantism. For consider, the endurantist explicitly holds the view that the manner
in which objects persist through time is radically different to the manner in which they extend
through space. Objects extend through space by having spatial parts at spatial locations, while
they persist through time by being wholly present at each time at which they exist.
Perdurantists, however, construe persistence through time as analogous to extension through
space, with objects persisting by having parts " temporal parts " at temporal locations. Given
this, it is not clear what would motivate the claim that acrosstime sparseness is radically
different to atatime sparseness. If there is such a case to be made, it is at least imcumbent
on the perdurantist sparsist to make that case.
So while this is perhaps one way to reconcile perdurantism with sparsism, it is not a
wholly attractive way, and nor, I imagine, is it an option that will find favour with many
sparsists. But if the sparsist rejects the assumption that every instantaneous object exists, then
we need to determine which instantaneous objects exist and why. Clearly the best proposal
would be to hold that there exist the various everyday persisting objects of our ontology, and
the instantaneous temporal parts of those objects. Adopting McCall's topdown approach then, *( p-++!! it might be thought that this is precisely what we can accomplish. As sparsists, we feel
confident that dogs exist. Given that dogs exist, we can conclude that the instantaneous
temporal parts of dogs exist. Thus dogs perdure in virtue of being composed of temporal parts,
and those temporal parts have natural borders in that they are temporally basic: they are
instantaneous. There are two problems for this view. First, we do not want sparsism to be
simply the view that particulars compose some object just if we think they do, that is, just if
that is what our intuitions tell us. Sparsism is supposed to be the view that there are some
informative composition criteria. On many criteria, composition involves complex causal
relations between the composing particulars (whether these be causal relations constituting a
life, or constituting a natural kind or sortal). Whatever these criteria amount to, they must
apply equally to all objects. But in general, instantaneous objects do not meet the usual sorts
of sparsist criteria for composition. Instantaneous objects are probably not members of sortals
or natural kinds, for the underlying properties that constitute those kinds are properties of
persisting objects. So too no instantaneous object exemplifies the property of having a life.
So the topdown solution does not seem hopeful: for it is plausible only if we think that
the composition criteria for persisting objects is different to that for instantaneous objects.
Perhaps so. But some account of the composition criteria for instantaneous objects would need
to be forthcoming, and this criteria could not simply be that particulars compose some
instantaneous object if that object is a temporal part of a persisting object. For that is not to
provide composition criteria, it is just to state which objects exist and which do not. Until such
criteria are forthcoming then, the topdown view is not at all compelling.
Moreover, the view is faced with an additional problem. On this view, there do not exist
any extended temporal parts. Now consider some person P, who has some blue experience.
P has the property of having a blue quale. Now suppose that blue experiences are not ever
X experiences of instantaneous objects. U _ q F = ԍAs for instance McKinnon (2003) strongly argues. U Having a blue experience, however, is surely an
intrinsic property of P. Unfortunately though, it is not a property of any temporal part of P,
since P has no extended temporal parts. Strangely then, although having a blue experience
would have been an intrinsic property of P if that experience had been instantaneous, since
P would have had a temporal part that had that property simpliciter, as things stand, P has no
such temporal part, and thus does not have that property simpliciter. So too for any
temporally extended
property. So this version of perdurantism is stuck with saying that
some apparently intrinsic properties are really disguised relations to times or some such. This
is even more alarming than straight endurantism, since it turns out that some apparently
intrinsic properties are indeed intrinsic in virtue of being properties of temporal parts, and
some other apparently intrinsic properties are not intrinsic, in virtue of failing to be properties
of any temporal part.
So where does that leave us? If the perdurantist is committed to DITP, and if that doctrine
is inconsistent with sparsism, then are we forced to conclude that sparsism and perdurantism
are inconsistent? Before we make such a declaration, we should first consider whether there
is some other way to salvage a perdurantist sparsism. In the next section we will briefly
consider whether it is perdurantism that ought to be jettisoned in favour of some other version
of four dimensionalism. Though this suggestion will be rejected, it does lead to the idea that
we should alter the definition of temporal part. This alteration yields a version of perdurantism
that is acceptable to the sparsist, and is, I will argue, the best reconciliation of perdurantism () G p-++!! with sparsism. But while it is the best reconciliation, ultimately it too must be rejected on the
grounds that it simply does not have the theoretical apparatus to explain the phenomena that
an account of persistence must explain.
X 1 6 a 5 Temporal Extension without Temporal Parts? ă
We might think that the best way to combine sparsism with four dimensionalism is by
rejecting perdurantism. This would need to involve more than, for instance, adopting Sider's
stage view which accepts the same ontology as perdurantism but merely disagrees about which
objects in that ontology ought to be the referents of our terms. Rather, it would require the
radical view that objects can be temporally extended and thus four dimensional, and yet have
X no proper temporal parts. Call such an object a temporal simple. This view has recently been
X
defended by Parsons. 4
_ q F ԍParsons (2000). 4 Parsons does not suppose that composite persisting objects such as
dogs and trees could be temporal simples, and indeed it is hard to see why a view that
countenanced this possibility would be preferable to three dimensionalism. For it would no
longer be possible to use the apparatus of temporal parts to explain change over time or to
provide an account of temporary intrinsics. If anything, it would seem to render to nature of
persisting objects all the more mysterious.
The idea of temporal simples does, however, suggest another possibility. For consider
what happens when we attempt to construct
a composite four dimensional object that lacks
proper temporal parts. For the perdurantist, a four dimensional object is composed of various
instantaneous objects which are themselves composed of particulars at a time. But suppose
that we think of a some object not as composed of instantaneous objects at times, but simply
as composed of particulars at times. That is, some persisting object O is composed of
X j particulars S at t, P at t, Q at t, S at t1, P at t1, R at t1 etc. Then have we just described a four
dimensional object with no proper temporal parts, or merely a persisting three dimensional
object?
X If we think that the particular S at t and S at t1 is the very same particular S, viewed at
X different times, then O is simply an enduring three dimensional object. If S at t and S at t1 are
strictly identical, then S endures, and so too with all of the other particulars. Then O is a
composite object composed of enduring particulars S, T, R, Q etc. Since by definition there
exists no object that wholly overlaps O at a time and is part of O, that is, by definition O has
no proper temporal parts, O must itself be an enduring three dimensional object.
X On the other hand, if we think that S at t and S at t1 are distinct particulars, then we think
X p that there exists Satt and Satt1. O is thus composed of Satt and Satt1: it has these
particulars as parts simpliciter. Thus there is an important sense in which O deserves to be
called four dimensional despite the fact that it is not composed of temporal parts as I defined
them earlier. As I defined a temporal part, x is a temporal part of y at t only if x wholly
X # overlaps y at t. Call this a maximal temporal part. O has no maximal temporal parts. However,
O is composed of four dimensional particulars. Each particular has temporal parts: S exists
at different times, and it does so in virtue of being composed of instantaneous temporal parts
X % Satt, Satt1 and so forth.
If an object O perdures just if O persists by being the mereological fusion of maximal
temporal parts, then the object just described does not perdure. It has been suggested, (* G p-++!! however, that we ought to broaden our definition of temporal part to include nonmaximal
X parts. E _ q F b ԍSee for example Merricks (1999). E Let us then define a nonmaximal temporal part as follows:
o x is an instantaneous nonmaximal temporal part of y at t just if 1) x is part of y 2)
x exists at, and only at t and 3) x is wholly overlapped at t by some part of y that
exists at t.ƶ
o x is an extended nonmaximal temporal part of y during interval T just if 1) x exists
at all and only times in T, 2) x is part of y 3) x is wholly overlapped by some part
of y at all times during T.ƶ
We can then say that an object O perdures just if O persists by being the mereological
fusion of maximal or nonmaximal temporal parts. Then the object O described above will
perdure in this sense, since it is composed of nonmaximal temporal parts.
X The critical question then, is whether the sparsist should think that S at t and S at t1 are
distinct particulars or are strictly identical. There is a case to be made for each view. On the
one hand, the sparsist thinks that we have a single unitary object just where we have natural
X borders. She certainly thinks that Satt would be a distinct particular in a world W where
there is nothing Slike at temporal instants that abut t. For in that world, Satt clearly has
natural temporal borders. In the actual world though, sparsist intuitions might steer one
towards holding that there is no object Satt, for that putative object has no natural borders
in virtue of being temporally abutted by other Slike particulars. Rather, we have one
particular, S, and to claim that in addition to S, there exist various instantaneous objects that
compose S, would be to claim that there exist objects with arbitrary borders.
There is something appealing about the intuition that something has a natural temporal
border only if it is not temporally abutted but like particular or particulars. After all, part of
the intuition that the putative temporal part of me that exists only for ten minutes today has
arbitrary temporal borders, is that it is temporally abutted by personlike objects. If that same
putative object existed in some world W and was not abutted by anything personlike, we
would be happy to concede that it has natural temporal borders in that world. Recall however,
that in the previous section we rejected the idea that any arrangement of particulars can
compose some instantaneous object, on the grounds that many such objects would have
arbitrary spatial borders. We noted though, that just as the spatial borders of a mereological
simple are natural, so too the temporal borders of an instantaneous object are natural: these
objects are temporally fundamental. So the most basic particular is one that is mereologically
X p simple, and temporally fundamental, np G _ q F # ԍWhere being temporally fundamental is either being instantaneous, or being of the shortest possible temporal
length eg. Planck length if it is not possible to be instantaneous. and the borders of this most basic particular are
natural. If the sparsist accepts this, then she can proceed to hold that some of these
instantaneous simples are causally related such that they compose perduring basic particulars
such as S. For the persisting object S is mereologically simple, but not temporally
X # fundamental: it is composed of the most basic particulars Satt, Satt1 and so forth. We can
then say that composite objects are composed of perduring particulars like S, and thus
ultimately of mereologically simple instantaneous objects. These composite objects therefore
have no maximal temporal parts, but only nonmaximal parts with natural borders. %+ p-++!! Ԍ So the sparsist requirement is fulfilled: there exist no objects that have wholly arbitrary
borders. Yet this view also seems to afford the perdurantist all of the metaphysical apparatus
needed to explain change and temporary intrinsics. Though we cannot say that object O is red
at t in virtue of having some maximal temporal part Oatt that is red simpliciter, we can say
that O is red at t in virtue of each of its nonmaximal temporal parts Satt, Ratt and so forth
being red simpliciter. Problem solved; sparsism and perdurantism reconciled.
Not so fast. This reconciliation too is problematic. For plausibly, macroproperties such
as being red, being cold, being conscious, being a person and so forth, are not exemplified by
instantaneous objects. Plausibly, they are not exemplified by any instantaneous maximal
X temporal part, but they are certainly not exemplified by any instantaneous nonmaximal basic
particular. Consider some object O that is red at t. We cannot, in fact, say that O is red at t
in virtue of ever basic particular that composes O at t, being red at t. For Satt is not red. Satt is too small to be red. What is red at t, is the totality of the arrangement of the particulars
at t. This arrangement, however, does not compose any object, for there is no maximal
temporal part of O at t. So there is nothing that exists at t and is red simpliciter, and we find
ourselves faced again with the problem of temporary intrinsics. So too, on this view there
exists no extended maximal temporal parts, and thus no such parts of persons. Since
consciousness almost certainly supervenes on some temporally extended temporal part, here
too there is a problem. Not only is it not the case that any most basic particular Satt is
conscious, but no arrangement of these most basic particulars at a time has the property of
being conscious: only some arrangement of these particulars over time could exemplify that
property, but no such arrangement composes any object. So only entire four dimensional
persons are conscious.
Of course, a proponent of this view might maintain that there is no real problem here.
After all, this is precisely the sort of difficulty certain sorts of eliminativists, (such as
eliminativists about beliefs or eliminativists about composite objects) face each day when our
X everyday language quantifies over nonexistent objects. G _ q F ԍCf. Unger (1979) and Unger (1990). G For eliminativists, talk that
quantifies over nonexistent dogs, for instance, is true just if there is some paraphrase in which
it is true that there are particulars arranged in a dogwise way, or some such. So too, we might
think, the sparse perdurantist can maintain that O is red at t just if there is some Owise
arrangement of particulars at t that is red simpliciter.
But this is all rather tortuous. For those who (plausibly) think that an arrangement of
particulars exemplifies some macroproperty such as being red just if that arrangement
composes some macroobject, the eliminativist solution will be no solution at all. Either the
arrangement of particulars at t composes some object, namely Oatt which is a maximal
temporal part of O and is red simpliciter, or the arrangement does not compose any object at
t and there is no object that is red simpliciter at t. There is no middle ground, and thus no way
to have one's sparsist cake and eat it too. So while there might be those who are sympathetic
to the eliminativist strategy and thus willing to accept this combination of sparsism and
perdurantism, they must surely be in a minority.
There is of course one other way that this version of sparsist perdurantism could deal with
macoproperties, and that is by adopting an endurantist analysis of properties. There are two +', G p-++!! X possibilities available, indexicalism, < _ q F y ԍCf. van Inwagen (1990). < according to which properties are temporally
X relativised, and adverbialism M G _ q F ԍCf Johnston (1987) and Haslanger (1989). M according to which the having of properties is temporally
relativised. For the perdurantist O
refers to the whole four dimensional object. So if O is
X red at t and blue at t1, following indexicalism we can say that O has the properties of being
X is redatt and blueatt1. Or following adverbialism we can say O has the properties of being
X red tly and blue t1ly. So although there is no temporal part of O that is red or that is blue, we
can still attribute these properties to O.
This is by far the best solution, and this is certainly the best combination of sparsism and
perdurantism. It is consistent and workable, but it has the cost of jettisoning much of the
theoretical apparatus that motivated perdurantism to begin with. Many perdurantists such as
Lewis are moved to embrace perdurantism because they believe that it best deals with the
problem of change, in that allows that temporary intrinsics are not disguised relations to
X b times. 9 b _ q F ! ԍLewis (1986) pg. 204.9 In adopting an endurantist analysis of properties, however, this version of
perdurantism is forced to concede that O does not have the property of being red simpliciter,
but rather has a property that is temporally relativised in one way or another. The extent to
which this sparsist perdurantism is acceptable then, will depend on how repugnant one finds
the endurantist analysis of properties. That is, it will depend on just how strong one's
intuitions are that intrinsic properties are not relations to times. The dilemma is that the more
one is attracted to the endurantist analysis, the less reason one has to prefer perdurantism in
the first place, and the more repugnant one finds the analysis, the less one will be happy to
accept this version of perdurantism. So while I cannot rule out this version of sparsism
perdurantism as being coherent, it does require the loss of much of the theoretical elegance
of traditional perdurantism, and might well not be a view that many perdurantists, even those
of a sparsist bent, will be happy to embrace.
X a& 6 The Brutality of Composition ă
A final possibility. Suppose the sparsist relinquishes the idea that there are any nonarbitrary informative criteria of composition. She might, for instance, follow Markosian in
X holding that composition is brute. 6 _ q F W! ԍMarkosian (1998). 6 Either arrangements of particulars compose an object or
they do not, and there is no further story to tell. If composition is brute, then both sparsist and
perdurantist intuitions can straightforwardly be reconciled. Some arrangements of particulars
compose objects that appear to have wholly arbitrary temporal borders, namely temporal parts.
Other arrangements of particulars fail to compose objects that would have had arbitrary
borders had they existed. But there just is no reason why the former objects exist and the
latter do not, for there is no principled reason why some arrangements of particulars compose
X T objects and others do not. Indeed, there can be no such reason.
Perhaps there are good reasons to think that composition is brute. Still, this move seems
wholly unsatisfactory when applied to the case at hand. The proposition that composition is
brute is arrived at by considering the cases where we take composition to occur and those #- 0 p-++!! where we do not, and arguing that there is no informative criteria that would pick out only
the former and not the latter as being instances of composition. It is by considering examples
of composition in the real world that we are lead to adopt sparsism. So it had better turn out
that most of our intuitions about when composition occurs are right, or the very motivation
for adopting sparsism in the first place will evaporate. It would, therefore, be disingenuous to
X claim that even though our sparsist intuitions tell us that temporal parts do not exist, we can
maintain that they do exist, and that no explanation for this inconsistency need be forthcoming
since composition is brute. On these grounds I could argue that there is some object that is
composed of my dog and Lincoln's foot, and that there is no object composed of your dog
and Jefferson's foot. Why? Well who knows, composition is brute!
X a, 7 A General Problem ă
The difficulty inherent in trying to reconcile sparsism and perdurantism is that sparsism
just does not afford a sufficient number of parts to meet perdurantist requirements. Because
the version of perdurantism discussed in section 5 eschews the existence of maximal temporal
parts, there exists nothing that is menow. That is, there exists no object that is wholly present
now and which exemplifies all of my momentary properties now. So too although there is
some object that is my heart, and which is a proper part of me, my heart is a four dimensional
object which is part of a four dimensional person. Since my heart is not composed of maximal
temporal parts, there is no object that is wholly present now and is part of menow. Neither
endurantism nor traditional perdurantism has this odd consequence. For the endurantist thinks
that both me and my heart are now wholly present, and thus that my wholly present heart is
now part of wholly present me. So there is a straightforward sense in which my whole heart
is part of me now. Perdurantists, on the other hand, although they disagree that my whole
heart is now part of me, do agree that there is some object that is wholly present now and is
part of me now, namely the maximal temporal part of my heartnow. The timeslice of menow has that entire object as a part. So neither are forced to conclude that there is no sense
in which my heart now is not a part of me at all.
This counterintuitive consequence of this version of perdurantism might lead one to
conclude that sparsism and perdurantism simply are not to be reconciled, and thus that one
must be abandoned. Since sparsist intuitions have a firm grip on us, there are those who will
jettison perdurantism in favour of these intuitions. But is this the correct response? In fact
when we consider sparsism stripped of any perdurantist overtones, we find that we are faced
with an analogous difficulty. Indeed, it turns out that the counterintuitive consequences of
sparsist perdurantism is just a specific version of a general problem faced by sparsism alone.
Consider a time honoured paradox we find in the literature on persistence: Tibbles the
X ! unfortunate cat. . %! _ q F .$ ԍThe example of Tibbles the cat is found in Geach (1980) section 110, but is formally the same as the case of
Dion and Theon originally created by Chryssipus, a stoic philosopher in 280206 BC. A discussion of the Dion and
Theon problem can be found in Burke (1994). . According to the paradox, there exists an object, Tibbles the cat, and some
proper part of Tibbles, call it Tib, which includes all of Tibbles but for her tail. The paradox
arises after Tibbles has her tail amputated, and we are asked to consider what we should say
about the relation between Tib and Tibbles postamputation. But what is this object Tib of
which I write? At time t, prior to the amputation, Tib is a proper part of Tibbles. Both three
and four dimensionalists alike accept this, though of course they gloss it differently. In
general, the existence of Tib has been granted without comment by all sides of the debate, +'. p-++!! X including three dimensionalists who are frequently sparsists. T _ q F y ԍCf. Baker (1997); Simons (1987); Thomson (1998).T The exception here is van
Inwagen who rejects the existence of Tib precisely because its existence is necessary in order
X to generate the paradox. 8 G _ q F ԍvan Inwagen (1981). 8 van Inwagen goes further than merely denying the existence of
Tib, he rejects the spatial analogue of the doctrine of arbitrary temporal parts, the doctrine of
arbitrary undetached parts (DAUP):
o DAUP: For every material object M if R is a region of space occupied by M at time
t and if subR is any occupiable subregion of R whatever, there exists a material
X object that occupies the region subR at t. ? _ q F ԍvan Inwagen (1981) pg .191.? ƶ
Tibblestype paradoxes do not require that one reject DAUP. But should the sparsist
accept DAUP? I think not. Whatever the composition criteria turn out to be, it is difficult to
see how Tib and putative objects like it, could be seen as meeting those criteria. For suppose
we follow van Inwagen in holding that particulars compose something just in case their
arrangement constitutes a life. Does the Tibwise arrangement of particulars compose
something that constitutes a life? Well in one sense it does. Since Tibbles can survive the loss
of her tail, Tib, if it exists, must be a sufficient supervenience base for life. But I think this
misses the point. For it is clear that the tailwise arrangement of particulars, though they are
X | not sufficient in and of themselves to constitute a life, are part of a life, namely the Tibbles
life. There is but one life there (with the exception of any cellular organisms that are floating
around of course but that is beside the point). There is not Tibbles' life, and then Tib's life,
one life which includes the tail, and the other which does not. But if there is only one life, and
that life if the life of Tibbles, then we cannot conclude that the Tibwise arrangement of
particulars composes anything.
Similarly, consider the view that particulars compose some object just if their arrangement
constitutes something whose underlying explanatory properties mark it out as a member of a
natural kind or sortal. Then we can see why the Tibbleswise arrangement of particulars
composes something: because that arrangement constitutes something that is a cat, and a cat
is a member of a natural kind. But is there any natural kind or sortal of which Tib, if it exists,
is a member? I cannot see that there is. Tib certainly would not be a member of a natural
kind, nor does it seem plausible to think that it is a member of a sortal. If this is so, then we
should conclude that the Tibwise arrangement of particulars do not compose anything.
None of this should come as a surprise, since Tib, if it existed, would flout the sparsist
intuition that objects have natural borders. For we arbitrarily defined Tib as that thing which
includes all of Tibbles but her tail. Sparsists should conclude that Tib does not exist. And if
Tib does not exist, then Tib is not a proper part of Tibbles. So when the sparsist says that Tib
is a proper part of Tibbles she says something that should be, by her lights, false.
The case of Tibbles is by no means an isolated one. Discussion of the relation of objects
to their proper parts often involves talk of proper parts that by the lights of the sparsist, do
not exist. Consider the case where we talk of a statue that has a little chunk of clay removed
from it. Three dimensionalists typically want to say that the aggregate of clay Clay
that $/ p-++!! constitutes the statue, Statue
prior to the removal of the chunk, cannot survive this removal
for Clay persists only if it is mereologically constant. On the other hand, they want to say that
Statue does survive the removal of the chunk, for its persistence conditions are such that it can
survive such an event. This story only makes sense, however, if the chunk of clay is a proper
part of the statue, and again, I see no reason to suppose that it is if one accepts sparsism.
It turns out then, that if the sparsist is right, we frequently quantify over nonexistent
objects. We talk of the part of Tibbles that is the small hunk of flesh on her right thigh. We
talk of the small piece of clay that fell off the statue. We talk of a scoop of ice cream
removed from the tub. We talk of the enamel chip that fell off the plate. And in all of these
cases we will often talk of the flesh, the clay, the chip and so forth as being proper parts of
the objects in question. But if sparsism is true, then I submit that we would be wrong to do
so. For I can see no basis at all on which to say that these are objects at all.
If sparsism is true, then sometimes we quantify over the nonexistent. So it is not merely
the sparsist perdurantist who is forced to embrace the eliminativistic paraphrasing of everyday
language. Just as there is no maximal temporal part of O that is red simpliciter, so too there
is no chunk of clay is part of O at t. When our everyday talk appears to be quantifying over
proper parts that do not exist, we are really quantifying over certain arrangements of
particulars. Thus although strictly speaking Tib does not exist, there does exist a Tibwise
arrangement of particulars. And we can truly say that none of the particulars that are arranged
Tibwise, are arranged tailwise. If this is so, then when I say that Tib is not part of the tail,
what I say is true. Similarly, when I say that Tib is part of Tibbles, what I say is true just if
there is some Tibwise arrangement of particulars such that each of those particulars is part
of Tibbles.
Again though, this relies on it being coherent to talk of the macroproperties of
arrangements of particulars despite the fact that there is no object that exemplifies those
properties. Perhaps this is not an insurmountable problem for the sparsist. What is shows,
however, is that the difficulties inherent in reconciling sparsism and perdurantism are merely
specific instances of difficulties that the pure sparsist faces with the notion of parthood. Not
only does the sparsist not have enough parts to meet perdurantist requirements, it doesn't have
enough parts to meet everyday folk requirements. It is usually held to be a major virtue of
sparsist views that they take the middle ontological road between eliminativism and
universalism, and thus avoid not only the plethora of odd objects posited by universalism, but
also the need for paraphrasing ordinary language required by eliminativism. But sparsism does
X k not avoid all such paraphrasing, and does not preserve all of our ontological intuitions, for it
entails that Tib and others of its ilk do not exist.
X ! 0a/ 8 Conclusion ă
Sparsism is inconsistent with any plausible version of perdurantism. Should we then reject
sparsism in favour of perdurantism, or perdurantism in favour of sparsism? The answer to this
question involves a delicate balancing act of weighing up the virtues of each. Perdurantism
has much theoretical elegance to recommend it. Sparsism has much intuitive appeal. But not
as much intuitive appeal as we might have thought, since by the lights of sparsism we
quantify over nonexistent objects quite frequently. Perhaps this difficulty provides sufficient
weight to tip the scales in favour of perdurantism over sparsism. Or perhaps sparsist intuitions
are so resilient that some will prefer to adopt an eliminativisttype strategy in dealing with
nonexistent objects such as Tib. The point is that some sacrifice will need to be made: it is \*0 p-++!! not possible to reap the theoretical elegance of perdurantism and also the ontological
parsimony of sparsism. Something has to give.
X H a0 References ă
X o Baker, L. R. (1997). Why Constitution is not Identity.
Journal of Philosophy 94: 599621%"
X o Burke, M. (1994). "Dion and Theon: An Essentialist Solution to an Ancient Puzzle." Journal
X of Philosophy 91: 129139.%"
X T o Geach, P.T. (1980). Reference and Generality: An Examination of Some Medieval and Modern
X ? Theories. Cornell University Press.%"
X o Haslanger, S. (1989). Endurance and Temporary Intrinsics.
Analysis 49: 119125.%"
X o Heller, M. (1990). The Ontology of Physical Objects: Four Dimensional Hunks of Matter.
X Cambridge: Cambridge University Press.%"
X K
o Johnston, M. (1987). Is There a Problem about Persistence?
The Aristotelian Society Supp
61: 107135. pp 113115.%"
X o Lewis. D. (1986). On the Plurality of Worlds. New York Blackwell Press%"
X o Lewis, D. (1991). Parts of Classes. Oxford: Blackwell.%"
X W o Markosian, N. (1994). The 3D/4D Controversy and NonPresent Objects.
Philosphical
X B Papers 23: 24349.%"
X o Markosian, N. (1998). Brutal Composition
Philosophical Studies 92: 211249.%"
X o McKinnon, N. (2003). Presentism and Consciousness.
Australasian Journal of Philosophy
81(3): 305323.%"
X N o Parsons, J. (2000). Must a Fourdimensionalist Believe in Temporal Parts?
The Monist
83(3) 399418.%"
X o Rea, M. (1997). Material Constitution. Rowman and Littlefield.%"
X o Sider, T. (2001). Fourdimensionalism: an ontology of persistence and time. Oxford University
Press.%"
X C o Simons, P. (1987). Parts: A Study in Ontology. Oxford: Clarendon Press.%"
X o Thomson, J.J. (1998) The Statue and the Clay. Nous 32: 149173.%"
X ! o Unger, P. (1979). "I Do Not Exist". in Perception and Identity ed G. F. Macdonald. London
Macmillan Press.%"
X O# o Unger, P. (1990). Identity, Consciousness and Value. New York: Oxford University Press.%"
X $ o Van Inwagen, P. (1981). The Doctrine of Arbitrary Undetached Parts
Pacific Philosophical
X % Quarterly 62 12337 reprinted in M Rea. (ed). Material Constitution. Rowman and
Littlefield 1997 pp191208.%"
X ' o Van Inwagen, P. (1987). When are Objects Parts?
Philosophical Perspectives 1.
X ( Metaphysics: 2147;%"
X 1* o Van Inwagen, P. (1990). Material Beings. Cornell University Press.%" 1*1 p-++!! Ԍ X o Van Inwagen, P. (1990). FourDimensional Objects.
Nous 24: 245255.%"
X a o Van Inwagen, P. (2000). Temporal Parts and Identity Across Time.
The Monist 83(3) 437459.%"
X o Wiggins, D. (1980). Sameness and Substance. Oxford: Blackwell Press.%"
X o Zimmerman, D. (1996). Persistence and Presentism.
Philosophical Papers 25: 115126.%"
b m Q v Kristie Miller
G Department of Philosophy
G University of Queensland
M ; Brisbane Australia
E 9
2 p-++!! N %" 5 a / D M& W B
a
" Are Functional Properties Causally Potent?
by Peter Alward`!%"] ă
y I d d d y / =
a a @
d d x !
d d x 3 @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 4955
# Are Functional Properties Causally Potent?
% Copyright by Peter Alward and SORITES -- , a 5 a v
s J Are Functional Properties Causally Potent?
X / Peter Alward
+
X
a Jaegwon Kim has recently 6 q F ԍKim (1997, 2000). 6 argued that a solution to the exclusion argument against the
intelligibility of mental causation is to be found if mental properties can be shown to be
X s
reducible to physical properties in an appropriate way.
6 s
G q F ԍKim (2000) p. 46. 6 In a departure from some of his
X \ earlier work, : \ q F ԍSee, e.g., Kim (1989).: however, Kim seems to recognize that reducibility by itself is not sufficient to
rescue mental causation. In particular, he attempts to secure the efficacy of mentality by means
of the deployment of the causal inheritance principle
and the identification of instantiations
X of mental properties with instantiations of their realizing physical properties. n q F y ԍKim (2000) pp. 546. Note: strictly speaking, this strategy is designed only for those mental properties amenable
to functionalization.
In this paper, I wish to argue that Kim's putative solution to the exclusion problem rests
X _ on an equivocation between instantiations of properties as bearers of properties and
X J instantiations as property instances. On the former understanding, the causal inheritance
principle is too weak to confer causal efficacy upon mental properties. And on the latter
understanding, the identification of mental and physical instantiations is simply untenable.
Recall: the exclusion problem arises for views according to which mental
properties/events are (i) causally efficacious vis!vis physical events and (ii) nonidentical to
physical properties/events. If the physical domain is causally closed, then every (caused)
physical event has a physical cause. And this physical cause threatens to exclude any mental
X ! cause of the physical effect in question. 6 ! q F $ ԍKim (2000) p. 37. 6
X According to Kim, functional properties are 2nd order properties. Q q F ' ԍKim (1997) p. 290. See also Kim (2000) p. 20.Q A 2nd order property
X i is the property of having a (1st order) property which satisfies some specification or other. For
X R example, my shirt has the 2nd order property of being my favourite colour because it has a
colour property ! blueness ! which meets the condition of being my favorite. Functional
X $ properties are 2nd order properties for which the specification is that of occupying a certain
causal role. So, for example, a functionalist might claim that being in pain is the property of
!3 - = p-p-p- instantiating a property which is caused by tissue damage and causes winces and groans. In
humans (as the philosophical literature would have it) the realizing property is that of Cfibre
activation.
Kim's solution to the exclusion problem relies on the Causal Inheritance Principle and
what might be called Instantiation Identity.
Kim characterizes the Causal Inheritance
Principle as follows:
t M b o If a secondorder property F is realized on a given occasion by a firstorder property H 8then the causal powers
of this particular instance of F are identical with (or are a subset of) the causal powers of H (or this instance
X of H). 4 ^ q F ]
ԍKim (2000) p. 544 %"
According to Instantiation Identity, each instantiation of a (functionalizable) mental property
X ,
is identical with an instantiation of the 1st order physical property which realizes it on the
X occasion in question; 8each instance of M is an instance of P1, or of P2, or 8, where the P's
X are M's realizers.
S G _ q F ԍKim (2000) p. 111. See also Kim (1997) p. 295. S Kim utilizes these theses as follows. From Instantiation Identity it follows
that mental causation does not involve two causes competing for efficacy. There is only one
cause ! the physical cause ! of any given physical event. Nevertheless, the Causal Inheritance
X Principle entails that mental properties inherit the causal powers of their 1st order realizing
X properties. U _ q F a ԍSee also Kim (1997) p. 295 and Kim (2000) p. 116.U The upshot is that 8functional mental properties turn out, on account of their
X multiple realization, to be causally heterogeneous but not causally impotent.
6
_ q F ԍKim (2000) p. 116.6
`Instantiation' talk is equivocal between talk of bearers of properties and talk of instances
of properties. A particular shirt might be said to be a bearer of the property blueness, whereas
the bluenessoftheshirt is an instance of the property. Kim's use of the term `instance'
suggests that the latter is what he has in mind. But at one point, he remarks, [we] may take
X instances
or instantiations
of properties as events, states, or phenomena,
6 0 _ q F ԍKim (2000) p. 41. 6 which
suggests that property bearers are what are at issue. In order to avoid terminological confusion,
I will henceforward articulate the distinction as one between property bearers and property
X I tokens.
Now it might be thought that, given his analysis of events, Kim would argue that being
a property bearer and being a property token amount to the same thing. After all, on Kim's
X | view, an event just is a property token, a complex consisting of a substance x, a property P,
X g and a time t ! [x, P, t] ! which exists just in case x has P at t. 7 g _ q F ' ԍKim (1999) p. 337. 7 Although tempting, this
interpretation of Kim's background metaphysic should be resisted. Events cannot be identified
with property tokens because events and property tokens differ in their modal profiles. Events
are the relata of the causal relation, and need (or, perhaps better, have) essences commensurate
X with this role. Even if one rejects a counterfactual analysis of causation, the causal relation 4 v
p-++!! remains counterfactual supporting (except in cases of causal overdetermination, preemption,
etc.). And Kim concedes that this requires treating the constitutive time of an event, and
X perhaps the constitutive property, as inessential to it.
n _ q F K ԍKim (1999) p. 344. Lewis (1986, p. 250) argues that even taking the constitutive substance to be essential is
problematic. At the end of the day, Kim is best
thought of as identifying events with functions from possible worlds to substancepropertytime complexes existing at those worlds, rather than with substancepropertytime complexes
X per se (although Kim might eschew the possibleworlds formulation I've given here).
For expository purposes, I will assume that property tokens also are functions from worlds
X to substancepropertytime complexes. Q _ q F ԍThey could, of course, be constant functions.Q In my view, however, they are (typically) distinct
X functions. More to the point, what I want to argue is that if the constitutive property of an
X event is an inessential feature of it then that event is distinct from a token of the constitutive
property. Even though the property token and the event share a manifestation at at least one
world ! that is, the value of each of the corresponding functions at that world is the same
substancepropertytime complex ! they will have distinct manifestations at other worlds.
There are a couple of reasons for thinking that the constitutive property of a property
token is an essential feature of it. First, intuitively, although a blue shirt, for example, is
presumably only accidentally blue, the blueness of the shirt is essentially blue. Or again,
suppose it is true of a walking event that was in fact a strolling that it might have been a
X k striding. 9 k _ q F ԍLewis (1986) p. 251. 9 Nevertheless, the strollingness of the event is, again intuitively, essentially a
stroll. Second, in contrast to events, the modal profiles of property tokens are governed (in
part) by the role(s) they play in property metaphysics. For example, in order for various
X & reductive programmes &D _ q F ? ԍE.g., tropetheoretical attempts to reconstruct properties (i.e., types) as classes of resembling tropes. in metaphysics to be even minimally promising, the modal relation
between property tokens and property bearers needs to systematically covary with the modal
X relation between properties proper (types) and their bearers. This means that at any world w1
X at which an object x has a property P at time t there must exist the corresponding property
X token ! a complex consisting of x, P, and t. And at any world w2 at which x lacks P at t, no
X such complex can exist. One might rejoin that this is compatible with the existence of the w1Ī X token at w2, as long as we do not insist that being P is an essential feature of the w1ĩtoken.
But if typetoken modal covariance is to be sustained, this would seem to require that we take
X n x to bear P at w2, contra hypothesis. z n _ q F *$ ԍI suppose one could insist that x does bear P at w2, but in a nonPish manner. z , nn _ q F % ԍOne might, of course, simply reject typetoken modal covariance and concede that any reduction of types to
q F & tokens, or vice versa, is a nonstarter.
Suppose we formulate Kim's putative solution to the exclusion problem in terms of
property tokens. This would require that we reformulate the Causal Inheritance Principle and
Instantiation Identity as follows: 5 p-++!! Ԍ X o CIPT: If a secondorder property F is realized on a given occasion by a firstorder
property H then the causal powers of this token of F are identical with the causal powers
of this token of H.%"
X 1 o IIT: Every token of a functional mental property is identical with the token of the 1st order
physical property which realizes it on the occasion in question.%"
X y Now CIPT does seem to yield a promising basis for the causal efficacy of functional mental
X b properties. nb _ q F ԍThis general strategy is defended in Robb (1997) and Robb and Heil (2003), and is criticized in Noordhof
(1998), although the issue is framed in terms of tropes rather than in terms of substancepropertytime complexes. After all, if each token of a property has causal powers then the property itself
does, even if its powers are highly heterogeneous. But it is hard to see why the Ftoken would
inherit the causal powers of the Htoken unless they were identical. As a result, on the token X formulation, the truth of IIT is required to underpin the transfer of causal power from realizing
to functional properties, as well as to ensure that mental causes do not compete for efficacy
with physical causes.
X N But IIT is simply untenable. The connection between functional mental properties and their
X 7
1st order realizing properties is contingent: not only could a functional property be realized by
X a number of distinct 1st order properties, a substance could bear one of these 1st order
X properties without bearing the functional property ! Lewis's madman is a case in point. 2 _ q F ԍLewis (1980). 2 , % _ q F ԍMy focus here is on core realizers rather than total realizers. Presumably, distinct mental properties will often
(normally?) share total realizers. As a result, they are not even candidates for standing in identity relations with
mental properties.
More to the point, a functional mental token cannot be identical to a realizing physical token
because there are possible circumstances in which that very token occurs but fails to occupy
the functional role at issue. If, as argued above, the constitutive property of a token is essential
to it, the functional mental token in question could not exist in such circumstances. And since
X identity holds of necessity, this entails that the mental and physical tokens are distinct. _ q F ԍTrue identity statements can, of course, be contingent, but not the identities themselves. , 9 U _ q F ԍEhring (1999, p. 24) argues that mental tokens and their realizing physical tokens cannot be identical on Kim's
picture because the constituent properties of mental tokens are properties of events whereas the constituent
properties of realizing physical tokens are properties of the constituent substances of these same events. Although
q F this criticism is prima facie compelling, it can I think be met by insisting that the constitutive property of an event
is automatically part of the character of that event. 9
Perhaps Kim's solution does better if formulated in terms of property bearers. This would
require that we again reformulate the Causal Inheritance Principle and Instantiation Identity:
X = o CIPB: If a secondorder property F is realized on a given occasion by a firstorder
property H then the causal powers of this bearer of F are identical with the causal powers
of this bearer of H.%"
X n o IIB: Every bearer of a functional mental property is identical with the bearer of the 1st
order physical property which realizes it on the occasion in question.%" W6
p-++!! Ԍ X Now IIB, unlike IIT, is more or less uncontroversial. n _ q F y ԍOne might again worry that the bearers of mental properties are events and the bearers of their realizing
physical properties are the constituent substances of events. But see note # 23 above. Moreover, as long as we assume that
X the only mental causes are the bearers of mental properties, IIB may well suffice to ensure that
X mental causes do not compete for efficacy with physical causes. Moreover, if IIB is true, then
X CIPB trivially follows from it; if the bearer of a mental property just is the bearer of its
realizing physical property, then of course the causal powers of the former and the causal
powers of the latter coincide.
X The trouble that arises for the bearerformulation is that the truth of IIB and CIPB does not
suffice to confer causal efficacy upon functional mental properties. In order to do so, these
theses would need to be supplemented with a principle to the effect that if all bearers of a
property are causally efficacious, then the property itself is causally efficacious. However, this
supplementation is untenable: it implies that any property whatsoever with at least one
(causally efficacious) bearer is itself causally potent. But not only would this undercut Kim's
X d own criticisms of anomalous monism, 0 d _ q F 7 ԍKim (1989). 0 it would entail that highly extrinsic/ relational
properties are efficacious. The property of being seventythree million light years distant from
me as I write this sentence may well have a (causally efficacious) bearer ! if it does not, I'll
just choose another distance. Nevertheless, this property is not (thereby) efficacious in its own
X right. n _ q F ~ ԍOne might attempt to avoid this latter worry by restricting the supplementation to intrinsic properties, but since
functional properties are relational, this would not suffice to secure the efficacy of the mental.
One final possibility would be to endorse a mixed formulation of Kim's solution to the
X P exclusion problem, perhaps by combining the uncontroversial IIB and the promising CIPT. One
X 9 difficulty with this suggestion is that the falsity of IIT brings the truth of CIPT into question.
X " If a 2nd order token is distinct from its 1st order realizing token, it is unclear why the former
should inherit the causal powers of the latter. Moreover, there seem to be clear counter X examples. Consider again my shirt, which has the 2nd order property of being my favourite
X colour because it has the 1st order property of blueness which meets the condition of being my
favorite. Included among the causal powers of the blueness token would be certain
photochemical powers. But insofar as the favouritecolour token is distinct from the blueness
X token, it would be implausible to suppose that it inherits these photochemical powers. } _ q F h! ԍOf course, the bearer of this 2nd order property will presumably have such powers. }
There is, however, a natural rejoinder to this last point. Kim did not claim that instances
X of 2nd order properties inherit all of the causal powers of their realizing 1st order instances. He
X claimed instead that they inherit either all or a subset of these powers. 5 _ q F <& ԍKim (2000) p. 54.5 As a result, he could
easily assert that the photochemical powers simply fall outside of the subset of powers that
the favouritecolour token inherits from of the blueness token. But this response immediately
raises the following question: exactly which of the causal powers of its realizing property
X o token does a 2nd order property token inherit? And the answer is not hard to find: the powers
X X specific to the condition that the 1st order token has to meet in order to realize the 2nd order X 7 A
p-++!! property. The favouritecolour token inherits from the blueness token exactly those powers it
has in virtue of being a token of my favourite colour. And a functional mental token inherits
from its realizing token those powers it has in virtue of meeting the functional specification
at issue.
X Now strictly speaking, this manoeuvre may well render the truth of CIPT consistent with
X the falsity of IIT, even if a certain puzzlement remains as to why any inheritance of causal
powers should occur, let alone such a tidy pattern of inheritance. But even so, it remains far
from clear that the mixedformulation yields a satisfactory solution to the exclusion problem.
An adequate solution requires establishing that (i) functional mental properties are causally
efficacious and (ii) mental causes do not compete for efficacy with physical causes. On the
mixedformulation, the latter is established by taking the bearers of mental properties to be
the only mental causes, and identifying mental and physical bearers. But if mental bearers are
the only mental causes, then mental tokens are not causes. Any suggestion that they are
nonetheless causally efficacious borders on unintelligibility.
X
a0 References ă
o Alward, Peter, `Excluding Tropes,' unpublished manuscript. %"
X h o Davidson, Donald, `Mental Events,' Essays on Action and Events, Oxford: Oxford University
Press, 1980. %"
X o Ehring, Douglas, `Tropeless in Seattle: the cure for insomnia,' Analysis 59: 1924, 1999. %"
X o Kim, Jaegwon, `Events as Property Exemplifications,' in Stephen Hales, ed., Metaphysics:
X Contemporary Readings, Wadsworth Publishing Company, pp. 33647, 1999. Originally
X published in Myles Brand and Douglas Walton(eds), Action Theory, Dordrecht: D. Reidel,
pp. 15977, 1976.%"
X 3 o """", `The Myth of Nonreductive Materialism,' Proceedings and Addresses of the APA, Vol.
63, No. 3, pp. 3147, 1989.%"
X } o """", `Does the Problem of Mental Causation Generalize?' Proceedings of the Aristotelian
X h Society, 97: 281297, 1997. %"
X o """", Mind in a Physical World, Cambridge, Mass.: The MIT Press, 2000. %"
X * o Lewis, David, `Events,' in Philosophical Papers: Volume II, Oxford: Oxford University Press,
pp. 24169, 1986.%"
X t o """", `Mad Pain and Martian Pain,' Readings in the Philosophy of Psychology, Vol., Ned
Block, ed., Harvard University Press, pp. 216222, 1980.%"
X " o Noordhof, Paul, `Do Tropes Solve the Problem of Mental Causation?' The Philosophical
X # Quarterly, 48:221226, 1998.%"
X
% o Robb, David, `The Properties of Mental Causation,' The Philosophical Quarterly, 47:178194,
1997%" %8 p-++!! Ԍ X o Robb, David and Heil, John, `Mental Properties,' American Philosophical Quarterly 40: 175196, 2003.%"
b J S v Peter Alward
G Department of Philosophy
G ; University of Lethbridge
L 4401 University Dr.
K Lethbridge, Alberta
Q _ Canada T1K 3M4
H 2 9 p-++!! N %" 5 a 7 M& f W B a " Subcontraries and the Meaning of `If8Then'
by Ronald A. Cordero`!%"] ă
y I d d d y 7 =
a a @
d d x 3
d d x : @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 5667
# Subcontraries and the Meaning of If8Then
"
Copyright by Ronald A. Cordero and SORITES -- , a 5 a v
X Subcontraries and the Meaning of If8Then
X - Ronald A. Cordero
+
X =
a pa- 0. Prefatory Note ă
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X j a! 1. Conditionals Composed of Subcontraries
Paradox seems to arise when conditional statements have subcontrary statements as
antecedent and consequent. I am going to argue here, however, that these conditionals can in
fact function as well as ordinary conditionals and that a careful consideration of such cases
can both resolve the apparent paradox and lead to a better understanding of the meaning of
if8then
in general.
Suppose, for a first example, that we know that our friend Jan never serves strawberries.
If we learn that she had guests for dinner last night, it would be perfectly possible for us to
say meaningfully, Well, if she served fruit, she didn't serve strawberries.
We may not know
X whether or not she served fruit, but we know what kind of fruit she didn't serve if she served
any fruit at all. Now, the statements She served fruit,
and She didn't serve strawberries,
fit the definition of subcontraries: they could be true simultaneously, but they could not be
false at the same time. She could have served fruit but not strawberries but could not have
served strawberries but not fruit. So we have a conditional statement composed of
subcontraries that we might very well use in replying to a question such as If she served
fruit, was it strawberries?
And anyone who remembered our reply and later learned that Jan
did in fact serve fruit could reason by the axiom (rule of inference) known as Modus Ponens
(((pDq) U p) 8 q ) to the conclusion that she did not serve strawberries.
Or suppose we learn, for a second example, that a certain figure drawn on the sidewalk
was either a triangle or a square. We of course know that if it was a square, it was not a
circle. So does it not follow that if the figure wasn't a triangle, it wasn't a circle?
jj 1. T V S # Given
X ( jj 2. S D C # Analytic a priori
jj 3. T&
&
# Assumed for CP
jj 4. S&
&
# 3,1; Disjunctive Syllogism
jj 5. C&
&
# 4,2; Modus Ponens <,: = p-p-p- Ԍ jj &
&
#
jj 6. T D C # 35; CP
Line 6 represents a conditional composed of subcontraries that could presumably be used to
transmit useful information. If someone learns the truth of (6) from a dependable source, and
later learns that the figure was not a triangle, they will know that it was not a circle either "
X even if they do not know what kind of figure it was.
It is worth noting that in the two examples considered so far, simultaneous falsity of
X antecedent and consequent is an a priori impossibility.
1. If Jan served fruit, she didn't serve strawberries.
2. If the figure wasn't a triangle, it wasn't a circle.
Because of what we mean by the terms fruit
and strawberries,
it cannot be that Jan failed
to serve fruit but did serve strawberries. And because of the way we use the words triangle
and circle,
the figure cannot have been both. But not all cases of conditionals composed of
X subcontraries are like this. We can also encounter conditionals composed of statements that
X are de facto or a posteriori subcontraries. This occurs with subcontraries that are the
X w contradictories of de facto contraries!statements which owing to some a posteriori fact of the
situation cannot both be true but can both be false.
Suppose, for instance, that there is only room for one person to park in the drive. Then
it cannot be true both that Jim is parked in the drive and that Mary is parked there too. But
it can be the case that neither Jim nor Mary is parked there. It follows that the statements
Jim is parked in the drive,
and Mary is parked in the drive,
are, in the circumstances
X e given, de facto contraries. And this makes their respective contradictories de facto
subcontraries:
jj Jim isn't parked in the drive.
jj Mary isn't parked in the drive.
In the circumstances specified, both can be simultaneously true, but both cannot be false at
the same time.
Now imagine, for the sake of having an example parallel to those given earlier, that we
learn that either Mary or Bill is parked in the drive. This means that if Mary isn't parked
there, Bill is. But, of course, if Bill is parked there, then Jim (a third party) isn't " because
of the limited space. So8if Mary isn't parked in the drive, Jim isn't parked in the drive. This
X Y! last conditional is composed of subcontraries but is potentially much harder to recognize as
X B" such than conditionals composed of a priori contraries. Anyone who knows what triangles and
circles are can see the subcontrariety of antecedent and consequent in If the figure wasn't
a triangle, it wasn't a circle.
But just knowing the meanings of the terms involved will not
enable one to see the subcontrariety of antecedent and consequent in If Mary isn't parked
X % in the drive, Jim isn't parked in the drive.
To see that subcontrariety, one would have to
X & have a posteriori knowledge about the situation (viz., the knowledge that there is just room
enough for one person to be parked there). '; p-++!! Ԍ X a- 2. Peculiarities ă
Thus there are cases in which we can meaningfully assert and reason from conditional
statements with subcontrary antecedents and consequents. Still, it cannot be denied that there
is something strange about conditionals of this sort. In fact, they turn out to have certain
features that are hard to characterize as anything other than paradoxical.
One unusual feature of conditionals composed of subcontraries is that falsity of the
antecedent requires truth of the consequent. This follows, of course, simply from the nature
of subcontraries. Two subcontraries cannot both be false, so it the antecedent and the
consequent are subcontraries and the former is false, the latter must be true. Consider our three
examples:
1. If Jan served fruit, she didn't serve strawberries.
2. If the figure wasn't a triangle, it wasn't a circle.
3. If Mary isn't parked in the drive, Jim isn't parked in the drive.
X With (1) if the antecedent is false!if Jan did not serve fruit, she certainly did not serve
X strawberries. And as for (2), if the figure was a triangle, it can't very well have been a circle.
X o Nor is the case regarding (3) much more complicated. If Mary is parked in the drive and the
circumstances are as originally described, space limitations make it impossible for Jim to be
parked there too.
Admittedly, this is not the way normal conditional statements behave. Ordinarily, knowing
that p D q and learning that p is false does not enable us to conclude that q is true.
A second unusual feature of conditionals composed of subcontrary antecedents and
consequents follows from the first: when such conditionals are true, they always have true
consequents. As just seen, if the antecedent is false, the consequent is true. But if the
conditional statement is true and the antecedent is true, the consequent must also be true
(Modus Ponens). So whether the antecedent is true or false, the consequent is true. Consider
the three examples once more:
1. If Jan served fruit, she didn't serve strawberries.
2. If the figure wasn't a triangle, it wasn't a circle.
3. If Mary isn't parked in the drive, Jim isn't parked in the drive.
If (1) is true, it is evident to anyone familiar with the meaning of fruit
and strawberries
that Jan must not have served strawberries: the true statement at hand asserts that she didn't
X ! serve any if she served fruit!and she obviously didn't serve any if she didn't serve fruit. So
assuming (quite reasonably) that she either did or did not serve fruit, one can conclude that
she didn't serve strawberries. Similarly, in the case of (2), the figure must have been a triangle
or not. But then since being a triangle would have kept it from being a circle, and since not
having been a triangle means it wasn't a circle, it can't have been a circle at all. The situation
X Z& with (3) is essentially the same, in spite of the fact that the subcontraries involved are de facto
X E' rather than a priori. If Mary isn't parked in the drive and (3) is true, Jim is not parked there.
But if Mary is parked in the drive, given the space limitations stipulated, it is also the case
that Jim is not parked there. So granting that Mary either is or is not parked in the drive, Jim
simply cannot be parked there. *< p-++!! Ԍ Once again, this is obviously not a property shared by normal conditionals. Ordinarily,
just knowing that p D q does not enable us to conclude that q .
Perhaps these two properties of conditionals with subcontrary antecedents and consequents
could be considered merely unusual rather than paradoxical, since they do not prevent this sort
of conditional from being employed in the way other conditionals are. There is, however,
another feature of this kind of conditional that really does border on the paradoxical: if the
X consequent of such a conditional is false, the antecedent has to be true. This is clearly the
case, since two subcontrary statements cannot both be false. But this is definitely not the way
conditionals are supposed to behave. Once more, the three examples:
1. If Jan served fruit, she didn't serve strawberries.
2. If the figure wasn't a triangle, it wasn't a circle.
3. If Mary isn't parked in the drive, Jim isn't parked in the drive.
If (1) is true and it turns out that Jan did serve strawberries, then she obviously served fruit.
Or (2), if the figure was in fact a circle, it certainly wasn't a triangle. And as for (3),
X discovering that Jim is parked in the drive would mean under the circumstances stipulated that
Mary isn't parked there!which is what the antecedent says.
Yet according to Modus Tollens (((p D q) U q) 8 p), falsity of the consequent is
X M supposed to mean falsity for the antecedent!not truth. So it would seem that there is
something seriously wrong with conditionals composed of subcontraries!unless in fact Modus
X ! Tollens does not apply to all conditionals.
Nor is the trouble limited to situations in which Modus Tollens would usually be thought
to apply. With conditionals of the sort under consideration, it also turns out that Transposition
((p D q) < (q D p)) cannot apply either. If p and q are subcontraries, a conditional of the
form p D q simply cannot be equivalent to q D p , since subcontrary statements are not
capable of being false at the same time.
1. If Jan served fruit, she didn't serve strawberries.
2. If the figure wasn't a triangle, it wasn't a circle.
3. If Mary isn't parked in the drive, Jim isn't parked in the drive.
How could (1) be equivalent to If Jan served strawberries, she didn't serve fruit,
? How
could (2) be equivalent to If the figure was a circle, it was a triangle,
? Or how, given the
limited space in the driveway, could (3) equate to If Jim is parked in the drive, Mary is
parked in the drive,
? In each case, if we suppose the original statement to be true, applying
Transposition takes us from something true to something false. And that is paradoxical enough
to cause concern. Either Transposition does not apply to conditionals composed of
subcontraries or else there is something really amiss with such statements.
X % Da 3. The Possibility of Avoiding Conditionals Composed of Subcontraries ă
In the light of these considerations, conditional statements with subcontrary antecedents
and consequents may seem to be so bizarre that one could be tempted to ee handle
them by
simply refusing to use them. One could, that is, decide to classify them as meaningless and
decline to use them in reasoning at all. And yet, as already seen, there really do seem to be
instances in which such conditionals are perfectly intelligible and do indeed permit valid *= p-++!! inferences to true conclusions. Moreover, it may not always be possible to recognize
conditionals composed of subcontraries when they are encountered. One might, that is,
encounter a conditional without realizing that it was of the sort that had to be considered as
meaningless and therefore rejected. To see this, consider two new examples:
4. If Lisa is in France, she is not in Altkirk.
5. If Jan served fruit, she did not serve calamondins.
X Regarding (4), it is clear that if I do not know where Altkirk is, I cannot tell whether the
conditional statement is composed of subcontraries. So even if I want to avoid reasoning with
conditionals composed of subcontraries, I will not know whether or not to avoid reasoning
with (4). Yet I can clearly base valid inferences on the statement: if I learn that Lisa is indeed
in France, I know that she is not in Altkirk!even without knowing where that is. And I can
infer by lefttoright
Material Implication ((p D q) 8 (p V q)) that she is either not in
France or not in Altkirk. The problem, of course, is that without knowing whether or not the
conditional statement is composed of subcontraries I cannot tell whether or not the inference
by Transposition to If she is in Altkirk, she isn't in France,
is valid.
X Since example (4) involves geographical location, it involves a posteriori truth: whether
or not a particular city is in a particular country depends on national borders, and the location
X [ of those is an empirical fact, national borders being notoriously mobile. But any subcontrariety
X F in example (5) could be an a priori matter: it could follow, that is, from the meaning of
calamondin
. Suppose I have never heard of calamondins but am assured by a reliable source
that if Jan served fruit she did not serve calamondins. I might well want to ask what
calamondins are!but suppose that for some reason I cannot. If I subsequently learn that Jan
did in fact serve fruit, can I not still validly infer that she did not serve calamondins!whatever
they may be? And could I not also infer (by Material Implication) that Jan either didn't serve
fruit or didn't serve calamondins? Again, however, unless I learn what calamondins are, I
cannot know whether (5) is composed of subcontraries or not!and so cannot be sure about the
inference by Transposition to If Jan served calamondins, she didn't serve fruit.
Thus even if we wanted to reject conditionals composed of subcontraries as meaningless
or unserviceable for valid inference, we would have the problem of knowing when we were
dealing with such conditionals. Sometimes we might not realize that we had encountered the
sort of conditional we had decided to reject. Moreover, when we did succeed in avoiding
X them, we would lose the benefit of their use in cases in which valid inferences could be made.
Ultimately, I submit, it may be best not to reject these conditionals but rather to admit that
under certain circumstances certain axioms!such as Transposition!simply do not apply.
But such an approach raises serious questions about the nature of conditionalization!about
the meaning of if8then
. How can it be that in the case of some true conditional statements,
falsity of the consequent means falsity of the antecedent, while in the case of other true
conditional statements it does not? Is Transposition wrong in its claim that If a, then b,
is
always equivalent to If not b, then not a,
? I believe that a fairly simple explanation of this
matter is possible, but I believe that the explanation requires a basic reconsideration of the
nature of conditionalization in the light of what has been seen about conditionals composed
of subcontraries. %(> .,, Ԍ X a! 4. Reconsidering the Meaning of If8then
ă
X _ What must If a, then b,
mean if it is the case that falsity of the consequent sometimes
requires falsity of the antecedent and sometimes requires truth? Certainly the meaning cannot
be as symmetrical as is indicated by Transposition, which asserts that (a D b) < (b D a)
. The relation between antecedent and consequent asserted in a conditional statement must
involve more in one direction!so to speak!than it does in the other.
X p In 1891 in the first issue of Rivista di matematica, Giuseppe Peano introduced a symbol
X [ for conditionalization, writing that he would use a [here he wrote a backwards c
] b
to
X F represent b is a consequence of a,
b can be inferred from a,
or if a, then b.
F 0 q F
ԍ Principii di logica matematica,
Rivista di matematica, 1 (1891): 110. See also note 5 to the article. His
X 1 symbol, a backwards c
for conseguenza, later became D
, the famous horseshoe
. I
think it is quite plausible to say that when we assert that if a then b , we are asserting that b
X is a consequence of a , that b can be inferred from a . But exactly what does that mean? I am
going to propose that it means that truth of a would permit a sound inference to b , in the
precise sense of a valid inference from true premises to a true conclusion.
Consider the relationship between a conditional argument and the corresponding
nonconditional argument.
a
bjj &
&
# a
cjj &
&
# b
&
&
#
djj &
&
# c D d
The argument on the right is a conditional argument simply because it has a conditional
statement as its conclusion. It is like the nonconditional argument on the left except for the
fact that c has become an antecedent in the conclusion, while d has become a consequent. If
the nonconditional argument is valid, the conditional argument is valid too, and vice versa.
A major advantage of the conditional argument is, of course, that its soundness does not
depend on the truth of c . If we know that both arguments are valid, and we know a and b are
true, we know that the conditional argument is sound, while we don't know whether the
nonconditional argument is sound or not.
In general, if we find ourselves in a situation in which we can see that an argument is
valid, but are uncertain about the truth of one of its premises8
jj e T
jj f T
jj g ?
jj
jj h &? G .,, Ԍwe can go to the sound corresponding conditional argument, keeping as premises only the
statements known to be true and concluding that truth of the (former) premise not known to
be true would permit an inference to the (original) conclusion:
k jj e &
&
T%"&
jj f &
&
T%"&
jj
jj g D h
kIn the conclusion of the example, we are saying that truth of g would permit a sound inference
(valid inference from true premises) to the truth of h .
X
My point is precisely that this is always what we are doing when we assert a conditional
statement!saying that the truth of one statement (the antecedent) would permit a sound
inference to the truth of another statement (the consequent). This is what I believe it means
to say, as Peano does, that the statement in the then
part of a conditional statement is a
consequence of the statement in the if
part.
At this point a change in symbolization is in order. Although Peano may well have
intended to represent the full sense of if8then
by his backward c
, the horseshoe it
evolved into represents only a disjunction between the consequent and the negation of the
antecedent ((a D b) < (a V b)) . Since I wish to represent the full sense of if8then,
I am
going to use a different symbol!the number
sign ( #
) to indicate conditionalization.
When I write a # b,
I will be representing if a, then b,
in what I take to be the full,
ordinary sense of if8then
. My claim about conditionalization then is that in asserting a #
b one is simply asserting that truth of a would permit a sound inference to the truth of b.
But this claim can be amplified. In saying that the truth of a would permit us to have a
sound inference to the truth of b (and thus a sound argument with b as its conclusion), we are
saying that a together with something already known to be true entails b . The meaning of
conditionalization, that is, can be expressed in terms of entailment. A symbolized version, if
one is wanted, might go something like this8
jj (a # b) < (y(x)(Sx U Tx U (ax8b))): ]]A G
with Sx
meaning x is a statement,
and Tx
meaning x is true.
Saying that truth of
a would permit a sound inference to the truth of b is saying in effect that there is a true
statement (It might of course be a lengthy conjunction.) which together with a would entail
b . We can refer to this something known to be true as premise x:
jj x T&
&
jj a ?&
&
x # T
jj &
&
jj b &
&
a # b
In asserting that if a then b, we are saying that b is entailed by a and x, where x is something
known to be true.
At this point concrete examples are needed. To begin with, let us consider a conditional
statement that is analytically true: *@ .,, Ԍ 6. If she drew a square, she drew a foursided figure.
What I am proposing is that (6) is a claim to the effect that knowing that she drew a square
would permit us to make a sound inference to the conclusion that she drew a foursided
figure!to have a sound argument with that conclusion. And this amounts to saying that there
is a true statement (something we already know) which together with the statement that she
drew a square would entail the conclusion that she drew a foursided figure. In this case there
is not much mystery about what that something (premise x) might be:
x T jj Every square is a foursided figure.: ]]A G
N T%"j
S # Fjj So if she drew a square, she drew a foursided figure.
The statement F is a consequence of S because F is entailed by the conjunction of S and the
known fact (x) that every square is a foursided figure.
Next consider a conditional statement that is not analytically true. Suppose for example
we say8
jj 7. If he was there, she was there.
On my analysis, statement (7) effectively asserts that learning that he was there would enable
us to know that she was there!that the antecedent and something we already know to be true
would entail the conclusion that she was there. But in this case it is not obvious what premise
x might be:
xjj T &
&
???@@' + 0 ,,5 : ]]A G T
&
&
H # S &
&
So if he was there, she was there.
The true premise could, as a matter of fact, be a number of different things. We might know
that they always go everywhere together. Or we might know that she promised to attend if he
did and that she always keeps her promises. Premise x simply has to be something known to
be true which, together with H, would entail S.
It might also be interesting to consider an example of a counterfactual conditional
statement:
jj 8. If we were in Tahiti now, we'd be warm.
I am saying that the claim in (8) is that truth of the statement about us being in Tahiti now
(which is acknowledged not to be true) would permit a sound inference to the conclusion
about us being warm. There is some true statement, that is, which taken together with the
statement We are in Tahiti now,
would entail We're warm.
But as in the case of example
(7), the actual nature of x could vary considerably. One possibility might be8
X R& x jj T &
&
It's 30$ C in Tahiti now, and we're warm when it's over 27$. [ T
&
&
T # W &
&
So if we were in Tahiti now, we'd be warm.
Other possibilities for x include We are always warm when we are in Tahiti,
and
Everyone is warm in Tahiti.
The corresponding nonconditional argument (unsound because ^+A .,, of a false premise), would have neither the past subjunctive nor the conditional verb forms
characteristic of counterfactual conditionals:
X H q jj It's 30$ C in Tahiti now.0 ,,5 : T
jj We're warm when it's over 27$.,,5 : T
jj We are in Tahiti. @@' + 0 ,,5 : F
jj
jj So we're warm.q
And what of conditional statements composed of subcontraries? We can begin by
considering example (1) yet again:
1. If Jan served fruit, she didn't serve strawberries.
According to the analysis being considered, (1) asserts that the truth of Jan served fruit,
would permit a sound inference to the truth of She didn't serve strawberries.
There is some
statement already known to be true, that is, which together with Jan served fruit,
would let
us have a sound argument with the conclusion that Jan didn't serve strawberries. Here, as in
the previous example, the nature of premise x could vary considerably. It might, for example,
be something like8
jj &
&
# When Jan serves fruit, it's always citrus fruit,
X ! x jj T &
&
# and strawberries aren't citrus fruits. G
N e e T [ T
&
&
#
J # S &
&
# So if Jan served fruit, she didn't serve strawberries.
Or it might simply be Jan never serves strawberries,
in which case the statement Jan
served fruit,
would indeed permit the sound inference!though only in the sense of not
blocking it:
Jan never serves strawberries. 0 ,,5 : ]]A G T
So if she served fruit, she didn't serve strawberries.
Or consider example (3) again: If Mary isn't parked in the drive, Jim isn't parked in the
drive.
Here the conditional statement is an assertion that truth of Mary isn't parked in the
drive,
would permit a sound inference to the truth of Jim isn't parked in the drive,
!that
Mary isn't parked in the drive,
together with something already known to be true would
entail Jim isn't parked in the drive.
And as the example was presented, the something
known to be true is a complex statement:
X ~% jj &
&
There's only room for one person to park in the drive,
x jj T &
&
and either Bill or Mary is parked there. G
N e e T [ T
&
&
M # J&
&
So if Mary isn't parked in the drive, Jim isn't parked there. )B .,, ԌHere premise x is a conjunction. And the antecedent of the conclusion plays an active role in
the inference to the consequent of the conclusion.
X H "a) 5. Resolving the Paradox ă
The crucial question now is whether or not this analysis of conditional statements in terms
of possible sound inferences allows for an explanation of the peculiarities observed in
connection with conditional statements composed of subcontraries. I believe that it does.
On the present analysis, to assert a # b is to assert that truth of a would permit a sound
inference to b . It is, in other words, to assert that something already known to be true
(premise x), taken together with a , would entail b :
jj &
&
T
jj &
&
(a U x) 8 b
If the conditional statement a # b is true, x must be true, since the truth of x is part of what
is asserted by the conditional statement. But then if b turns out to be false, it must be that a
is false. There could be no other explanation for the falsity of b . (Compare this to the logic
X of a reductio ad absurdum proof, in which the falsity of the conclusion indicates the falsity
of the premise assumed to be true. Or compare it to the testing of a theory, in which the
falsity of a prediction indicates the falsity of some part of the theory.) This is, in fact,
essentially the truth stated by Modus Tollens:
jj ((a # b) U b) 8 a
If a conditional statement is true, and its consequent is false, its antecedent has to be false.
Yet with conditional statements composed of subcontraries, the consequent and the
antecedent, because they are subcontraries, cannot both be false. If the consequent is false, the
antecedent has to be true. But this means that if the consequent of a conditional statement
X # composed of subcontraries turns out to be false, the conditional statement itself has to be false
X (because it has a true antecedent and a false consequent). As noted above, true conditional
statements composed of subcontraries always have true consequents. So it can never be!when
the conditional statement involved is composed of subcontraries!that (a # b) U b . Thus
Modus Tollens will never be applicable with conditionals composed of subcontraries: it will
X simply never be the case that the conditional statement and the negation of its consequent are
both true at the same time.
1. If Jan served fruit, she didn't serve strawberries.
2. If the figure wasn't a triangle, it wasn't a circle.
3. If Mary isn't parked in the drive, Jim isn't parked in the drive.
If Jan in fact served strawberries, it cannot be that if she served fruit, she didn't serve
X % strawberries. If the figure was a circle, it must be false that if it wasn't a triangle, it wasn't
a circle. And if Jim really is parked in the drive (and no more than one person can be parked
there), then it can't be the case that if Mary isn't parked there, Jim isn't either. In general, if
a and b are subcontraries, and b is not the case, it cannot be that truth of a would permit a
sound inference to b . Thus the peculiarities of conditionals composed of subcontraries do not
require any restriction on Modus Tollens!since where such conditionals are involved, Modus
Tollens can never apply. And this is, of course, true even when conditionals composed of
subcontraries are not recognized as such: i+C .,, Ԍ 4. If Lisa is in France, she is not in Altkirk.
5. If Jan served fruit, she did not serve calamondins.
Since Altkirk is presently in France, (4) will never be true when Lisa is in Altkirk. And since
calamondins are a sort of citrus fruit, (5) cannot be true if Jan did in fact serve them.
With Transposition, however, the matter is different: some restriction on the use of the
axiom may be required. To assert that a # b is to assert that truth of a would permit a sound
inference to b . But according to Transposition as it is usually stated, asserting that a # b is
equivalent to asserting b # a . Yet when a and b are subcontraries, a and b are
contraries, so truth of b cannot possibly permit a sound inference to a . Evidently the claim
of Transposition that (a # b) < (b # a) cannot hold when antecedent and consequent are
subcontraries.
On the analysis of conditional statements being proposed here, falsity of the consequent
in the case of a conditional statement composed of subcontraries would mean (1) that the
antecedent is true and (2) that the conditional statement is false. And that would indicate that
the something else thought to be true (statement x) was not. Consider again one of the
fleshedout examples about Jan and the strawberries:
jj &
&
When Jan serves fruit, it's always citrus fruit,
x &
&
and strawberries aren't citrus fruits.
&
&
J # S &
&
So if Jan served fruit, she didn't serve strawberries.
Here x and J together entail S . If we discover that Jan did in fact serve strawberries, we
know for sure that J is true (she did serve fruit), so we can conclude that x must be false.
X J Perhaps she doesn't always serve citrus fruit. And that explains the falsity of the conditional
statement: there turns out not to be something true which together with J entails S . Truth
of J would not permit a sound inference to S .
In order to take all this into account, Transposition will have to be stated with a
qualification preventing its application in such cases. And to allow for the inclusion of this
restriction, the axiom will have to be formulated as a statement of oneway entailment with
a reference to possibility:
jj ((a # b) U _(a U b)) 8 (b # a)
Evidently the restriction is not particularly complicated: it simply rules out cases in which
antecedent and consequent cannot both be false at the same time. If truth of a would permit
X " a sound inference to b , and a and b can both be false at the same time, falsity of b would
permit a sound inference to a .
Actually, it may be best not to write the restricted axiom with the usual symbol for
X & possibility. Since _ is commonly employed to indicate logical possibility, it may be better to
use a different symbol, perhaps a black diamond ` , to indicate possibility in general:
jj ((a # b) U `(a U b)) 8 (b # a)
X ) Written with this symbol, the axiom screens out both cases involving a priori impossibility
X * and cases involving a posteriori impossibility. *D .,, Ԍ 1. If Jan served fruit, she didn't serve strawberries.
2. If the figure wasn't a triangle, it wasn't a circle.
3. If Mary isn't parked in the drive, Jim isn't parked in the drive.
With the revised form of Transposition, (1) does not entail If she served strawberries, Jan
X didn't serve fruit,
because it is impossible a priori that she didn't serve fruit but did serve
strawberries. Nor does (2) entail If it was a circle, the figure was a triangle,
since it is
X impossible a priori that the figure was both a triangle and a circle. And (3) does not entail If
Jim is parked in the drive, Mary is too,
because as the situation has been described it is
X impossible a posteriori that both are parked in the drive.
With the qualification indicated, Transposition does not have to be abandoned as a valid
X
rule of inference. There is no need to question its ability to get us from truth to truth.
0 q F q
ԍIn this connection, see for example Donald Nute, Topics in Conditional Logic (Dordrecht: Reidel, 1980) 85.
But does this not leave us with a problem already raised!that of situations in which we
X @
do not know whether antecedent and consequent are subcontraries? Suppose we come to know
somehow that8
9. If Jones didn't call, Smith didn't call.
But suppose we are ignorant of whether just one or more than one person called. We have no
X idea, that is of whether or not Jones didn't call,
and Smith didn't call,
are de facto
subcontraries. What can we do? Does Transposition apply or not? Is there nothing to be
inferred? I think the answer is clear from an alternative formulation of Transposition in its
restricted form:
jj (a # b) 8 (`(a U b) # (b # a))
We can infer from (9) that if it is possible that both called, Jones called if Smith did:
jj 1. J # S
jj 2. `(J U S) # (S # J) + 1; Transposition and Double Negation
X | a/ 6. Conclusion ă
If conditional statements are understood as claims to the effect that truth of one statement
would permit a sound inference to the truth of another, the fact that there are conditional
statements with subcontraries for antecedents and consequents poses no insuperable problem
for logical analysis.
b N v Ronald A. Cordero
G Department of Philosophy
9 The University of Wisconsin at Oshkosh
I Oshkosh, WI 54901, USA
L $E G .,, N %" 5 a N f k5 W B a " Does Frege's Definition of Existence Invalidate the Ontological Argument?
by Piotr Labenz`!%"] ă
y d d d y N =
a a @
d d x :
! d d x F @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 6880
Does Frege's Definition of Existence Invalidate the
. R Ontological Argument?
% Copyright by Piotr Labenz and SORITES -- , a 5 a v
s Does Frege's Definition of Existence Invalidate
s . ) ' the Ontological Argument?
X
/ * Piotr Labenz
+
V
, a Affirmation of existence is nothing but the denial of number nought. Because existence is a
X
X property of concepts, the ontological argument for the existence of God breaks down. 5 q F ԍFrege (1950: 65).5
X N %" a 1. The purpose and arrangement of the paper ă
In his attempt to build a basis for mathematics, Frege proposed definitions of some very
X A basic concepts that proved to be of import not only to the Grundlagenproblem, but to other
areas of philosophy, ontology in particular, as well. Frege himself seems to have noticed that,
X as is indicated by his remark that the definition of existence he gave in the Grundlagen der
X Arithmetik < G q F ԍi.e. Frege (1950).< shows the invalidity of Saint Anselm's ontological argument for the existence of
God. As often is the case with Anselmian problems, this claim seems " given the definition
" obvious at first glance, but upon closer inquiry " quite philosophically perplexing and not
obvious at all. Hence, in this paper we shall expound that definition and investigate how can
it be used to criticise the ontological argument. In order to do this, we will also need to give
some attention to the context of Frege's system, possible criticisms of his definition of
existence, and logical structure of Anselm's argument.
To begin with, we will introduce some of the basic ideas of Frege's system, which form
the rudiments of modern predicate calculus as well as of Fregean ontology. Of course we shall
not attempt any interpretation of Frege's system as a whole in this paper, but rather adopt its
X standard account, and only focus on our subject J q F Q$ ԍFor a throughout interpretation of Frege, see e.g. Dummett's works (1981, 1983, 1991). For Frege's logic, see
q F % Boche9ski (1961: 291292, 320322 et al.). In this paper we will focus on the philosophical issue and not on
exegesis of Frege's writings (which, even though clearly written, are obsolete and sometimes ambiguous, and hence
prone to differing interpretations). Nor will we touch upon any related issues, such as whether Frege was a realist
or a nominalist " see papers in Klemke (1968), Biriukov (1964), or for implications of the definition of existence for
traditional metaphysics " see Angelelli (1967: 225227). . (It seems acceptable to speak of `Fregean
ontology' as clearly Frege did conceive of the results of his linguisticological inquiries as
applying to the real world, that is, describing not merely relations between expressions, but d F = ...ee X between actual objects as well n b q F y ԍThis is proven by Wells (1954: 537 ff). It would suffice to support it by recalling the vast literature devoted to
q F 0 Frege's ontology, notably Cocchiarella (1972: 181 ff), Klemke (1968), and the monograph by Williams (1981). . In any case, only this allows using a definition of existence
stemming from pure languageanalysis in discussing an ontological issue.)
Then, we will present Frege's definition of existence itself, together with arguments in
support of adopting it; and next, its criticisms. This discussion will conclude with certain
results that will be important for further analysis of how the definition relates to the
ontological argument. Next, we will proceed to exposing the structure of the ontological
argument for the existence of God and considering several questions significant in its analysis.
Finally, we will put the results of the analysis together and look at what do they imply for the
topic question of the paper, and conclude with an answer to it.
X
a$ 2. Frege's definition of existence
X |
\a* 2.1. Object and concept ă
The crucial, for us, idea in Frege's system is the distinction of objects and concepts, and
the latter of first and second order; or, more precisely, distinction of names of objects, names
X
of firstorder concepts and names of secondorder concepts. We will follow Forgie & %
b q F ԍForgie (1972: 254256) gives an exceptionally lucid account of this, which we will thus follow in this section. Also
see e.g. Walker (1965: Ch. 2.), Grossmann (1969: Ch. 2.), Munitz (1981: 82104). References to relevant passages
in Frege can be found there.& by
abbreviating `names of objects' as `Aexpressions', `names of firstorder concepts' as `B1 X expressions' and `names of secondorder concepts' as `B2expressions' b q F c ԍForgie (1972: 254). The following definitions of A, B1 and B2expressions are due to Forgie as well. . An Aexpression is
such that can be the grammatical subject of an utterance, but cannot be the grammatical
predicate; or which is a complete declarative sentence. The latter is because sentences,
according to Frege, denote Truth or Falsity, which are objects, hence sentences denote objects,
like other Aexpressions, rather than as, loosely speaking, expressing facts (which is somewhat
X confusingly different from the nowadaysstandard namesentence distinction T b q F ԍWhich has been introduced by Ajdukiewicz (1935).T . Examples of
Aexpressions: `horse', `Socrates', `the teacher of Plato', `God', `7', `Socrates is mortal',
`5+7=12'.
Now, Aexpressions can serve as arguments for functions, that is, expressions having
argumentplaces, such as: `!!! is mortal', `the capital of !!!', `!!! + !!! =12' (where
`!!!' is the argumentplace) and so forth. If an argument, i.e. an Aexpression(s) is (are)
substituted for argumentplace(s), an Aexpression is obtained from the function (strictly
speaking, from the name of the function). If the Aexpression resulting from substituting an
Aexpression into a given name of function denotes Truth or Falsity, then the name of the
function is a B1expression, that is, a name of a firstorder concept; e.g. `is mortal', `8+8=12',
but not `is the capital of. (By substituting an Aexpression into it, a saturated expression is
obtained from an unsaturated Bexpression; `saturated' meaning complete, selfstanding, able
X n of being meaningful X n U b q F * ԍSee e.g. Biriukov (1964: 25 ff) on saturation.X .) n G
.,, Ԍ Finally, B1expressions can be the arguments of secondorder functions. If the Aexpression resulting from substituting a B1expression into a given name of secondorder
function denotes Truth or Falsity, then the name of this secondorder function is a B2expression. Examples of B2expressions are adjectives of number: `there are 460 of', `there
X are as many8as' and V b q F ԍMunitz (1981:9899) following Dummett (1973: 262).V , as will be shown, existence. They name secondorder concepts, because
these concepts are being assigned to another concepts, rather than no objects. Frege's
illustration of this point is that whereas `is thoroughbred' is a B1expression, the argument of
which is the name of an object, say `horse', `there are four' is a B2expression, the argument
X H of which is a concept, say `thoroughbred horses'
nHG b q F d ԍOf course, `thoroughbred horses' is a saturated concept, and an Aexpression, obtained from the unsaturated
firstorder concept, the function ` "" is thoroughbred', which is a B1expression. 9 H b q F
ԍFrege (1950: 59, 64).9 .
X a' 2.2. Definition of existence ă
X
Now, Frege defines existence as the negation of number zero n
D b q F ԍFrege (1950: 65). See also wider exposition in Williams (1981: Ch. 3.) and comparison with other definitions
of existence in Labenz (1999: 34). . To say that x exists is
X
to say that there is a nonzero number of xĩs; that is, to say simply that there are xĩs.
Therefore, `exists' is a B2expression, meaning in fact `there are more than zero of'. This is
a key claim of Frege's, denying existence to be a firstorder concept, which it might at the
first sight appear to be.
A single argument for the claim that existence is a secondorder predicate in Frege might
be the weak naturallanguageanalogy argumentation for the claim that number in general is
X a secondorder predicate 9
b q F R ԍFrege (1950: 59, 64).9 . However, we have proposed another argument on more
X ontological lines 5 A
b q F ԍLabenz (1999: 3).5 . Now, let `P' stand for the concept of `there are 460 of'; what falls under
it is, e.g. `the members of the Polish Diet'; let `Q' stand for the concept of `there are 0 of'.
In case of a parliamentary crisis leading to the dissolution of the Diet, Q would apply to `the
members of the Polish Diet'; and there would be no such members. If `P', `Q' were B1expressions, then in that case `Q' would have a nonexistent argument. This, however, would
lead to the serious ontological problem of socalled `Plato's Beard' " predication about
nonexistent objects. Although this is not a proper place for discussing `Plato's Beard', we can
X ' assert this difficulty should be avoided n' b q F % ԍFor a good analysis of `Plato's Beard' problem (which originates from Plato's Parmenides), see Jadacki (1981)
and Williams (1981: 3741). . And this can only be done by treating `P', `Q', and
hence also `exists', as B2expressions.
Moreover, let us remark that the secondorder concept of existence applies (semantically)
X X to the object, not to the firstorder concept, under which that object falls y X> b q F k+ ԍThis ought to be reflected in the model (of logic) of an ontology using this concept.y . So, if we XH .,, substitute `thoroughbred horse' into ` "" exists', we assert the existence of a horse (and a
thoroughbred one), and not of thoroughbredness. This is significant, as otherwise this secondorder concept of existence would lead to difficulties with fictional entities. Let the firstorder
concept be that of a unicorn, the secondorder that of negation of existence. Then, what is
meant to be nonexistent is not the concept of unicorn, but unicorns " the object, not the first X order concept n b q F ԍGrossmann (1969: 6970) recognizes the problem, claiming that Frege confuses the two possibilities: asserting
the existence of a concept and of an object. .
X An opposite view is held by Munitz 9 b q F
ԍMunitz (1974: 7880).9 , supported by some evidence from Frege's writing.
He claims that existence, being a secondorder concept, applies to the firstorder concept rather
than to object, and is to be read: `is instantiated', rather than `exists'. Hence `a thoroughbred
horse exists' would be `thoroughbredness is instantiated'; similarly, `unicorns do not exist'
would be `beingaunicorn is not instantiated'. This is to be so, because, according to
X y
Munitz 1 y
b q F ԍIbidem.1 , the B1expression `exists' is equivalent to quantifying existentially an Aexpression; and what Frege means is to define existence in terms of the quantifier rather than
as a predicate. But even if this interpretation is correct, still it is undeniable that what is
concerned is the existence of the object, not merely nonemptiness of the concept. When we
say `a thoroughbred horse exists', we do not only say that `thoroughbredness is instantiated',
X but that it is instantiated by a horse (rather than by, say, a hound, or a language r D b q F ԍFowler in The Modern English Usage calls German a thoroughbred language.r ) as well.
Therefore, the analysis holding the existential quantifier equivalent to secondorder concept
X being generally correct X b q F ԍAn exception will be pointed at in the next section.X , still the latter applies to objects the previous quantifies, and not to
their properties " which are quite irrelevant for the quantifier, and hence must be so for the
secondorder concept of existence too. Thus Munitz is right (and quite insightful, perhaps) in
that analysis, but he contradicts himself saying that existence applies to firstlevel concepts,
X | not objects 6 | b q F ԍMunitz (1974: 78).6 .
Finally, an issue that should be made clear is whether on Frege's definition existence is
a predicate or not. On the face of it, Frege expressly and repeatedly says that it is a
X property I - b q F $ ԍFrege (1950); see Labenz (1999: 13).I , hence a predicate. However, it has been widely claimed " after Kant " that in fact
it is essentially not a fullfledged, real predicate like `thoroughbred', `mortal' etc., but merely
a logical (i.e., behaving like a predicate syntactically, but not being one semantically) and non X h determining one (i.e., such that does not enlarge the argument's connotation nh b q F
) ԍLike `identical with itself', `thoroughbred or not thoroughbred', etc. In Polish there is a proverbial expression for
that: `buttery butter'. ). X h* b q F g+ ԍSee Williams (1981: 1741), Shaffer (1962: 309311).X And this hI .,, has been put forward as the gist of both Kant's and Frege's criticisms of the ontological
X argument b q F b ԍCommonly at least; and while rightly so about Kant, we will consider whether rightly about Frege, too. . Moreover, the secondlevel predicate has been interpreted as equivalent to first X order quantifier C G b q F ԍMunitz (1974), Williams (1981).C , thus further confusing the distinction between predicate and quantifier
definitions of existence. The discussion on whether existence is a predicate by far exceeds the
X scope of this paper; n b q F c ԍA classic text on this widelydiscussed issue is Moore (1936); a interesting treatment is given by Sommers
(1973). therefore, we should content ourselves with a following view, perhaps
not very satisfactory, but seeming in accordance with most expositions. For Frege, then,
existence is a secondorder predicate " which may be understood as paraphrase of firstlevel
quantifier " but not a firstorder predicate. The latter claim will be discussed critically in the
following section.
X a. 2.3. Criticism ă
However, there are arguments against treating existence as a secondorder predicate.
Firstly, there is a problem with the existence of individual objects " that is, such, that we do
not need to assert any additional concept besides that of existence of them " such as, say
X `Andrzej Goota' D b q F ԍThis objection has been raised by Forgie (1972: 259261) and Grossmann (1969: 6769). Walker (1965: 32)
ignores it, apparently not regarding it as a problem at all, quite counterintuitively. Munitz (1974: 7885) soothes the
problem away, on the ground of quantifier interpretation of Frege's definition of existence, reducing it into the
Russelian theory of descriptions. . It seems obvious that we can say `Goota exists' (even if not much more
exciting could be said about him), and that there are not two concepts there, but one, that of
existence, and it is a firstorder concept applied to the object of Goota. Perhaps, though, a
Fregean would answer that `Goota' itself is a concept (however preposterous might that
sound), and uttering `Goota exists', we actually say `something, which is Goota, exists', and
X N existence remains a secondorder predicate quite well nN
b q F / ԍOf course this would be an answer somewhat in the manner of Russelian theory of descriptions; Russell (1905).
See Munitz (1974: 8486). . That sounds somewhat odd, but
might perhaps be agreed upon. However, let us put forward a weaker claim than about the
existence of Goota, namely: `something exists'. Now, this is true (and entailed by the
existence of Goota, or of whatever), and cannot be analysed in terms of secondorder concept
X of existence nf b q F -# ԍOne might try some exquisite wordconstructions to save Fregean approach here, e.g. analyse `something' as
`whatever thing', `whatever ""' being a B1expression. However, this seems quite unfeasible. . This precludes the attempt to eliminate the problem by the means of a theory
X of descriptions. While (yx) P(x) " where `P' is the property of being a given individual, say,
X Goota " could be treated as -(P(x)) or so, (yx) x cannot " we can only rephrase it to some
X F(x): a firstorder predicate, that is, concept of existence is needed here (and similarly (?p) p;
X rephrasing into (yx) x=p is of no help, and abovesuggested rephrasing into (yx) P(x) is in fact
doubtful, as will be shown below).
Therefore, it is impossible to dispense with a firstorder concept of existence, because
some utterances about existence cannot be formulated without it, but using only the second J .,, ԫ X order one. Apparently, Frege himself has noticed it, but was not able to account for it k q F y ԍForgie (1972: 260); the passage referred to there is Frege (1960: 108).k .
Rather, he would say that it is meaningless to talk about the existence of individual objects;
X they can be `real' 3 ! G / q F ԍWirklich.3 , but cannot exist. To say that Goota exists is, according to Frege,
X meaningless " / q F z ԍFrege (1960: 50); see Grossmann (1969: 64é68) for a discussion of this view in relation to Russell and Moore. . This is so, because `Goota' is not a property (i.e., ` "" is Goota' is not a B1expression); it does not make sense to talk about `falling under Goota
' (meaning, of course,
the concept of Goota) in the same way as `falling under thoroughbredness
'. Therefore the
existence of individual objects cannot be expressed by the means of a secondorder concept
of existence.
However, an attempt to relieve that problem has been made by C. J. F. Williams in his
X interpretation of the Fregean doctrine < # / q F ԍWilliams (1981: 81107).< . He claims that the solution is to treat individual
objects as `unique instantiations' of certain properties, or sets of properties that unambiguously
X y
point to these individuals 9 $ y
0 / q F ~ ԍWilliams (1981: 106).9 (for Goota, these might be e.g. ` "" lost against Lennox Lewis's,
` "" lost against Mike Tyson' etc.). Then, in accordance with our above analysis of Frege,
existence as secondorder concept can be asserted of such individual. Though, we can easily
see that this counterargument is not much more than a simple negation of the claim that being
an individual is not a property; it is to be, in fact, a property or conjunction of several
properties that determine it in the above manner. Now, it seems that not much can be done
about that: these are two opposing views and apparently there are no convincing arguments
X to falsify either % n / q F ԍAs Leszek Koakowski has once noticed, it is a common fate of all the philosophical problems of past twenty
five centuries than no party is ever convinced by the opponents' arguments. . Thus we might conclude by only recalling that, firstly, Frege himself held
that to assert existence of individuals is meaningless and, secondly, there is no satisfactory
account for `something exists' in Fregean terms.
Moreover, even, if all propositions about existence were expressible in terms of Frege's
definition, it still would be not a sufficient and compelling reason to accept it. As has been
shown above, some utterances are better " more simply and intuitively " analysable using the
firstorder concept of existence. Now, there is apparently no good reason to discard firstorder
formulations in favour of secondorder ones, which Frege would do. Different utterances can
X express the same thought by the means of quite different predicative structures ^ & - / q F % ԍForgie (1972: 259). Forgie names this claim `principle P'.^ . Hence there
is no good reason to restrict ourselves to secondorder concept of existence, if some utterances
have predicative structures better explicable by other means.
Finally, let us consider one more apparent objection. As we have already said, Frege's
secondorder concept of existence applies to objects rather than firstorder concepts. But what,
if we wanted to ascribe existence to a concept " for instance to say: `the wisdom of the people
exists', meaning the existence of the wisdom, not of the people? We would seemingly need K & .,, to use a thirdorder concept to account for it, as the secondorder concept ascribes existence
to the object only, not to the concept describing it. However, according to the definition of
Aexpression, a saturated name of a concept (e.g., `the wisdom of the people') can be
perfectly well treated as an Aexpression, and hence an argument of a B1expression, to the
referent of which existence can apply. Thus no higherthatsecondorder concepts are needed
here.
X a. 2.4. Conclusion ă
Therefore, we can say that Frege's doctrine of existence cannot be defended in its full
strength. It cannot account for the sentence `something exists', and deals rather inadequately
with sentences about individuals. On the other hand, it disallows the use of firstorder concept
of existence, which would solve this problem. Of course this does not mean that Frege's
definition of existence is thoroughly wrong. For instance, it might be enough to follow
Munitz's approach and combine Fregean definition with a quantifier definition of existence
in order to eliminate these difficulties. However, to do that would require exact and extensive
analysis, being too ambitious task to be attempted here. For our purposes it will be enough
to recapitulate the following about the Fregean definition. Firstly, it applies to objects
characterized by some firstorder concept. Secondly, it could be thus applied to individual
objects, by the means of a theory of descriptions, but in a rather awkward manner and
contrary to Frege's contention that is meaningless to do so. Thirdly, it claims existence to be
a secondorder predicate, which means it not to be a real predicate.
X a 3. The criticism of the ontological argument
X
a% 3.1. Formulation of the argument ă
To begin with, there is not a single ontological argument, but numerous versions of it,
which are not necessarily equivalent. The inventor of the argument, St. Anselm of Aosta
X & himself gave (at least) two logically distinct formulations of the argument ' & / q F ԍIn Proslogion Ch. 2 and Ch. 3 " Anselm (1979); the distinctness of the proofs is shown by Malcolm (1960). , and other
philosophers, from Descartes and Leibniz to Alvin Plantinga have put forward other
formulations. Now, Frege in his brief remark does not refer to any particular version of the
argument; therefore we will consider here the most wellknown formulation, that is, that of
X Proslogion, Chapter 2, which is what he most probably meant when writing about `ontological
argument for the existence of God'.
Furthermore, even this single formulation of the argument " originally formulated in
eleventhcentury Latin " has indeed numerous paraphrases in terms of more modern technical
X philosophical vocabulary or various logical systems ( G / q F $ ԍSome of these are Koodziejczyk (1998), Kelly (1994), TichU (1979), Adams (1971), Plantinga (1966). . We will not attempt to offer an exact
analysis of that sort; rather, as Frege probably intended his remark against the simple common
explication of the argument, we will present such elementary, but still interesting and general
enough, account. As our end is analytical rather than historical, we will generalize rather than
follow Anselm's text exactly (this will be essential for further discussion). Hence, the
argument has the following form:
I. A. 1. a.(1)(a) i) a) (1)(1)(1)(1)(1)(1)(1)(1) b/
jcM"z *!#%2(*, 5a
iq!y )" b X ,!! God is by definition the being than which no greater being can be thought of (or,
conceived).%"! &L ( .,, Ԍ X X ,!! A being, which is thought of and exists in reality \ ) Z q F y ԍAs opposed to `in thought', so simply " which exists.\ , is greater than an otherwise
identical being, which is thought of but does not exist in reality.%"!
X ,!! I think of God, so God is thought of.%"!
X ,!! If (a) He existed in reality besides being thought of, he would be greater, than if (b)
he didn't exist in reality.%"!
X ,!! But He is the greatest being, so (a) is the case rather than (b), that is: God exists in
X reality. Q.E.D.%"!
The key point as far as Fregean (and Kantian) critique is concerned is (2), and more
specifically the claim that to exist is greater than not to exist, entailed in (2). In Anselm's text
the argument is formulated in terms of adding another property, and is formulated more like:
`a being having all the God's properties except existence is less perfect than a being having
all the God's properties including existence'. Then the critics straightaway claim that existence
is not a perfection, that is, in more modern terms, a real predicate, that is, in Fregean terms,
X
a firstorder concept ? *
G Z q F ԍThus Plantinga (1966: 538).? , so this step is, according to them, invalid and hence the argument
fails.
However, we have seen, in the above formulation of the argument, no resort to the
notion of God's properties. Obviously in intuitive terms, there is a notion of `adding' or
`subtracting' existence from the set of properties of God in the argument. But it is not
X indispensable for the argument at all Z + Z q F q ԍThis observation has been made by Forgie (1972: 251). Z . (It has been argued that the existential use of
quantifier is excluded in the case of the ontological arguments, for it involves quantification
over objects both existing really and only thought of, so it is necessary to use the existence
predicate. Though, it is not a valid objection, as it seems enough to use predicates
X V distinguishing real and intensional objects, without an existence predicate , nV Z q F ԍThen existence could be expressed by the quantifier, in the manner of Quine (1969), only the universe would
have to include both real and intensional objects. .) Therefore
probably Frege was thoroughly incorrect in his remark, basing it on the mistaken belief that
X ( existence's being a property, and a firstorder property, is essential for the argument - n( Z q F ! ԍOn the ground of the aboveconsidered Frege's doctrine of existence, Plantinga's (1966: 54) `desultory gesture'
against Frege in terms of problems with nonexistents is quite besides the point. .
(Besides, not uncommonly is the assertion that existence is not a predicate deemed to prevent
X us from defining God into existence : . A
Z q F & ԍE.g. Allen (1961: 59).: . However, it seems equally inappropriate to define, say
Goota into being victorious, or a horse into thoroughbredness; and it is not a reason to claim
that being victorious and thoroughbredness are not predicates.) Still, it might be worthwhile
inquiring whether Frege's critique would invalidate the argument if it did rely on the
assumption that existence is a predicate; we will, then, try to apply Frege's definition to the
argument and see whether it would result in showing the latter's fallibility. M . .,, Ԍ X *
a# 3.2. Is `God' a name or a predicate? ă
Apart from the abovegranted (for the purpose of the inquiry only) proposition that the
ontological argument relies on existence's being a predicate (we have not granted it must be
firstorder). Then, having in mind the above discussion, there seems to be a point worth
consideration: is `God' a proper name (an Aexpression) or a predicate (a B1expression)? In
the first case we would run into all the abovementioned difficulties with reconciling Frege's
definition and existence of individuals; in the second, on the contrary, we might use the
secondorder concept of existence perfectly well.
It has been claimed problematic both to regard `God' as a logically proper name, that
X is, name with reference but without sense, and as a predicate / n Z q F ԍAllen (1961: 6163). Unfortunately, most of his argumentation is based on premises we have not granted, or
otherwise doubtful. . Of course the argument that
to treat it as a name would be defining into existence misses the point, for there obviously are
proper names of nonexistent objects as well, e.g. `Gandalf'. Nor is the argument that existence
cannot be asserted of the referents of proper names, because existence is a secondorder
X property, sound " it would be a petitio principii to accept it here, of course.
X " However, it might be claimed z 0 " Z q F ԍSomewhat inspired by Allen (1961: 62), but we have changed the argument significantly.z that `God' could not be a property just because of
what the ontological proof seems to rest on: that the concept of God involves necessary
existence and certain properties (omniscience, omnipotence, benevolence etc.). Then, it seems
X unfeasible to say: `x is God iff x exists necessarily and is omniscient etc.', because it suggests
X that x has the properties of being omniscient etc., and of being God, too, which is obviously
a misunderstanding. Thus, being God seems not to be a property (just like being Goota, as
has been indicated above), which agrees with the intuition rejecting predicates that have, by
definition, only one instantiation.
Though still, these are not conclusive, however intuitively appealing, arguments either
way. From a logical point of view " which is the decisive one " both solutions are possible;
therefore we will consider both in further analysis.
X a 3.3. Can secondorder predicates be used freely? ă
X t An argument has been put forward 4 1 t Z q F ԍMavrodes (1966).4 that secondorder predicates are such that cannot
be ascribed freely to objects " as some firstorder predicates can " but are a matter of fact
independent of the languageusers' decisions. For instance, we can define a `gavagai' to have
the firstorder property of being gray, but we cannot influence the secondorder property of
whether it exists, or of how many gavagais are there, etc. We can conceive of an object's
X having firstorder predicates, but not secondorder predicates (thus it is meaningless to say `x
can be conceived as existing').
However, there are other accounts of secondorder properties quite different from this.
X 4" For instance, Cocchiarella 8 2 4"D Z q F M* ԍCocchiarella (1969).8 , following medieval logicians, has proposed a logic in which
firstorder predicates (and relations) are such that entail existence of the object (so called `e #N 2 .,, ԫattributes'), and secondorder predicates are such that do not: for instance, respectively, `is
thoroughbred' and `is thought to be thoroughbred'. Now, of course, this assumes a quite
different interpretation, or rather use of secondorder predicates, and entirely different
approach (e.g., fictional objects on this account have " and necessarily so " only secondorder
properties, while on the previous they have only firstorder properties).
Therefore, we can see that the account claiming secondorder predicates not to be
freely ascribable to objects is not compelling. It is possible to adopt a contrary position; both
can be well formulated in logical terms. This points to a more general conclusion. It is not
unusually the case that there are several distinct, and often mutually contradictory, possibilities
of formulating a philosophical point. There might be a heated discussion over these, involving
arguments in favour and against particular solutions of the problem. However, as long as a
solution is not proven to be either selfcontradictory or contradicting something we otherwise
hold " all arguments can be at most suggestive, and not conclusive. As long as a solution can
be consistently formulated, there is no compelling reason for rejecting it. This conclusion
obviously applies to several abovediscussed issues, too; hence we will now not attempt
defending particular solutions of these, but rather try to see how the various possibilities
influence the ontological argument.
X e } a 3.4. Application of the results to the argument ă
In order to present the results of the above analysis, we will collect them in the form
of the following table, showing possible outcomes. It has to be noted that the table has been
constructed in a simplified way, outright excluding some impossible combinations (e.g. that
existence cannot be a predicate and firstorder predicate can express existence).
@
! d d x F AZ Z =O @ @ -- H (1) Is existence's being a property essential for the ontological argument? H = E AZ Z =O @
aZ Z 6O WH @@ E H H@ H= X 6 Yes. (2) Can existence be a predicate (real or merely logical " i.e., of any
order)? X 6 No.
The
whole
critique
is besides
the
point. H@ H 6 O
aZ Z 6O WH @@ Z Z ~ O WS9H @@@ O H@ H H@ @ H6 X ~ Yes. (3) Is `God' a name or a firstorder predicate? g X ~ No.
The
whole
critique
is besides
the
point. & H@ @ H ~ Y Z Z ~ O WS9H @@@
Z Z &O W$ /
9H @@@@ Y H@ @ H @ @ @ H~ X & A name. (4) Can a firstorder predicate express
existence? ) X & A firstorder predicate . (5) Can
secondorder predicates be used
freely? ) ) @ @ H & )O 2 .,, Ԓ h
Z Z &O W$ /
9H @@@@ Z Z P WU'9H @@@@@@ h " A A IA A A A A I " X Yes. The
argument
holds, critique
fails. X No. The
proof is
fallacious. (*) X Yes. Then the
critique fails, the
argument is not
invalidated. X No. The critique
is right, the
argument is
invalidated. A A I
Now, we have established that (1) is false, but we have suspended that result so that
further inquiry would be possible. The answer to (2) has been shown to be `yes'. The answers
to (3) and (5) have been shown to be possible either way. It has been shown, too, that answer
to (4) is possible either way, and that Frege's answer was `No'. And, moreover, Frege would
also (need to) answer to (3) that `God' is a firstorder predicate, for, as it has been shown,
existence, according to his doctrine, applies to objects characterized by a firstorder concept
(here: ` "" is God'). Needless to say, he would say `No' to (5).
(A short comment should be made on the case (*). If `God' is a name, and firstorder
predicates do not express existence, and existence can be expressed by a predicate, then
existence must be expressed by a secondorder predicate. But then, the firstorder predicate
is not the concept of God, for it would be redundant to predicate it of an object named `God'
" though, perhaps, sill possible, but then the case would fall under ` God
is a predicate'.
Then, it must express something else; we do not want to engage in philosophical fiction and
speculate, what. In any case, then the argument would aim at the existence of that mysterious
concept rather than God, and be thus fallacious. Therefore we will neglect the case (*).)
Therefore, we can conclude the following. Granted the false assumption that
existence's being a property is essential for the ontological argument, and granted the
assumptions, for which no conclusive proof is offered, that `God' is a firstorder predicate and
that secondorder predicates cannot be used freely " indeed, then, Frege's definition of
existence invalidates the ontological argument for the existence of God. Hence in order to
uphold this critique one would have to make the case for all the assumptions, which would
be quite an ambitious task, especially as far as the problem whether it is indispensable for the
ontological argument that existence were a predicate is concerned. The assumptions present
interesting problems in themselves, and imply several other problems, e.g. whether a firstorder predicate must be a real predicate etc. However; if all the mentioned assumptions are
not granted " and, let us repeat, seemingly no convincing reason for doing so has been given
" then the critique falls short of its objective, and Frege's definition of existence is quite
unharmful to the ontological argument for the existence of God.
X 0a/ Bibliography ă
X P! X R. M. Adams, `The Logical Structure of Anselm's Arguments', The Philosophical Review,
LXXX, 1 (Jan., 1971)%"
X # X K. Ajdukiewicz, `Die syntaktische Konnexit t', Studia Philosophica, I (1935)%"
X $ X R. E. Allen, `The Ontological Argument', The Philosophical Review, LXX, 1 (Jan., 1960)%"
X \& X I. Angelelli, Studies in Gottlob Frege and Traditional Philosophy, Reidel, Dordrecht 1967%"
X ' X Anselm of Aosta [of Canterbury], Proslogion with A Reply on Behalf of the Fool, tr. and ed.
M. J. Charlesworth, Univ. of Notre Dame Press, Notre Dame 1979%"
X * X B. V. Biriukov, `On Frege's Works on Philosophical Problems of Mathematics', [in:] idem,
X * Two Soviet Studies on Frege, tr. and ed. I. Angelelli, Reidel, Dordrecht 1964%" *P 2 .,, Ԍ X X I. M. Boche9ski, A History of Formal Logic, Univ. of Notre Dame Press, Notre Dame 1961%"
X a X N. Cocchiarella, `A Second Order Logic of Existence', Journal of Symbolic Logic, XXXIV,
1 (Mar., 1969)%"
X X """, `Properties as Individuals in Formal Ontology', Nous, VI, 2 (May, 1972)%"
X X M. Dummett, Frege: Philosophy of Language, 2nd ed., Harvard Univ. Press, Cambridge 1983%"
X m X """, Frege: Philosophy of Mathematics, Harvard Univ. Press, Cambridge 1991%"
X X """, The Interpretation of Frege's Philosophy, Harvard Univ. Press, Cambridge 1981%"
X / X J. W. Forgie, `Frege's Objection to the Ontological Argument', Nous, VI, 3 (Sept., 1972)%"
X
X G. Frege, The Foundations of Arithmetic, tr. J. L. Austin, Blackwell, Oxford 1950%"
X X """, The Translations from the Philosophical Writings of Gottlob Frege, ed. and tr. P. T.
Geach, M. Black, Blackwell, Oxford 1960%"
X ; X J. Grossmann, Reflections on Frege's Philosophy, Northwestern Univ. Press, Evanston 1969%"
X X J. J. Jadacki, `On the being of what seems not to be', Dialectics and Humanism, VIII, 4
(1981)%"
X C. J. Kelly, `Circularity and Amphiboly in Some Anselmian Ontological Proofs: A Syllogistic
X Inquiry', Nous, XXVIII, 4 (Dec., 1994)%"
X 0 X E. D. Klemke (ed.), Essays on Frege, Univ. of Illinois Press, Urbana 1968%"
X L. A. Koodziejczyk, `O tak zwanym ontologicznym dowodzie istnienia Boga uwag kilka'
[Some Remarks on the SoCalled Ontological Proof for the Existence of God],
X c Filozofia Nauki, 2324 (34/1998)%"
X X P. Labenz, `Definicje istnienia' [`Definitions of Existence'], Filozofia Nauki, 33 (1/2001)%"
X % X N. Malcolm, `Anselm's Ontological Arguments', The Philosophical Review, LXIX, 1 (Jan.,
1960)%"
X o X G. I. Mavrodes, `Properties, Predicates and the Ontological Argument', The Journal of
X Z Philosophy, LXIII, 19 (Oct., 1966)%"
X G. E. Moore, `Is Existence a Predicate?', [in: W. Kneale, G. E. Moore, `Symposium: Is
X Existence a Predicate?'], Aristotelian Society: Supplementary Volume, XV (1936)%"
X ! X M. K. Munitz, Contemporary Analytic Philosophy, Macmillan, New York 1981%"
X f" X """, Existence and Logic, New York Univ. Press, New York 1974%"
X # X A. Plantinga, `Kant's Objection to the Ontological Argument', The Journal of Philosophy,
LXIII, 19 (Oct., 1966)%"
X & X W. V. O. Quine, `Existence and Quantification', [in:] idem, Ontological Relativity and Other
X & Essays, Columbia Univ. Press, New York 1969%"
X ]( X B. Russell, `On Denoting', Mind, XIV, 56 (Oct., 1905)%"
X ) X J. Shaffer, `Existence, Predication and the Ontological Argument', Mind, LXXI, 283 (Jul.,
1962)%" *Q 2 .,, Ԍ X X F. Sommers, `Existence and Predication', [in:] M. K. Munitz (ed.), Logic and Ontology, New
York Univ. Press, New York 1973%"
X J X P. TichU, `Existence and God', The Journal of Philosophy, LXXVI, 8 (Aug., 1979)%"
X X J. D. B. Walker, A Study of Frege, Cornell Univ. Press, Ithaca 1965%"
X X R. D. Wells, `Frege's Ontology', The Review of Metaphysics, IV (1951) [reprinted in Klemke
(1968)]%"
X V X C. J. F. Williams, What is Existence?, Clarendon, Oxford 1981%"
b S > v Piotr Labenz
3 L Institute of Logic, Language and Computation
H University of Amsterdam
P
L K R 2 .,, N %" 5 a 2 ? [m W B C a " Why Prisoners' Dilemma is Not a Newcomb Problem
by P. A. Woodwarde e T `!%"] ă
y d d d y ? =
a a Y Z Z P WU'9H @@@@@@
d d x S Y A-- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 8184
! d
Why Prisoners' Dilemma is Not a Newcomb Problem
$ ) Copyright by P. A. Woodward and SORITES -- , a 5 a v
s & Why Prisoners' Dilemma is Not a
s m . NewcombProblem
s 0
(It's Not Even Two Newcomb Problems Side by Side.)
X . P. A. Woodward
+
, a In a brief discussion David Lewis argues that [c]onsidered as puzzles about
rationality, or disagreements between two concepts thereof8
Newcomb's Problem is 8one
X and the same problem8
(Lewis, p. 235) as that posed by the Prisoners' Dilemma. n q F ԍPage references to Lewis are to David Lewis, Prisoners' Dilemma Is a Newcomb Problem,
Philosophy and
q F Public Affairs, 8, No. 3 (1979): 235240.
X l Prisoners' Dilemma,
he claims, is a Newcomb Problem " or rather, two Newcomb
Problems side by side, one per prisoner
(Lewis, p. 235). Lewis concludes his discussion with
the following paragraph:
t M X Some have fended off the lessons of Newcomb's Problem by saying: Let us not have, or let us
not rely on, any intuitions about what is rational in goofball cases so unlike the decision problems
of real life.
But Prisoners' Dilemmas are deplorably common in real life. They are the most down X toearth versions of Newcomb's Problem now available (Lewis, p. 240).
This paragraph makes it clear that Lewis thinks that we can learn what is rational in
common
Prisoners' Dilemma decision situations by studying Newcomb's Problem.
To show that the Prisoners' Dilemma (PD) is a (pair of) Newcomb's Problems (NPs)
Lewis lists the three elements which, in a nutshell
capture the decision problem
faced
by each prisoner in the PD (Lewis, p. 236). Two of the three elements are, he claims, the
same for the decision problem faced by the agent in the NP, only one of the elements differs,
and he attempts to show that it differs only in its inessential trappings
(Lewis, p. 235). Only
slight modifications to the element not in common with regard to those inessential trappings
are required to show that the two problems are the same problem. It is assumed, in Lewis's
discussion and in most discussions of PD and NP, that each prisoner in PD and the player in
the NP is an egoist. Thus, what counts as rational
is what maximizes the prisoner's or the
player's own self interest.
Let's consider Lewis's modifications to the decision element not shared by the puzzles.
The three decision elements in the PD are as follows:
X ,!! (1)M M I [one of the prisoners] am offered a thousand [dollars] " take it or leave it.
[Taking it amounts to confessing in usual presentations of PDs.]%"!
X ,!! (2)M M Perhaps also I will be given a million [dollars]; but whether I will or not is
causally independent of what I do now. Nothing I can do now will have any effect )S = ...ee on whether or not I get my million. [In typical PDs, a reduced sentence is
analogous to getting a million dollars.]%"!
X ,!! (3)M M I will get my million if and only if you [the other prisoner] do not take your
thousand [dollars] (Lewis, p. 236).%"!
Lewis thinks that (1) and (2) are shared by the agent in a Newcomb Problem, but that
(3) is replaced by
X ,!! (3') I [the agent] will get my million [dollars] if and only if it is predicted that I do
not take my thousand [dollars] (Lewis, p. 236).%"!
Lewis points out that it is inessential whether or not the prisoners (in PD) choose
simultaneously, or one after the other (assuming that no information regarding the choices is
shared until after the choices have been made), or whether the prediction (in NP) has been
made before, during, or after the one faced with the problem (the agent) decides to takes his
thousand dollars or forego it, again assuming that the agent's prediction plays no role in the
agent's decision (see Lewis, pp. 236237).
Lewis also points out that it is inessential to the NP that 8any prediction8should
actually take place. It is enough that some potentially predictive process should go on, and that
whether I get my million is somehow made to depend on the outcome of that process
(Lewis,
p. 237). Thus, he claims, NP is characterized by (1), (2), and
X (3'') I will get my million if and only if a certain potentially predictive process (which
may go on before, during, or after my choice) yields the outcome which could warrant
a prediction that I do not take my thousand (Lewis, p. 237).%"
But, claims Lewis, that potentially predictive process
could be a replica of the agent
(i.e., Lewis himself) thus (3'') is correctly replaced by
X (3''') I will get my million if and only if my replica does not take his thousand
(Lewis, p. 238).%"
The Predictor in NPs is usually taken to be nearly perfect at making this type of
prediction, thus a replica of the agent that matches the agent perfectly in respects relevant to
the decision 8will have more predictive power than a less perfect replica8
( Lewis, p. 238).
But, as Lewis points out, a nearly perfect predictor is not necessary to the problem. The
disagreement between conceptions of rationality that gives the problem its interest arises when
the reliability of the predictor, as estimated by the agent, is quite poor " indeed even when
the agent judges that the predictive process will do little better than chance
(Lewis, p. 238).
The replica
may be simply another person placed in a similar situation as the agent
facing the NP. Thus, (3''') can be replaced by
X ,!! (3)M M I will get my million if and only if you [i.e., the other person in a similar
situation] do not take your thousand (Lewis, p. 239).%"!
Thus, argues Lewis, [i]nessential trappings aside, Prisoners' dilemma is a version of
X 5' Newcomb's Problem, quod erat demonstrandum
(Lewis, p. 239).
Some who discuss Newcomb's Problem, claims Lewis, think it rational to decline the
thousand if the predictive process is reliable enough " and some who discuss Prisoners
Dilemma think it rational to not take the thousand they are offered, if the two partners are
enough alike. These are, according to Lewis, two statements of the same view, a view often Q+T .,, labeled expected utility.
In following this strategy the agent maximizes his expected utility.
Lewis claims that he thinks it rational to take the thousand in both problems because no matter
what the predictor does, and no matter what his partner does, he would be better off than if
he didn't take it " better off by a thousand dollars. This is the dominant strategy.
Following
a dominant strategy, if there is one, the agent performs that action which will maximize his
utility whatever the other agent does (or whichever possible situation turns out to be actual).
Since, thinks Lewis, in NPs the dominant strategy seems to be (more) clearly what is
rational for an egoist, and since the two problems are (according to Lewis) the same problem,
we ought to accept the dominant strategy as being more rational in the all too common PDs
we run into in real life; this is the claim that Lewis's last paragraph (quoted above p. 80)
amounts to.
But notice, in typical presentations of NPs the predictor is said to be very good, and
evidence of that success is drawn from the fact that he (or it) has been nearly flawless in the
past when he (it) has made similar predictions. As Nozick (in an early discussion of NPs)
describes the predictor
t M X you know that this being has often correctly predicted your choices in the past (and has never, so
far as you know, made an incorrect prediction about your choices), and furthermore you know that
this being has often correctly predicted the choices of other people, many of whom are similar to
X L you, in the particular situation to be described below [i.e., a familiar Newcomb Problem situation]. D %L i q F ԍRobert Nozick, Newcomb's Problem and Two Principles of Choice,
in Paradoxes of Rationality and
q F | Cooperation: Prisoner's Dilemma and Newcomb's Problem, ed. Richard Campbell and Lanning Sowden (Vancouver:
The University of British Columbia Press, 1985), p. 107.D
Given Nozick's telling of the story, which is consistent with Lewis's version, the NP
is really the last of a series of NPs. The PD, as usually described, and as Lewis describes it,
is taken to be a single playing of the game. The partners in the PD are usually given a
onetime opportunity to make a decision, the results of which will involve more or less jail
time. Described as the last of a series of NPs the game has a unique character in which the
dominant strategy has a certain strong appeal; and that appeal may transfer over to a single
play of the PD.
Lewis has noted that PDs are deplorably common in real life
(Lewis, p. 304, quoted
above, p.80). Robert Axelrod has claimed that the
t M X Prisoner's Dilemma is simply an abstract formulation of some very common and very interesting
situations in which what is best for each person individually leads to mutual defection, whereas
X J everyone would have been better off with mutual cooperation. nJ i q F " ԍRobert Axelrod, The Evolution of Cooperation (New York: Basic Books, Inc., Publishers, 1984), p. 9. References
to Axelrod are to this volume.
Such situations include international trade situations in which two nations must decide
whether to erect trade barriers; situations in which legislators must decide whether to support
other legislator's bills; situations in which corporate executives must decide whether to
cooperate with other executives; and situations in which manufacturing companies (with but
one competitor) must decide on a price for their products (see Axelrod, Part I). Gregory Kavka
has noticed that situations in which two nations must decide whether to maintain an expensive 6#U .,, X and dangerous arsenal of nuclear weapons, or to disarm, can be described as a PD. l i q F y ԍGregory Kavka, Space War Ethics,
Ethics 95 (1985): 67391.l Virtually
any situation in which an individual must decide how much to exploit, for personal gain, a
commonly held resource (such as pasture land, the air, rivers, etc.) can be described as a PD.
What is common to these real life
PD situations is that they are not single play
games. They are situations in which the PD is faced numerous or an indefinite number of
X times by each player, often with the same partner.
They are iterated PDs.
1 G i q F ԍIt should be noted that national disarmament of nuclear weapons is a process, when it disarms, a nation builds
down
its nuclear arsenal. Such a process can be reversed, thus a nation disarming in not similar to, e.g., a
cornered outlaw throwing out his weapon
to the surrounding police. The latter is, perhaps, the last play of a game,
the former involves numerous plays, it is similar to an iterated PD. The importance of this point was suggested to
me in discussion with my cousin, economist Peter Woodward.1
It's clear that in an iterated NP, in which it is not known how many times the agent
will face the decision, if the predictor is a little better than chance at making the prediction,
the agent is better off foregoing the thousand each time he plays then he would be by taking
it each time. Moreover, it is clear that the better the predictor is the better off the agent is by
foregoing the thousand (on each play) compared with how well he would be by taking it (on
each play). The agent would do better by taking his thousand only if he could correctly predict
X when the predictor will be wrong. G i q F s ԍFor example, if the agent makes the decision 10 times, each time following Lewis's strategy of taking the
thousand and the predictor is 60% accurate, the player will receive $4 million, 10 thousand dollars. But if the player
foregoes his thousand each time and the predictor is 60% accurate, the player will receive $6 million; that is $1.99
million more. If, on the other hand, the predictor is 90% accurate, the player will receive $7.99 million more by
foregoing the thousand each time then he would by following Lewis's strategy and taking it.G Thus, following the dominant strategy, which was so
attractive to Lewis, is not in the player's best interest in iterated PDs. Hence, the common PDs
are not (one play) NPs ! not even two such NPs side by side. Further more, if the NP were
iterated
the situation would not be similar to the situation faced by someone playing iterated
PD because, although in PD both players have a rational egoistic rank ordering of possible
outcomes and can gain mutual benefit by cooperating with each other, the predictor in NP
does not have such a rank ordering of possible outcomes, and it's not clear what counts a
benefiting the predictor " he seems not to be an egoist. The best that we can do as far as
figuring out the predictor's preference ordering is to conclude that he prefers that the agent
get something rather than nothing, but he prefers that the agent not get everything. Whether
the agent gets $1 million or $1 thousand does not seem to matter to him, as long as the agent
does not get $1,001,000. We should not, therefore, be too quick to follow the dominant
strategy in common
iterated PD situations.
b # Q v P. A. Woodward
G F East Carolina University
L r V E .,, N %" 5 a ; [m W B a " A Paradox Concerning Science and Knowledge
by Margaret Cuonzoe e T `!%"] ă
y d d d y ; =
a a @
d d x S
d d x W @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
' Issue #17 " October 2006. Pp. 8594
# 7 A Paradox Concerning Science and Knowledge
# Copyright by Margaret Cuonzo and SORITES -- , a 5 a v
s A Paradox Concerning Science and Knowledge
X - K Margaret Cuonzo # V 2 P kC 4P# X01 Í Í. 4 X1* Í Í. 4 * q F * ԍThis article is the result of research done at an NEH Summer Seminar at Virginia Tech, 1999. Special thanks
are due to Deborah Mayo and the NEH Summer Seminar participants. I also thank Long Island University for
granting me released time to write this article. In addition, an early version of this article was presented at Long
Island University's Faculty Forum. I thank the faculty members present for their comments. X1* Í Í. 4 X01 Í Í. 4 # 4o \ P C +XP#
+
X , a @a/ Introduction ă
As a means of clarifying a particularly vexing philosophical issue, it is often helpful
to put that philosophical problem in the form of a paradox. This is so for Quine's and
Duhem's problem regarding the laying of blame
that occurs when an experimental result
conflicts with a scientific hypothesis. The Quine/Duhem problem is easily translated into a
standard version of a philosophical paradox, and the various approaches to the problem
correspond to unique solutions. Moreover, the paradox involving isolating the faulty
assumption(s) when there is a result that conflicts with a hypothesis is a specialized case of
a more general paradox, the skeptical paradox discussed by, among others, Rene Descartes in
X H his Meditations on First Philosophy. As I'll show below, both the famous skeptical paradox
of Descartes' Meditations and the Quine/Duhem paradox have a similar type of solution.
X a Watchmakers, Doctors, and Scientists: The Quine/Duhem Problem ă
Pierre Duhem and Willard Quine raise a problem for what is traditionally known as
the hypotheticodeductive model
of scientific hypothesis testing. On this traditional model,
a scientific hypothesis is tested by deducing an observable consequence of the hypothesis, and
then empirically observing whether this consequence actually is the case. That is,
X ,!! 1)!! H e%"!
X ,!! 2)!! Not e%"!
X ,!! 3)!! Therefore, Not H%"!
On this model of scientific testing, a logical consequence e is derived from the hypothesis H,
and then e is observed. In the event that e turns out to not be the case, then on this model, the
X " hypothesis is shown to be false. The relevant rule of inference here is modus tollens. Here's
a rough example of the kind of reasoning involved. Assume that I am a scientist and my
X # hypothesis H is that drinking coffee causes cancer. ^ %#l q F * ԍThis example of an experimental hypothesis is admittedly rough. A more precise hypothesis is that coffee
drinkers are more likely to develop cancer. I phrase the hypothesis this way for effect. If this is troubling to some
readers, they can substitute the cubeshaped earth hypothesis discussed later in the paper.^ A logical consequence of my hypothesis
is that, when a group of people with similar health histories and habits are divided into the $W } = ...ee coffee drinkers and noncoffee drinkers, there will be a significantly higher occurrence of
cancer in the coffee drinkers than the noncoffee drinkers. The e in this case is the claim that
there will be a significantly greater occurrence of cancer in the coffee drinkers than in the
noncoffee drinkers. Suppose then I do the experiment and discover no significant difference.
On the traditional hypotheticodeductive model of scientific testing, this evidence proves
conclusively that the hypothesis that coffee causes cancer is false. If, though, a significant
difference is shown, this does not conclusively prove the truth of the hypothesis. It only shows
that the hypothesis has passed one test. This is because, strictly speaking, there could be some
other factor that brings about the greater occurrence of cancer than the drinking of coffee.
Pierre Duhem made the following critique of this model of scientific testing:
t M
X The [scientist] can never submit an isolated hypothesis to the control of experiment, but only a
whole group of hypotheses. When experiment is in disagreement with his predictions, it teaches him
that one at least of the hypotheses that constitute this group is wrong and must be modified. But
X I experiment does not show him the one that must be changed (185).
Duhem's problem is with the first part of the model, (1). His claim is that no
hypothesis can be separated from an indefinite set of auxiliary hypotheses. In our coffee
example, an auxilary premise might be that the test was done on a group of people with
similar health histories, or that there was no mistake in the counting of the instances, etc.
Taking into consideration this indefinite set of auxiliary premises, we then have:
X ,!! 1)!! {H, (A1, A2, A3, 8 , An)} e%"!
X ,!! 2)!! Not e%"!
X ,!! 3)!! Not {H, (A1, A2, A3, 8 , An)}%"!
If Duhem is right, then all that the conflicting result shows is that one of the set of the
main and auxiliary hypotheses is mistaken. What the result does not show any longer is Not
H. In our example, the lack of a significant difference is no longer conclusive grounds for
rejecting the hypotheses that coffee causes cancer.
Duhem sometimes explained his problem by comparing the scientist to a doctor, and
X contrasting the scientist with a watchmaker T W q F J ԍSee, for example, the discussion in Mayo (1997).T . A watchmaker, when faced with a watch that
does not work, can look at each part of the watch in isolation, going from piece to piece, until
the defect is detected. The doctor, on the other hand, cannot examine each of the parts of an
ailing patient's body in isolation. Instead she must detect the seat of the illness only by
inspecting the effects produced on the whole body. Similarly, the scientist cannot separate out
each of an indefinite set of auxiliary premises to test each in isolation. From this, though, a
paradox arises, one concerning how it is reasonable to lay blame
on a main scientific
hypothesis or one of its auxiliaries.
X # 0a0 The Paradox ă
On its standard definition, a paradox is a deductive argument with seemingly true
premises, employing apparently correct reasoning, with an obviously false or contradictory
conclusion. Consider, for example, a version of the famous skeptical paradox:
X ,!! 1)!! I can know that I live in Brooklyn, only if I can know that I am not a brain in a
vat.%"! )X G .,, Ԍ X ,!! 2)!! I cannot know that I am not a brain in a vat.%"!
X ,!! 3)!! Therefore, I cannot know that I live in Brooklyn.%"!
This is an argument with seemingly true premises, employing apparently correct reasoning,
but with what looks like an obviously false conclusion. Premise one merely states that it is
a precondition for my knowing that I live in Brooklyn that I know a more fundamental truth,
namely that I am not a brain in a vat. If I were a brain in a vat, then I would not necessarily
be in Brooklyn. Premise two is the claim that I cannot know that I am not a brain in a vat.
No evidence I could find would count as completely convincing evidence for the hypothesis
that I am not a brain in a vat, because it is possible that the evil scientist who has been
keeping me in this vat has produced in me the experience of getting this evidence. Notice that
this is not the claim that my being a brain in a vat is likely, but rather a claim about the
remote possibility that this is so. In addition, the reasoning involved in the skeptical paradox
is straightforward. The argument has the form:
X ,!! 1)!! P, only if Q%"!
X ,!! 2)!! Not Q%"!
X ,!! 3)!! Therefore, Not P%"!
However, the conclusion is implausible. The paradox could be, and often is, phrased as
precluding all knowledge, even of the most obvious truths. Since the premises are seemingly
true, the reasoning is straightforward, and the conclusion seems obviously false, the skeptical
paradox meets each of the requirements for being a philosophical paradox.
X Now consider the following simplified form of an argument I will flesh out below:%"
X ,!! 1)!! No hypothesis can be tested in isolation from an indefinite set of auxiliary
hypotheses.%"!
X ,!! 2)!! In order to show that a hypothesis is mistaken, it is necessary to isolate that
hypothesis from its set of auxiliary hypotheses.%"!
X ,!! 3)!! Therefore, no hypothesis can be shown to be mistaken.%"!
I will call the above argument the simple Quine/Duhem paradox.
The first premise of the
argument is the claim that whenever a test of a hypothesis is made there is an indefinite set
of auxiliary hypotheses that must go along with the hypothesis. For another example, imagine
an experiment is designed to test the hypothesis that the earth is a cube by observing the
shadow it casts during an eclipse. The hypothesis is that the earth is a cube, and an entailment
of this is that the earth will leave a square or diamondshaped shadow. But the earth's shadow
is an entailment only if certain other preconditions are met. For example, the experimenter
assumes that: the light will not be such that it turns the shadows of cubes into circles; the
instruments used to identify the shadow are functioning properly; we're all not brains in vats;
etc. The first premise implies that the cube hypothesis, in order to be tested, must accompany
these and a potentially infinite set of other hypotheses.
X The second premise is a statement of the moral of the Quine/Duhem problem:%"
X ,!! 1)!! {H, (A1, A2,8,An)} > e%"!
X ,!! 2)!! Not e%"!
X ,!! 3)!! ~{H, (A1, A2,8,An)}%"! R+Y .,, ԌThe above argument is not a paradox in itself, but rather an illustration of the kind of
deductive reasoning available for laying blame
on the hypothesis. As the argument states,
the negative experimental result (2) only shows that there is something wrong with the set of
hypotheses (H, A1, A2,8,An) and not necessarily with H itself. It is this type of argument that
licenses premise two, the statement that there must be some way of isolating H if one is going
to be able to show it mistaken. Premise two implies that for the cube hypothesis to be shown
mistaken, the hypothesis must be separated from its auxiliary premises. Assume that the cubeshaped earth experiment is performed and the shadow is circular. In this case, the circular
shadow only shows that one of the set of hypotheses that includes H and an indefinite set of
auxiliary hypotheses is mistaken. What it does not show is that H is mistaken. The conclusion
of the simple form of the paradox is that no hypothesis can be shown to be mistaken. In the
case of the cube hypothesis, this hypothesis cannot be shown to be mistaken either.
X Using the cube hypothesis, we have the following version of the paradox:%"
X ,!! 1)!! The hypothesis that the earth is a cube cannot be tested without being conjoined
to an indefinitely large set of auxiliary hypotheses.%"!
X ,!! 2)!! If the hypothesis that the earth is a cube cannot be tested without being conjoined
to an indefinitely large set of auxiliary hypotheses, then the hypothesis that the
earth is a cube cannot be shown to be mistaken.G %"!
X ,!! 3)!! The hypothesis that the earth is a cube cannot be shown to be mistaken.%"!
The conclusion is intuitively implausible given the obvious falsity of the hypothesis.
Moreover, in the simple form of the paradox, the conclusion is that no hypothesis can be
shown to be mistaken. This is even more implausible.
The last thing to consider in respect to my showing that this is a genuine paradox is
the reasoning employed. In both the cube version and the simple form of the paradox the
X reasoning is straightforward. In the cube version we have a standard use of modus ponens. In
the simple version, we have an argument of the form: No S is P; All Q are P; Therefore, No
S is Q. Nothing is out of line here. Thus, the Quine/Duhem Paradox meets all the
requirements of a standard philosophical paradox: it has apparently true premises; its
conclusion is highly implausible; and the reasoning involved is straightforward.
The Quine/Duhem problem, phrased as a paradox, is really a special case of the
skeptical paradox discussed earlier. The skeptical paradox claims that the incredibly strong
precondition for my knowing p cannot be attained, and hence I cannot know p. For the Q/D
Paradox, the set of auxiliary premises has the same function as the precondition in the
skeptical paradox: because they cannot be ruled out, they keep us from knowing the status of
the hypothesis. The conclusion of the skeptical paradox is that we cannot know some obvious
hypothesis (we can't prove H), while for the Q/D paradox, the conclusion is that no hypothesis
can be shown to be mistaken (we can't prove Not H).
The many solutions to the Quine/Duhem problem can be seen as solutions to the
Quine/Duhem paradox. For brevity's sake, I am going to restrict my discussion to the most
famous and/or the most plausible solutions. One famous response to the Quine/Duhem
problem is given by the popular Kuhnian approach. On this account the moral of the problem
raised by Quine and Duhem is that it is only whole theories, systems, or what Kuhn terms
paradigms
that are shown to be mistaken. The Kuhnian account, in other words claims that
the conclusion of the simple Quine/Duhem paradox is true, and that no hypothesis can be *Z .,, conclusively shown to be the one at fault. Only whole theories broadly construed, or
paradigms, are open for rejection. And a major factor in the choice of one paradigm over
another is the opinion of the scientific community itself. In The Structure of Scientific
Revolutions, Kuhn maintains that:
t M X As in political revolution, so in paradigm choice " there is no standard higher than the assent of
the relevant community. To discover how scientific revolutions are effected, we shall therefore have
to examine not only the impact of nature and logic, but also the techniques of persuasive
X ] argumentation within the quite special groups that constitute the community of scientists (1962:
94).
Only whole paradigms get accepted or rejected based upon the assent of the scientific
community. We mistakenly think that the conclusion No hypothesis can be shown to be
mistaken
is false, when in fact it is true. Only whole theories can be rejected in the way we
normally think that hypotheses are rejected.
There are reasons for avoiding this kind of approach. The first is that the response is
too drastic. Consider the cubeshaped earth hypothesis again. Does it really make sense to say
that such an obviously mistaken hypothesis cannot be shown false? Commonsense and the
obvious falsity of the hypothesis suggest that a solution to the paradox should be looked for
elsewhere. Another reason for looking elsewhere is that the successful account of this problem
is going to give some kind of analysis of the concepts involved. The whole systems
theorist, by taking the stance that it is only whole theories that are capable of being rejected,
does not address the issue of what happens to the particular hypothesis, except perhaps by
saying that as a result of the paradigm shift, it would be accepted or reject. So the approach
switches the issue from the evaluation of hypotheses to the evaluation of theories. More
generally, a criticism often leveled against the Kuhnian approach is that it reduces theorychoice to a kind of mob psychology,
where what counts as successful is determined by
sociological and not logical or evidential factors.
X
Ka0 Bayesianism ă
Another place to look for an account of evidential appraisal is statistics. In fact, a
group of philosophers of statistics claim to have a solution to the Quine/Duhem problem.
These philosophers, known as Bayesians,
get their name from the statistician Thomas
Bayes. According to the Bayesians, an answer to the Quine/Duhem problem can be given if
the hypotheticodeductive model of scientific testing is replaced with another model. On the
Bayesian model, evidence e confirms a hypothesis H to the extent that a scientist's degree of
belief in H is higher given evidence e than what it was or would be without this evidence.
Probability is a measure of the subjective degree of belief ranging from 0,
complete
disbelief, to 1,
complete certainty. The scientist's degree of belief in the hypothesis without
the evidence is called the prior probability
of the hypothesis, and the scientist's degree of
belief in the hypothesis after the evidence is called the posterior probability.
So if the
posterior probability of H is greater than the prior probability of H, the extent to which H is
confirmed is the difference between the posterior and prior probabilities. To figure out the
posterior probability of H, Bayesians use a version of Bayes' Theorem:
!! P (H | e) =
& UU- P (e| H) P (H)
!! M M
!! M M P (e | H) P (H) + P (e | not H) P (not H) [*[ .,, Ԍ This is read, The probability of H, given e, is equal to the probability of e given H
times the probability of H, over the probability of e given H, times the probability of H, plus
the probability of e given not H times the probability of not H.
Certain factors need to be
known before the laying of blame
can take place: (i) the prior probabilities in H and notH;
(ii) the likelihood, which is P(e | H); and (iii) what's called the Bayesian catchall factor,
which is P (e | notH). Once you have these, then you just plug them into Bayes' theorem to
get the posterior probability.
Here's an example: Consider a situation in which an experiment is done and the result
seems to contradict the hypothesis in question. One response to this negative result would be
to reject the hypothesis. Think back to the coffee example. Another response would be to look
at the auxiliary premises. Suppose for simplicity's sake that there is only one auxiliary
premise, that the subjects in the study have similar health histories. In this case, the main
hypothesis H is that coffee causes cancer, and the auxiliary hypothesis A is that a test of the
hypothesis will involve people who have similar health histories. In this simplified example,
hypothesis H and auxiliary A entail e, a significant difference in cancer occurrence, but note
is observed. The Bayesian account shows when A is more likely to be blamed than H, or vice
versa. Assume that there is a great deal of evidence for H, that there is hardly more evidence
for the truth of A than there is evidence for A's falsity. In our example, perhaps the health
histories of the subjects weren't checked very thoroughly so it is possible that the coffee
drinkers also smoke, while the nondrinkers don't. This is a situation in which it is more likely
that A is the where the problem lies, rather than H. Bayesians solve this type of problem by
plugging in values into Bayes' theorem. (i) First, they assign a lower prior probability to A
than to H. For example, A, being only slightly more probable than NotA, it would have a
prior probability of around .6. H, on the other hand, would have a very high probability, say
.9. (ii) With regards to the likelihood, the Bayesians assign a far greater likelihood for the
negative result to happen when notA is true rather than notH. For example, P(note | a) =
x, and P (note | nota and H) =50x, and P(note | nota and notH) = 50x. I'll spare you the
X calculations n W q F ԍThe appendix of this article has a stepbystep explanation of these calculations. These are also worked out in
Mayo (1997) and Dorling (1979). . The result is that after plugging these numbers into Bayes' theorem, the
probability of H is only slightly decreased, going from .9 to .897, whereas the probability of
A plummets from .6 to .003. So, given their prior probabilities, and the likelihood of getting
the note result when either A or H is not true, it follows that the scientist has good reason
to reject A, while still preserving H.
In a nutshell, the Bayesians, by analyzing the subjective degrees of belief of the
scientists, and plugging these probabilities into Bayes' theorem, attempt to give an account
of when the rejection of the main hypothesis is warranted, and when, instead an auxiliary
premise is what must be rejected. The Bayesians solve the Quine/Duhem paradox by rejecting
the second premise, the one which claims that in order to show that a hypothesis is mistaken,
it is necessary to isolate that hypothesis from its set of auxiliary hypotheses. As long as we
know the prior beliefs in the auxiliary premises then we can determine whether they should
be rejected.
Although this is the standard view of scientific reasoning about Duhemian problems
in the philosophy of statistics, the Bayesian account has certain problems. The most important
objection concerns the Bayesians' reliance on the prior degree of belief of the scientist in his
or her hypothesis before the hypothesis is tested. Assuming that scientists have such degrees (\ .,, of belief, and also assuming that these beliefs can be quantified into degrees, it is undesirable
that the prior beliefs be taken as central to reasoning in science. Such subjective beliefs are
highly variable, changing not only from person to person, but in addition, in the same person
from moment to moment. For example, if a scientist's belief varies even slightly during a day,
the justification for the acceptance or rejection of a hypothesis will be altered. Something
seems not right with this subjectivist account. The problem, I suspect, lies in the conflation
of a scientist's confidence in his or hypothesis with the evidence that it is true. As Deborah
Mayo asks in the title of an article critiquing the Bayesian approach, What's Belief Got to
Do with It?
Mayo claims that:
t M X 8scientists do not succeed in justifying a claim that an anomaly is due not to H but to an auxiliary
hypothesis by describing the degrees of belief that would allow them to do this. On the contrary,
scientists are one in blocking an attempted explanation of an anomaly until and unless it is provided
with positive evidence in its own right. And what they would need to show is that this evidence
X succeeds in circumventing the many ways of erroneously attributing blame (1997: 228229)
Here Mayo is critiquing the white glove
treatment given by the Bayesians as to how
to solve such problems. To determine when an anomolous result requires that the main
hypothesis or some auxiliary hypothesis to be rejected requires more than the subjective
degrees of belief of the scientist prior to the experiment. What is required is evidence that the
auxiliary is the faulty assumption, and an account of why this evidence isn't mistakenly taken
as evidence that the auxiliary is to blame.
X R
a$ Another Approach: Error Statistics ă
So far we have discussed two approaches to the Quine/Duhem problem, the Kuhnian
and the Bayesian account. Both approaches have serious problems. Like the Bayesian account,
the solution I'll argue for rejects the hypotheticodeductive model of explanation and instead
gives a probabilistic account of scientific reasoning. This approach takes as its starting point
something Deborah Mayo in Error and the Growth of Experimental Knowledge calls, error
statistics,
which includes everyday concepts in statistics like significance tests and confidence
intervals.
On the error statistical account of hypothesis testing:
X Data e produced by procedure ET provides good evidence for hypothesis H to the
extent that test ET severely passes H with e.%"
And:
X H's passing test ET (with result e) is a severe test of H just to the extent that there is
a very low probability that test procedure ET would yield such a passing result, if
hypothesis H is false.%"
Instead of focusing on the probability of the hypothesis itself, the error statistician focuses on
the reliability of the test of the hypothesis, particularly the probability that the test would pass
H if H were false. If there is a low probability that a test would pass a hypothesis when that
hypothesis is false, then passing that test is a good indication of the truth of the hypothesis.
To criticize attempts to explain away anomolous results, the error statistician employs
blocker strategies.
These strategies criticize such attempts on the grounds that (a) such
explanations fail to pass severe tests; or, worse, (b) their denials pass severe tests. In addition,
an anomolous test result may be legitimately blamed on an auxiliary hypothesis A by showing
that Not A passes a severe test. +] .,, Ԍ Consider, again, the example of the hypothesis about coffee and cancer, and the
anomolous result that there was no significant difference between the coffee drinkers and noncoffee drinkers. In order to successfully blame the anomolous result on the auxiliary
hypothesis that the subjects in the experiment have the same health histories and habits, the
scientist would have to severely test the denial of the auxiliary premise. In this case, a severe
test must pass the hypothesis that the two groups don't have similar health histories or habits.
If there were a strong probability that the two groups were radically different in health habits,
then the auxiliary premise should be blamed and not the hypothesis.
X a" Getting Back to the Quine/Duhem Paradox ă
Mayo seems to claim that the error statistical approach solves the Quine/Duhem
problem. This claim, I fear, is not completely correct. The moral of the Quine/ Duhem
problem is that there is no complete guarantee of when the main hypothesis is to be rejected
or when an auxiliary is at fault. And even on Mayo's model of hypothesis testing there is no
such guarantee. As is the case with all statistical accounts, there remains the (admittedly
remote) possibility that a false hypothesis passes a severe test. This is admittedly not likely,
but all that is needed to run into trouble with the Quine/Duhem problem is a nonzero
probability that a hypothesis could be passed by a severe test. This should not be a surprise,
given that the Quine/Duhem Paradox is a special case of the skeptical paradox.
Although it was not intended this way, the error statistical account can be turned into
a more restricted solution to the paradox. On this solution, the paradox arises because of a
faulty conception about evidence. The assumed feature of evidence which leads to paradox
is its conclusiveness, that is, that the evidence provides complete proof of the falsity of an
empirical hypothesis. No conclusive grounds can be given to reject any empirical hypothesis.
What can be given, though, is overwhelming good reason to do so. In fact, in the case of
some evidence, it would be irrational not to accept this evidence as cause for the rejection of
an hypothesis. Notice that this is a restricted solution because the claim is that there can be
no new notion of evidence appraisal that provides certain grounds for rejecting an empirical
hypothesis. The claim is that an alternative notion of evidential appraisal should be given, one
that gives up the idea of complete proof and instead provides indefinitely high probabilities
about the truth or falsity of a claim.
Here is a summation of my response to the Quine/Duhem paradox. Unlike the Bayesians and
(I believe) error statisticians, I claim that there can be no thoroughgoing solution to the
paradox, because to give one would involve showing, with Cartesian certainty, where to lay
blame when an experimental result conflicts with a scientific hypothesis. This, I submit,
cannot be done. What can be given is an alternative version of what counts as good evidence.
And it is here where the error statistical approach is most useful. Moreover, such an account
is all that is really needed as an account of where it is best to lay blame.
X #
a# Application to the Skeptical Paradox ă
Earlier, I claimed that the Quine/Duhem Paradox is a special case of the more famous
skeptical paradox. If this is so then the solution to one paradox will provide a solution to the
other. Applying the solution to the skeptical paradox involves giving up the Cartesian ideal
of certainty, at least with respect to empirical truths. Knowledge of the truth of the proposition
I live in Brooklyn,
if that is to mean certain knowledge, cannot be attained. What can be
attained is overwhelming proof to that effect. A more informal application of error statistics )^ .,, is relevant here. On this account, we can derive overwhelming evidence of the truth or falsity
of our beliefs by submitting these beliefs to severe tests as well.
For example, consider my belief that the blue patch on my hand yesterday was caused by
touching a rail in the subway that had wet paint on it. In this case, there is a fairly reliable
test of whether this in fact is so. I can go to the subway station and check to see if the rail
is freshly painted, and if the color matches the color on my hands. The test is reliable because
there is a low probability of there being fresh paint of exactly the color on my hands, while
I acquired the paint stain elsewhere. Yes, it is in principle possible that I got the stain another
way, perhaps there's a mailbox on my corner that's just been painted as well. But the
probability of this is quite small. Notice the claim is not that there is conclusive proof of
where I received the stain, but rather overwhelming proof. And this, generally, is all an
ordinary notion of proof that is needed. So, like the solution to the Quine/Duhem paradox, the
solution to the skeptical paradox is a restricted one. There is no successful account of
knowledge that assumes that we can have certain knowledge of empirical truths. However,
there can be overwhelming evidence of such truths. And this is all that is needed of an
account of empirical knowledge.
X | a0 Conclusion ă
In sum, the Quine/Duhem Paradox, and by extension, the skeptical paradox have a
restricted solution, one the gives up the Cartesian ideal of complete certainty and conclusive
proof. Our ordinary notions of proof and knowledge are problematic. However, alternative
notions to evidential appraisal and knowledge can be given. These more restricted notions do
all that is needed of our ordinary notions.
X Xa Appendix: Explanation of Calculations for the Bayesian Solution ă
It is given that A and H entail e, but note is observed. Also the probability of
observing note, while A and H are true is 0. That is, P(note | A and H) = 0
We assigned H a very high probability, P (H) = .9, whereas we have assigned A a
prior probability which makes it only slightly more likely than not. P (A) = .6. We also
assumed that H and A are statistically independent. That is, the probability of H does not
change the probability of A, or vice versa.
With regards to the assumed likelihoods, the probability of note being observed, given
that A is true and notH is assumed to be a very small number, x (for example, .001). That
is, P(note | A and notH) = x. On the other hand, we assumed the likelihood of note being
observed given notA being the case is assumed to be 50 times more likely, 50x. So: P(note
| notA and notH) = 50x and P (note | notA and H) = 50x.
X We then plug these numbers into a simplified form of Bayes' Theorem:%"
X P (H | note) = P (note | H) P (H)%"
X P (note)%"
P (note) = P (note | H) P (H) + P (note | notH) P (notH)
P (note | H)M M = P (note | A and H) P (A) + P (note | notA and H) P (notA)
X ,!! !! M M = 0 + 50x (.4)%"!
X ,!! !! M M = 20.6x%"! *_ .,, Ԍ X P(note) = 20x (.9) + 2.06x = 20.06x%"
X For the posterior probability of H,%"
X P (H | note) = 20x (.9)%"
X !! ______ = .897
& %"
!! M M 20.06x
&
For the posterior probability of A,
X P(A | note) = P(note | A) P (A)%"
X P (note)%"
X P (note | A) = P (note | A and H) P (H) + P (note | A and notH) P(notH)%"
X = 0 + x (.1) = .1x%"
P (A | note) = .06x
X ,!! ______ = .003%"!
X ,!! 20.06x%"!
Whereas the probability of H is hardly changed, the probability of A plummets.
X 2 5a0 Works Cited ă
X X Descartes, Ren). (trans. 1980). Meditations on First Philosophy, trans. Cress. Indianapolis:
Hackett.%"
X Dorling, John. (1979). Bayesian Personalism, the Methodology of Research Programmes, and
X Duhem's Problem.
Studies in the History and Philosophy of Science. vol 10; pp. 177187.%"
X X Duhem, Pierre. (trans. 1954). Essays in the History and Philosophy of Science, trans. Ariew
and Barker. Indianapolis: Hackett.%"
X X X Kuhn, Thomas. (1962). The Structure of Scientific Revolutions, first ed. Chicago: University
of Chicago Press.%"
X Mayo, Deborah. (1997). Duhem's Problem, the Bayesian Way, and Error Statistics: What's
X Belief Got to Do with It?
Philosophy of Science, vol. 64; pp. 222244.%"
X X Mayo, Deborah. (1996). Error and the Growth of Experimental Knowledge. Chicago:
University of Chicago +Press.%"
b 6# P y v Margaret Cuonzo
G Department of Philosophy
8 Long Island University, Brooklyn Campus
F ' &` .,, N %" 5 a V , 5 W B a " Between Platonism and Pragmatism: An alternative reading of Plato's Theaetetus
by Paul F. Johnson`!%"] ă
y d d d y V =
a a @
d d x W
! d d x a @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
& e Issue #17 " October 2006. Pp. 95103
Between Platonism and Pragmatism: An alternative reading
v M . of Plato's Theaetetus
# F Copyright by Paul F. Johnson and SORITES -- , a 5 a v
s # Between Platonism and Pragmatism: An
s 0 ! Alternative Reading of Plato's Theaetetus
X
- Paul F. Johnson
+
t M
l I was in a camp near Bayeux after the Normandy landing. A letter from Wittgenstein telling me he was reading
v M s U Plato's Theaetetus: Plato in this dialogue is occupied with the same problems that I am writing about.
O o " M. O'C. Drury,
; a Whether it is entirely fair to appropriate the work of the late Wittgenstein for the
pragmatic tradition, as contemporary writers such as Richard Rorty and Robert Brandom have
tried to do, is an interesting and important question. Whether a place can be found in the
pragmatic tradition for Plato is a question less frequently raised and may seem bizarre given
the the historical dislocalities it involves. But if Wittgenstein could claim to have been
working on the same problems as Plato, and if Wittgenstein can be claimed to have been
working in the pragmatic tradition, perhaps it is worth the effort of trying to describe the
X j general contours of the problems in the Theaetetus in a way that might have piqued
Wittgenstein's interest, and to articulate by this means a philosophical theme which could be
seen to stretch from one end of our tradition of discourse to the other. This is what I shall
attempt in this paper.
X There are two claims about the Theaetetus which I shall need to assume as generally
accepted among scholars today which provide the basis for my argument, both of which are
neatly expressed by Robin A.H. Waterfield in the interpretive essay which accompanies his
translation of the text. No scholar would today deny,
writes Waterfield in 1987, that
X , Theaetetus is one of Plato's later compositions 8 and indeed belongs to a period when [he]
was having doubts about some of his earlier ideas and assumptions.
(1987, p. 132) Despite
X ! the consensus that it is a later work, the Theaetetus shares with virtually all of the early,
Socratic dialogues a sort of ceremonial throwing up of the hands at the end with the dull
acknowledgement that no positive result has been achieved. Here is the closing exchange
between Socrates and Theaetetus (210a8 b12):
X SOCRATES: And nothing could be sillier than for us, who are engaged in an
inquiry into knowledge, to say that it is correct belief accompanied by
knowledge (of uniqueness or whatever). Therefore, Theaetetus, knowledge can
be neither perception, nor true belief, nor true belief with the addition of a
rational account.
X THEAETETUS: Apparently not.
X SOCRATES: Well, are we still pregnant? Is anything relevant to knowledge
still causing us pain, my friend, or have we given birth to anything? m,a = ...ee Ԍ X X THEAETETUS: I most certainly have: thanks to you, I've put into words more
than I had in me.
X SOCRATES: And does our midwifery declare that everything we produced was
still-born and that there was nothing worth keeping?
X THEAETETUS: Absolutely.
The interlocutors claim to have reached no other result than to realize how little they
really know about the topic they were talking about, and it seems odd that Plato, in a late
dialogue, would revert back to the aporetic mood of the earlier ones. But it is highly
significant, in connection with the second of Waterfield's claims about Plato having second
X thoughts about the Forms, that there is no mention anywhere in Theaetetus of the Forms, the
quintessential platonic doctrine which provides the center of gravity for so much of what is
taken to be Plato's settled position on epistemological and metaphysical issues generally. Now,
a facile, but not implausible reading of the dialogue would urge us to put these two features
of the dialogue down next to each other and draw the conclusion which is then only a short
X $ step away: it is precisely because there is no mention of the Forms that Socrates and his
friends must reach their inconclusive, aporetic result, and it is precisely to reinforce the
centrality of the Forms in his theory of knowledge that Plato leaves Socrates and Theaetetus
in the lurch once more.
I think we can do better than this. Let us accept the authority of Waterfield and
X ) suppose for the sake of argument that the Theaetetus is indeed a later composition and the
lack of any mention of the Forms is indicative of Plato's own misgivings about the whole
Theory of Forms. On these terms and conditions, I think it is possible to reach a much less
aporetic result and a far more constructive set of claims concerning Plato's thoughts on the
nature of knowledge. I propose a reading of the dialogue in which Plato can be seen to be
developing an alternative line of approach to the question of knowledge. I shall argue that
various features of this approach bear a striking resemblance to a contemporary school of
thought loosely affiliated under the term pragmatism
and associated with the work of such
prominent contemporary philosophers as Robert Brandom, Richard Rorty and JGrgen
Habermas. I also find a strong thematic resonance between the results obtained from this
X E alternative reading of the Theaetetus and prominent features in the work of Wittgenstein. I'll
build my case for a pragmatic reading of the dialogue by showing the affinity with some
claims of Wittgenstein as interpreted by Brandom. These lines of affinity could be extended
to include features of the work of Rorty and Habermas but that is work for another day.
The first two thirds of the dialogue is given over to a discussion of Theaetetus's first
attempted definition of knowledge, that knowledge is perception. Socrates praises Theaetetus
for his boldness in asserting his best opinion, but he is worried that this thesis, simple though
it is to state, is really an expression of a whole vast theoretical position that it will take some
trouble to articulate. In order to contextualize this thesis in its broader theoretical setting,
Socrates enters into a long discussion of what he takes to be the metaphysical background
against which the claim can be made proper sense of. Socrates achieves this broader
theoretical perspective by combining the doctrine of Heraclitus " that everything is flux " with
the doctrine of Protagoras " that man is the measure of all things. This conflation of doctrines
is famously controversial, and it is a fair and important question to ask whether the argument
which ultimately defeats the identity thesis
(that knowledge and perception are the same
thing) would go through if we were more careful than Socrates is to separate the distinctively
Heraclitean from the Protagorean claims. The reason for his running the two together is M+b .,, perhaps not hard to specify. Protagoras offers a theory about the nature of human
understanding and knowledge, and it is essentially Protagoras' theory of knowledge that
Theaetetus introduces; in order for that theory to be fully intelligible, we should also need
some account of the sort of world we are living in so that we can say what it is we have
X knowledge of, and how that world calls forth or instigates the peculiar form of knowledge we
humans are subject to. Heraclitus provides this second theoretical desideratum. The two parts
of the account work beautifully together to provide something like a totalizing theory about
what knowledge is (perception) in the context of a world of flux (that which is to be known).
Rather than concentrate on the argument which Socrates pursues to show that the
identity thesis is mistaken, I would like to highlight what may appear to be a minor complaint
voiced by Socrates against the more comprehensive theory that we get when we run Heraclitus
and Protagoras together as Socrates does. It would be a minor complaint except that Socrates
returns to it several times in the discussion, and frequently enough to suggest that Plato
intends us to take it more seriously than the sometimes flippant, sometimes ironic tone would
otherwise warrant. I am talking about the frequent mention of the language we use in common
life, the language we use in casual conversation or in the marketplace; and the language we
must unavoidably invoke in order to get a philosophical discussion of knowledge off the
ground in the first place. Socrates points out over and again, that if Protagoras is right in his
ideas about knowledge, then ordinary speech is grossly inadequate to the task of expressing
or capturing knowledge. After explaining how Protagoras would account for our perception
of things which are large or white or warm, Socrates says (154b) that everyday speech, my
friend, carelessly uses words which, from the Protagorean viewpoint and others which
approximate to it, are extraordinarily absurd.
A little later he claims that if Heraclitus is right
then the verb `to be' should be deleted from all contexts, despite the fact that habit and
ignorance often force us to employ it, and did so even in our recent discussion.
(157b)
Socrates puts into the very mouth of Protagoras, in the famous speech he presents on the great
sophist's behalf, the claim that the habitual use of words and expressions 8 are the means
by which most people confuse one another in all sorts of ways, because they can be
manipulated at will.
(168b) And reverting back to the Heracliteans he says (183b) that those
who hold this theory need to set up another language since at present they don't have
expressions which fit the theory, except `not like this either'.
Leaving aside the question of how appropriate it is to combine the Heraclitean
metaphysics with the Protagorean epistemology, or whether the successful attack upon the
latter would be effective if the former elements were excised from the theory, what seems
clear is that either theory separately, or both of them combined produce a result which
conflicts with our ordinary way of talking. We have nouns which help us to identify perduring
objects in space; predicate terms which enable us to impute essential and accidental properties
to such objects. And we get on pretty well with the world and with one another by invoking
the ordinary words we inherited and learned how to use at our mother's knee, quite bereft of
the benefits of abstruse philosophical theory. What the discussion of Heraclitus and Protagoras
puts before us is the possibility that the language and parlance of common life is wholly
mistaken in the representations of the world which it makes available to us. The combined
theory, if correct, would require a wholesale revision of ordinary language, a sort of linguistic
legislative omnibus bill that would purge, innovate, twist and leverage our vocabularies into
a better fit with the world as it is disclosed to us through the more accurate lens of
X ) philosophical theory. On my alternative reading of the Theaetetus in which this seemingly
minor complaint is brought into the foreground, we may interpret Plato as raising the question *c .,, whether any such global revision of ordinary language is even conceivable, and, by extension,
as testing the possibility that ordinary language is basically alright just as it is, and that it
provides the unavoidable starting point for any inquiry into the nature of knowledge. There
is, at any rate, something seriously wrong with any theory of knowledge that would require
us to abjure the language we toss around between ourselves in our comings and goings in the
affairs of daily life. Surely we make mistakes in our quotidian dealings with one another, but
can we be wholly mistaken?
The identity thesis is defeated, however, on altogether different grounds which lie to
the side of our present concerns (they are normative grounds, and a separate case could be
constructed to account for the force of Socrates' argument which would reinforce the
pragmatic reading of the dialogue I am attempting here). A second, very important segment
of the dialogue follows hard upon the collapse of the identity thesis in which the linguistic
issues are taken an important step further. What the foregoing discussion raises is the
possibility that the world could be, in its reality, altogether different from the way we
represent it in our language. Each one of us believes any number of distinct things about the
world " that today is Thursday, that the earth revolves around the sun, and that the sun will
rise tomorrow " and most of these beliefs, at least the ones we have occasion to make explicit
to ourselves, are available to us in the form of sentences in the language we speak. But if, as
the Heraclitean/ Protagorean theory seems to imply, our language is systematically and
globally distorting of the truth about the world, then all the beliefs we entertain in the terms
of that language are also distorted, and probably wrong. This next segment of the dialogue is
devoted to just this topic " the possibility of false belief " and Socrates confesses that it is
a topic which has puzzled him for a long time. In working through this socratic puzzlement,
Plato may be seen to be testing an epistemological hypothesis, exploring a line of approach
to the question of knowledge that is very different from the the one which issues in his Theory
of Forms during his middle period. It is also a line of approach that has much in common
with what is discussed by recent authors under the rubric of formal pragmatics. The
contemporary pragmatists that I have in mind are Robert Brandom and Jurgen Habermas, but
I'll confine my attention to Brandom's brand of pragmatism, about which, more in a moment.
At the heart of Socrates' bewilderment, in his initial expression of it, there lies an
equivocation between two different ways of talking about beliefs. In the argument which plays
out between 187e and 189c, the objects of belief are described as both the way things are,
and items of knowledge,
or, as we might put it, the facts out there in the world and the
propositions we use to express them. Now, in order for the argument to go down the track it
does and reach the conclusion that it is not possible to believe what is not, either about
anything which is or in any absolute sense,
and that therefore false belief is different from
believing what is not,
we need to ascribe the same form of being
to both the things that
are
" that is, objects in the world, the facts which they constitute " and the items of
knowledge
which are the objects of our beliefs " the propositions in which the facts are
X k$ expressed. The proposition itself has to be something in order to be an object of belief. Owing
to the equivocation between facts and propositions, it appears impossible to say that there
could be propositions to which no facts correspond, or propositions which represented
non-being. The question, How can one believe what is not?
is the question, How can one
have nothing (non-being) as an object of belief?
When I believe something false, surely there
X ( is some object before my mind. So how can that something not-be, that is, be false?
The assumption which guides the first leg of the argument is for each and every item,
that it is either known or not known.
The problem which the argument discloses arises D+d .,, through our platonizing
of propositional contents, imputing to propositions themselves a
form of being which is the same as the being of ordinary objects or facts. The assumption that
every item is either known or not known posits a set of items
, some of which are known,
some of which are unknown, but all of which have being. This assumption is shown in very
short order to lead to unsatisfactory results, and Socrates proposes to replace the assumption
with another one, to conduct the inquiry not on the basis of what is known and unknown,
but on the basis of what is and what is not.
This is progress, or at least an open possibility
because we here resolve the conflation of ontological and epistemological posits in favor of
an unequivocally ontological way of putting things. But even this won't work because it
requires us to say that false belief occurs when someone thinks and what he thinks is not
true.
(188e1) Even if we grant the reality of the proposition one holds before one's mind in
a state of belief, making sense of false belief still engenders the gap between thinking a
thought which is
(in the ontological sense) and thinking a thought which is not true.
What someone thinks " the thought or belief or proposition, whatever you want to call it "
has being, but this account of false belief requires us to attach the property of falsity to an
existent entity. But truth and falsity are not ontological properties at all: we cannot speak of
true being or false being. So, putting our feet firmly on the side of ontology will not allow
X y us to make sense of false belief because falsity cannot be given an ontological rendering.
This is no trifling matter for Plato. The Theory of Forms posits a realm of being
precisely for the sake of giving an account of genuine knowledge " as opposed to the lesser
grades of cognition like understanding and imagination and perception. In his middle period,
Plato resolves the epistemological question by moving quite deliberately onto ontological
grounds. The Forms are real " more real than the things we can perceive with our senses "
and knowledge consists in the mind's putting itself in contact with these entities (to use a
X P deliberately vague locution). In a series of dialogues from Meno to Sophist to Parmenides,
Plato raises a whole range of problems which the Theory of Forms must address if it is to be
fully satisfactory and prevail against the skeptical and relativistic position of the sophists "
people like Protagoras and Gorgias and Thrasymachus and Callicles " whom Plato regarded
as a desperate threat to the viability of the Athenian polity. Many of the problems associated
with the Theory of Forms come to this: as long as there is a gulf between human cognitive
capacities " and language as an essential feature of those capacites " on the one hand, and the
reality of the world on the other " whose ontological status is utterly indifferent to language;
as long as there is a gulf between the domain of truth and the domain of being, there will
remain the possibility that our language simply does not get us over to reality, that it in fact
constitutes an unbridgeable gap, and that knowledge is impossible. This is not a conclusion
X U that Plato could countenance. Perhaps by the time of his writing the Theaetetus " and again
assuming that it is indeed one of his later works " Plato was ready to entertain a different
approach to the question of knowledge, an approach which circumvents the ontological issues
altogether and goes, rather, to the heart of the matter: the question of language.
Notice what happens next in the dialogue. Notice, in particular, the subtle shift of
terminology which occurs at 189e 190a. Thinking is described here as a discussion that the
mind has with itself, and a belief is called a statement, but one which is not made aloud and
to someone else, but in silence to oneself.
Something vitally important has happened here.
The question of knowledge had hitherto taken the form of a relation that could be expressed
in various ways: between the mind and the forms; between items of knowledge
and the
things that are
, between thoughts or beliefs and the world. The question of knowledge is
posed here in terms of a relationship between a speaker and an auditor " both of whom in this *e .,, preliminary redescription of the nature of thought happen to be the same person. Rather than
testing for the truth of our beliefs by holding them up to the world, and thereby requiring us
to compare one thing which is inherently linguistic in nature with something else which is
inherently non-linguistic in nature, we may test for the truth of a belief, now described as a
statement one makes to oneself, by placing it within the context of the other beliefs we hold,
the other statements we are prepared to assert " to ourselves or to others. Knowledge does not
consist in the mind's taking possession of something external to itself, and the mark of truth
is not the correspondence of a belief to a fact but the consistency of one statement with a
collection of other statements. Socrates asks Theaetetus (190c) Do you think that anyone,
X 1 sane or insane, seriously says to himself and tries against the odds to convince himself that
what is a cow is a horse and that what is two is one?
(Emphasis added.) No, he continues,
on the assumption that believing is making an internal statement, no one whose mind has a
grasp on both of two things, and who therefore makes statements " that is, has beliefs " about
both, could state and believe that what is different is different.
The impossibility of believing
that a horse is a cow has nothing to do with the fact that the inherent properties of the one
are ontologically incompatible with the inherent properties of the other; the point is that you
X cannot say that a horse is a cow, either to yourself in the private inner dialogue of thought or
to anyone else in public discourse because to do so would be to violate the protocols of
speech, contradict the way we do, as a matter of fact, use these words in the parlance of
common life. Plato puts into the mouth of Socrates the recurrent complaint that the abstruse
theories of Heraclitus and Protagoras do violence to our ordinary ways of talking; he seems
now to be flirting with the possibility that the ordinary ways of talking have a certain authority
and firmness about them and may serve as a sort of testing grounds for the tenability of any
given statement that one has come to hold as a belief. I must reject as false any statement
which is inconsistent with other statements to which I am already committed or with
statements that other people may be able to convince me I should be committed to. The
internal logic of our language " or rather the common life speech practices that we all fully
well know how to engage " provides all the criterion we should ever need to gauge the
veracity of our beliefs. A false belief might then be characterized as a statement I had had
occasion to assert to myself provisionally but one which, unbeknownst to me prior to my
testing the statement in thought or in discourse with others, was inconsistent with other
statements to which I was more deeply committed.
This interpretation is compatible, indeed supportive, of what Socrates goes on to say
a little later in the dialogue, at a point where he and Theaetetus feel compelled to
acknowledge their failure to get through to an adequate statement of what false belief consists
in. But haven't I just said what a false belief would be on the assumptions that Socrates has
X >! given us to work with? Not quite. All we've got at this point is an indication of the method
we should have to deploy in order to test the veridicality of any of our beliefs. We cannot
determine which of our beliefs are false one-by-one, or simply by inspection because falsity
is not a quality a belief has all by itself. The falsity is only something that comes out in the
process of discussion, as part of an activity that people engage in, either individually or in
groups. What Socrates says later on in the dialogue is this: The fact is that a satisfactory
understanding of knowledge is prior to the possibility of knowing about false belief.
(200d)
What I should like to suggest now is that Plato has in fact given us that satisfactory
understanding of knowledge,
but not in the form that we may have expected to find it, given
all the talk about definitions of knowledge
in the early going of the dialogue, and all the
labored efforts by the droves of philosophers in our own day to identify Plato's theory of
knowledge.
Plato does not give us a theory of knowledge, he gives us a method for its C+f .,, X pursuit; he does not say what knowledge is, he shows us how it may be attained. The dialogue
itself is an illustration of the method, an argument by exemplification. At the end of the
dialogue, Socrates and Theaetetus concede that they have not reached any acceptable
X characterization of knowledge, but clearly they have learned something. And that, as Socrates
says, is itself a handsome reward.
(187c)
Putting the matter as I have, claiming that Plato does not say, but only shows us what
knowledge " or at least the pursuit of it " consists in, is intended to resonate with the
X language of Wittgenstein's great Tractatus. One of the most important teachings of that work
is that we cannot explicitly say or articulate wherein consists the power and authority of the
logic which structures our language, we can only show it in the use we make of that language.
We cannot expect a fully explicit theoretical statement of the nature and workings of logic
because we cannot use the language which presupposes and is built upon a consolidated
logical superstructure to represent in linguistic terms the very nature and design of that
superstructure. In a sort of adumbration of G?del, Wittgenstein maintains that any such logical
theory would be either incomplete or self-referentially incoherent. Plato seems to be hovering
X # in the same general vicinity with his claim, in the Theaetetus, that we should have to have
X some understanding of what knowledge is before we could even undertake an inquiry into the
nature of knowledge or, more explicitly, that we should need to have a satisfactory
understanding of knowledge
before we can construct an account of what false belief is. And
the two books have a similar paradoxical ending: Socrates and Theaetetus conclude that they
do not know what knowledge is, but they have learned something worthwhile along the way
to finding out even that much; Wittgenstein tells us that he has not really given the reader a
clear statement of what logic is or what it does, but anyone who has followed his meaning
will have got the message anyway, and can toss the ladder away.
My intention here, however, is not simply to point out interesting similarities between
the claims of these two great thinkers, but to try and say something interesting about the
nature of knowledge itself. It is actually Plato who has, I think, said something intensely
interesting about the nature of knowledge. But I have advertised an intention to show the
transition from platonism to pragmatism. That Plato has moved away from platonism is at
X [ least suggested in the lack of any mention of the Forms in Theaetetus, and, I hope, in the
significance I have imputed to that lack in the foregoing discussion. But where is the
pragmatism in any of this?
Let me note, first of all, that a charge of platonism is frequently alleged against the
X w Wittgenstein of the Tractatus, and also against Wittgenstein's inspiration for that work, Frege.
In the case of the early Wittgenstein, the platonism consists in the claim that the logical
X K! structure of our language which manifests itself in our talk is just sort of there, and is, in some
unspecified, and perhaps unspecifiable sense, real. The same transition that reveals itself when
X # we read the Theaetetus as one of Plato's later works in relation to his more dogmatic middle
period would provide a nice template for describing Wittgenstein's development from his own
early work to the later. That is a topic for another day. But in order to characterize the end
point of the emerging trajectory as, in some sense of the word, pragmatic,
let me defer once
more to the authority of people who have thought long and hard about these topics, and who
claim, and argue at length, that Wittgenstein's later position may be described in decidedly
pragmatic terms. Richard Rorty and Robert Brandom have both claimed the late Wittgenstein
for the pragmatic tradition, and have teased out certain aspects of his work to incorporate into
their own, more clearly defined pragmatic positions. I shall draw on several aspects of
X R+ Brandom's position, as laid out in his important book, Making it explicit (1994), to sketch out R+g .,, one conception of pragmatism that is current today, the one that I think can be seen lurking
X in the Theaetetus according to the reading that I recommend. Brandom constructs this
conception of pragmatism on the basis of several doctrines and arguments he draws from the
late Wittgenstein.
To arrive at his own understanding of Wittgenstein, Brandom presents an extended
analysis of the argument of Saul Kripke in the latter's by now classic study of Wittgenstein
on Rules and Private Language.
According to Brandom's Kripke, Wittgenstein's discussion
of rules in relation to our everyday linguistic practices leads us to a dilemma: in our attempt
to show how our practices are rule-governed we are either driven into an infinite regress " we
need not only rules to structure our moves in the language game, but rules to show us how
the rules are to be applied, and so on up " or forced to construe the rules so loosely as to
permit virtually any move in the game to count as complying with them. On either horn of
the dilemma there is no good way to show how the rules govern the behavior. Kripke is
pleased to settle for a skeptical
resolution of the problem: there is no way to know, in his
famous example of the mathematical operator quus,
whether a given individual is applying
the rules we would all accept as the ones which govern arithmetic or some bizarre permutation
on those rules whose bizarreness is concealed only by the fact that the individual seems to get
the same results we do when we apply the rule plus.
Reading Wittgenstein through this
Kripkean lens, Brandom draws the conclusion that the practices of everyday life " including
both arithmetic and ordinary conversational practice " have a certain intuitive clarity and
authority about them which stand in no need of validation by our teasing out the rules which
govern them. We can, to be sure, articulate and make explicit
to ourselves the general
forms and the rules which are implicit in our discursive practices, forms like the conditional
statement and rules like modus ponens. The mistake is only in thinking that these general
forms and rules are somehow prior to, or enjoy a privileged status over the practices
themselves. No, the practice is there first, practice has precedence over logic. If we try to
think of logic, or mathematics as revealing of a rational structure which is simply there
, a
structure possessed of some kind of ontological reality to which our thought, and our language
and our knowledge must be made to conform, we are well on our way to committing the same
X sort of metaphysical, platonizing excess which Plato may, in the Theaetetus, have had occasion
to regret in his own earlier work.
This, Brandom argues, is the proper way to understand Wittgenstein's teaching in the
X Investigations, and, I would add, it is an interpretation which is reinforced by what
X Wittgenstein says in On Certainty. Sooner or later our attempts at ever more basic
explanations come to an end; somewhere, as Wittgenstein says, our spade is turned.
Somewhere we just have to say: this we understand well enough to get on with, this we may
use as our beginning point for further explanations if we will. We have a pretty sure grip on
how to use the word knowledge
in our every day comings and goings, even if we cannot
give a theoretically satisfactory account of what it is. So much the worse for theory. We are
perfectly justified in taking our linguistic practices as a starting point in our philosophical
inquiries, and accepting their normal, effective operation as explanatorily basic.
This claim constitutes one of the most important features of the new pragmatism, and
it is also, I submit, an important part of the argumentative strategy which is trying to emerge
in the course of Socrates' discussion with Theaetetus. Rather than look for explanatory closure
in any metaphysical or transcendental domains, we ought to content ourselves with
explanations which incorporate and legitimate the practices which everyone understands well
enough, and which only a philosopher or a madman would suggest are so seriously in H+h .,, disrepair as to make them unsuitable for service. The theories of Heraclitus and Protagoras
throw our ordinary language into confusion, Socrates says, and to accept their theories would
require us to revise our normal ways of talking. But, Socrates also seems to suggest, you,
Heraclitus and Protagoras, have given us no independent grounds for thinking that your
theories are so much to be preferred as to warrant the massive effort that would be required
to wrangle our linguistic practices into line with them. What Brandom imputes to the
X v Wittgenstein of the Philosophical Investigations is a compelling argument which demonstrates
that no such independent grounds can ever be found. Socrates, and Wittgenstein, are calling
us back from the etherial heights of philosophical cloud-cuckooland and inviting us to begin
our philosophical inquiries at a point where we may expect to get real traction with problems
that actually matter to us, and within a context which precludes our going off the rails and
making all kinds of wild and exaggerated claims that only make us look ridiculous. This
contempt for the excesses of metaphysical speculation and disdain for the pretensions of
theory and its presumption to instruct us about how we ought to talk, is what gives the scoring
X rasp to Socrates' ironic and sarcastic tone of voice in the Theaetetus. It also motivates the
X
painstaking and fiercely analytical researches of Wittgenstein in the Investigations. And the
recommendation that we come back down to earth and reinstitute philosophy as an instrument
for practical research, that we eschew and resist the temptations and blandishments of
metaphysical abstraction, is at the heart of the new pragmatism as elaborated by Brandom.
X a0 References ă
X ' X Rush Rhees (ed.), 1981. Ludwig Wittgenstein: Personal Recollections. Totowa, NJ: Rowman
and Littlefield.%"
X q X Plato's Theaetetus, translated and with an essay by Robin A.H. Waterfield. Penguin Books.
1987.%"
X X Wittgenstein, L., 1922. Tractatus Logico-Philosophicus. C.K. Ogden (trans.). London:
Routledge & Kegan Paul.%"
X X _________, 1958 (2nd ed.). Philosophical Investigations. G.E.M. Anscombe and R. Rhees
(eds.), G.E.M. Anscombe (trans.). Oxford: Basil Blackwell.%"
X O X _________, 1969. On certainty. G.E.M. Anscombe and G.H. von Wright (eds.), Denis Paul
and G.E.M. Anscombe (transs.), Oxford: Basil Blackwell.%"
X X Kripke, S., 1982. Wittgenstein on rules and private language. Cambridge, MA: Harvard
University Press.%"
X X Brandom, R., 1994. Making it explicit. Cambridge, MA: Harvard University Press.%"
b D" P v Paul F. Johnson
L St. Norbert College
N De Pere, WI 54115
I ( '%i .,, N %" 5 a 1 , C W B _
a " Blob Theory: Nadic Properties Do Not Exist
by Jeffrey Grupp`!%"] ă
y d d d y 1 =
a a @
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A d d x j @ ---- " y M ( SORITES ($2"($), ISSN 1135-1349
- http://www.sorites.org
& = Issue #17 " October 2006. Pp. 104131
# { Blob Theory: Nadic Properties Do Not Exist
$ Copyright by Jeffrey Grupp and SORITES -- , a 5 a v
s E Blob Theory: Nadic Properties Do Not Exist
X / Jeffrey Grupp
+
X , a a. 1. Introduction ă
X
Discussion of, and reference to, properties pervades contemporary metaphysics, physics
X
(e.g., the electron has charge), and ordinary language (the lion is sublime). But only a
specialized group of a few hundred philosophers discuss the very specific details of what is
X involved in a particular having a property. Discussion of properties in the literature is of
course ubiquitous, as when physicists discuss the properties of space or particles, or when
philosophers discuss modal properties, mental properties, the properties of God, the properties
of space and time, the properties of matter or ordinary objects, and so on. And discussion of
the nature of properties " are they universals or tropes, platonistic or nonplatonistic, physical
X N or nonphysical? " is of course also common in the literature. But discussion of the very
X 9 precise details of what exactly is involved in a property's being possessed by a particular is
restricted to a minority of philosophers. Examples of a few of the philosophers who are
involved in the discussion of the philosophy of property possession are Bertrand Russell, D.
X C. Williams, 3 q F o ԍWilliams, 1998.3 A. J. Ayer, D. W. Mertz, 0 G q F ԍMertz, 2003.0 David Armstrong, @ q F ԍArmstrong, 2001, 1997, 1989.@ Michael Loux, 5 q F X ԍLoux, 2001, 1998.5 HectorNeri
X Casta9ada, and Keith Campbell, 9 0 q F ! ԍCampbell, 1990, 1981.9 George Bealer, 1 q F # ԍBealer, 1982.1 Doug Ehring, 2 v
q F *% ԍEhring, 2001. 2 Reinhardt Grossman, 3 q F & ԍGrossman, 1992.3
X Kristopher McDaniel, 3
q F Y) ԍMcDaniel, 2001.3 Dean Zimmerman, 4
_ q F * ԍZimmerman, 1997.4 E.J. Lowe, ; q F , ԍLowe, 2002, 2001, 1998.; John O'LearyHawthorne and Jan j = ...ee X Cover, G i q F y ԍO'LearyHawthorne and Cover, 1998. G Panyot Butchvarov, 5
G i q F ԍButchvarov, 1979.5 J. P. Moreland, 3 i q F ԍMoreland, 2001.3 James Van Cleve, 4 i q F b ԍVan Cleve, 2001.4 Albert Casullo, 2 0 i q F ԍCasullo, 2001.2
X William Vallicella, 5 i q F ԍVallicella, 2002.5 Peter Simons, 1 v
i q F 4
ԍSimons, 2000.1 Michael Jubien, 1 i q F ԍJubien, 1997.1 John Lango, 0
i q F z ԍLango, 2002.0 Arda Denkel, 1 _ i q F ԍDenkel, 1997.1
X H. H. Price, 0 i q F ԍPrice, 2001.0 Nicholas Wolterstorff, 8 i q F L ԍWolterstrorff, 1970.8 Francesco Orilia, 1 H i q F ԍOrilia, 1998.1 M. Glouberman, 5 i q F ԍGlouberman, 1975.5 Cody
X Gilmore, 2 i q F ԍGilmore, 2003.2 Chris Swoyer, 1 1 i q F ԍSwoyer, 1999.1 and Jonathan Schaffer, 3 i q F d ԍSchaffer, 2003.3 to name a few.
If the arguments I give in sections 2 4 of this paper are correct, they may lead to
vindication of a position that has been called blob theory
: there are no nadic properties.
The point of this paper is to show that blob theory may be the correct account of reality, due
to hitherto undiscussed problems that I will point out to do with property possession in bundle
X theory and substance theory, nw i q F
& ԍI will use substance
in this paper to denote any ordinary object that is not a property, that is not a bundle,
and which exemplifies nadic properties. and that are so serious that I do not see a way to avoid blob
X theory. %o i q F M) ԍIn addition to vindicating blob theory, the arguments of this article may vindicate a sort of atomism similar to
the atomism that some of the ancient Greek philosophers might have been arguing for, where atoms were
considered to be without attributes, without intrinsic nature, numerically distinct but metaphysically indiscernible. * .,,ee Ԍ If properties do not exist, then minds must invent experiences of properties in the mind in order to account
for the experience of a reality one might believe to have properties. To my knowledge, this idea"the idea that the
ordinary, commonsense, macroscopic reality of surfaces and patches of color is largely an invention of the mind
in its incorrect representations of particulate reality"is in line with quantum theory, since quantum theory is mostly
about empty space between particles, pointmasses (which have no size, and thus no color), and so on.
Furthermore, on this account, any mind, also being a propertyless item, would be an atom (or an activity
of a number of numerically distinct atoms, if propertyless atoms can be involved in activities.) The idea that a mind
is a philosophic Greek atom is suggested by (though apparently not strictly argued for) by Yandell (1999, 114).
I will discuss these issues a bit more in parts of this article below, especially the conclusion. k .,, Ԍ In this paragraph I will give a brief synopsis of how I argue that problems to do with
the theories of property possession lead to blob theory. As I will discuss in later sections, a
X property must be a property of something. There cannot be properties that are not possessed
X by something that they are properties of, for if there were, they would be properties that are
not properties of anything (they would be properties that are not properties). That is how I
come to the conclusion that properties do not exist, for the following reasons. If it could be
argued that there are no currently available coherent theories of property possession, then the
best account of reality we have is one where particulars do not have properties. If that is the
case, then if there are any properties, they can only be properties that are not properties of any
particular. But if there cannot be any such properties, as just mentioned, and as will be argued
X below, then there cannot be any properties whatsoever. Therefore, by focusing on problems
to do with property possession, I am able to come to the conclusion that properties do not
X
exist.
i q F ԍIt may be the case that some time down the road a philosopher will develop an entirely new theory of property
possession which my arguments do not attack and thus blob theory's correctness is not vindicated after all. But
since there is, to my knowledge, no such alternative theory of property possession in the works, I ignore this
possibility in this article.
Before discussing problems with property possession, in this introduction I will give
some introductory comments about blob theory (subsection 1.1), I will discuss the noncommonsensical nature of blob theory (subsection 1.2), and I will discuss property possession
in theories of objects as given to us by metaphysicians (subsection 1.3).
X k a. 1.1 Blob Theory ă
X Blob theory follows from what some have called extreme nominalism
: i q F f ԍWhat I mean by extreme nominalism (EN) is given by Armstrong:
[ > X There are a number of varieties of Nominalism, but a useful broad division of Nominalisms is into the extreme and the
\ > moderate varieties. An extreme Nominalist denies that there are universals, and furthermore denies the existence of
[ > P! objective properties and relations. The moderate Nominalist agrees that there are no universals, but does hold that there
[ > ! are properties and relations. They are particulars. (Armstrong, 1997, p. 21.)
X nadic
properties do not exist. According to blob theory, reality is unstructured (it is an unstructured
X blob). Moreland lucidly describes blob theory:
t M X Among other things, EN [extreme nominalism] is what has been called a blob theory regarding concrete
particulars. A blob theory of ordinary concrete particulars is consistent with a mereological analysis of those
particulars as wholes constituted by separable parts; but a blob theory renders concrete particulars
structureless entities with no internal differentiation of properties and relations within those concrete
particulars. In this sense, EN treats concrete particulars as simples and thereby fails to acknowledge that the l .,, redness, circularity, size and other features of [for example, a red ball] are real entities that are neither
X identical to each other nor to [the ball] as a whole. ! n i q F : ԍMoreland, 2001, 74. Notice that Moreland refers to concrete particulars as simples in blob theory. This will be
discussed more in the conclusion where I discuss that blob theory leads to a specific sort of philosophic atomism. %"
(Extreme nominalism is not identical to Quinnean nominalism. The primary difference
between the two is that extreme nominalism is identical to blob theory, and blob theory
implies that, as I will discuss in subsection 1.2, nearly all our statements are false, whereas
this is not the case at all in Quinnean nominalism.)
In sections 2 4, I discuss hitherto unnoticed problems to do with substance and
bundle theories. I do not discuss blob theory in detail. Rather, my goal is only to argue for
it by revealing hitherto unnoticed problems to do with property possession in analytic
metaphysics.
X T
a 1.2 Blob Theory in Opposition to Common Sense and Macroscopic Perception ă
Maddy writes that it is quite clear
that blob theory does not describe our world:
t M
X 8[A] purely physical world8 lacking all8 properties8 is a world entirely without individuating structure: The
Blob. Even if such a world is possible, it seems quite clear that it is not our world, and thus, that it would
X be of no particular interest to the physicalist. 5 " } q F g ԍMaddy, 1990, 273.5 %"
In opposition to Maddy, I will present novel arguments which indicate that (i)
metaphysicians apparently have not presented an account of property possession that is
X coherent, and for that reason, (ii) the position that there are no properties may be the best
theory of reality we currently have. I will also argue in the conclusion, and in opposition to
Maddy, that blob theory may be of utmost interest since it may be supported by theories of
philosophic atomism.
Extremely counterintuitive theories of reality, arriving at conclusions similar blob
theory have been discussed or argued for by many philosophers from Parmenides up to the
present. For example, in an excellent paper, O'LearyHawthorne and Cortens discuss
philosophical positions similar blob theory:
t M X In this paper, we wish to motivate a radical cluster of metaphysical pictures that have tempted
philosophers from a variety of traditions. These pictures share one important theme " they refuse
to accord countable entities any place in the fundamental scheme of things. Put another way, they
all suggest that the concept of an object has no place in a perspicuous characterization of reality.
v M Such pictures suffer from a number of fairly obvious prima facie difficulties. They seem to fly in
the face of common sense. They seem to suggest that just about everything we say is false. They
seem to gesture at a noumenal reality that human language is unable to describe. And so on. Our
aim is to meet such difficulties head on and, by doing so, vindicate this sort of radical picture as
X one that deserves to be taken seriously. M # } q F {& ԍO'LearyHawthorne and Cortens, 1995, 143.M
In Unger's earlier works, he also has argued for positions that arrive at conclusions
much like blob theory:
v M # X My main aim in this paper is to help foster a positive attitude toward a thesis of radical nihilism.
According to this thesis, none of the things which, it seems, are most commonly alleged to exist
do in fact exist: neither rocks nor stones, not tables nor chairs nor even people; perhaps most 0%m D # .,, importantly, neither you nor I exist. The positive attitude toward this thesis is that it is worthy of
X serious consideration. 5 $ } q F : ԍUnger, 1980, 517.5
If the reader is dubious about the project undertaken in this paper, due to the obvious
disagreement that blob theory has with ordinary experience, I suggest that the reader still
consider the arguments of this paper, for reasons I discuss next.
Serious consideration of accounts of reality that are entirely against our ordinary
macroscopic experience of reality (perhaps even to the point of resembling a philosophy of
the blob) is not uncommon in quantum physics, especially in theories of quantum gravity
(string theory, Mtheory, loop quantum gravity, etc.). (Quantum gravity theorists are not to be
X equated with quantum theorists in general, but rather they are to be thought of as a very
specific subgroup of quantum theorists). Many theories of quantum physics, especially
quantum gravity, are very much at odds with our commonsense understanding of reality, and
our macroscopic ordinary perception of reality, including being in direct opposition with the
X rarely questioned and entirely commonsensical classical concepts of distance, time, space,
X
motion, and so forth. B % %
G } q F ԍFor a discussion on many of these issues to do with the opposition quantum physics has with the concepts of
space, time, distance, and motion, see Quentin Smith (2003). Also, the theory of atomism I argue for late in this
article may also not involve time, space, distance, and motion. B On this issue, Petitot and Smith write:
t M X The rise of mathematical physics has long been seen by many as dictating a dismissal of the
phenomenal world " the world of macroscopically organized in objectual forms, shapes, secondary
qualities and states of affairs " from the realm of properly ontological concerns and as dictating a
X * concomitant psychologization' of phenomenal structures. & n*X } q F W ԍPetitot and Smith, 1997, 233. In their article, Petitot and Smith attempt to argue against the position that
quantum mechanics is strongly opposed to common sense.
Things are apparently so counterintuitive at the most fundamental quantum domain of
the Planck scale that many quantum gravity theorists are now telling us that at the
fundamental level nature is timeless
and spaceless
. Consider what Greene, a leading
quantum gravity theorist, tells us about noncommutative geometry, the mathematics that in the
X - future might be found to describe the smallest level of reality that physicists study: On scales
as small as the Planck length a new kind of geometry must emerge, one that aligns with the
X new physics of string theory. This new geometrical framework is called quantum
X geometry.
9 ' i q F s" ԍGreene, 1999, p. 232.9 Greene continues:
v M M X 8[R]esearch on aspects of Mtheory8 has shown that something known as a zerobrane " possibly
the most fundamental ingredient of Mtheory, an object that behaves somewhat like a point particle
at large distances but has drastically different properties at short ones " may give us a glimpse of
the spaceless and timeless realm8 [W]hereas strings show us that conventional notions of space
v M S cease to have relevance below the Planck scale, the zerobranes give essentially the same
conclusion but also provide a tiny window on the new unconventional framework that takes over.
Studies with these branes indicate that ordinary geometry is replaced by something known as
v M noncommutative geometry8 In this geometrical framework, the conventional notions of space and
X [ of distance between points melt away, leaving us in a vastly different conceptual landscape. A ( [ U } q F + ԍGreene, 1999, p. 379 A [ n
( .,, Ԍ X If reality is propertyless, then it is unstructured (undifferentiated) since, to give just
X one example, without partwhole relations (which are polyadic properties) there are no parts
X and wholes, and without parts and wholes, there is only one entity. ) n } q F O ԍOr, as I will argue in the conclusion, there are numerically distinct unstructured (propertyless) entities that are
metaphysically indistinguishable"such as in some ancient accounts of atomism. , * % } q F ԍI argue for the position that Western metaphysics leads to the position that reality can only be composed of one
entity in two other articles (Grupp forthcoming), and where I use entirely different arguments than those used in this
paper. The account of
reality at the quantum domain given by Greene more closely resembles an unstructured reality
more than it resembles a reality with structure. For this reason, the philosophy of blob theory
X may (notice I wrote may
) be a description of reality more in accord with the fundamental
X | quantum domain than the metaphysical realist + &| i q F `
ԍIn the Cambridge Dictionary of Philosophy (Cambridge University Press, 1995) pp. 56263, Butchvarov
describes what is meant by metaphysical realism
:
X Metaphysical realism, in the widest sense, [is] the view that (a) there are real objects (usually the view is
concerned with spatiotemporal objects), (b) they exist independently of our experience or our knowledge
of them, and (c) they have properties that enter into relations independently of the concepts which we
q F understand them or of the language with which we describe them. Antirealism is any view that rejects one
or more of these theses, though if (a) is rejected the rejection of (b) and (c) follows trivially8%"
X !! In discussion of universals [properties], metaphysical realism is the view that there are universals8%"
X descriptions of reality are. Greene's passage
X e of course does not establish that quantum reality = propertyless reality. Greene's passage only
X P perhaps suggests that a description of quantum reality may be more like a propertyless reality
than it is like a structured reality. This may be an important point for readers who are dubious
about blob theory, since despite the obvious conflict with commonsense, ordinary perception,
and our macroscopic understanding of reality that blob theory involves, there may nevertheless
be reasons for the reader to consider the philosophy of the blob, since, at the very least, blob
theory might be an account of reality that is aligned more with the fundamental level of reality
than any metaphysical realist account is.
I will not discuss quantum theories in this paper further than the comments just given,
other than one more comment in the next paragraph. The goal of this paper is only to argue
that blob theory is a serious theory that deserves consideration due to the novel arguments I
present in sections 2 ! 4 against the established theories of property possession. In this paper,
X I only discuss problems with property possession, and I make no attempt to describe how or
X why humans experience an ordinary, common sense reality of properties if blob theory is the
X correct account. 3 , ! i q F # ԍSee endnote 30.3 I do not discuss why or how, if reality is propertyless, humans have
experiences of properties, and have the experience of a world composed of properties grouped
in a way where the properties give rise to ordinary objects. This paper is not about any issues
except hitherto unnoticed problems to do with property possession in theories of ordinary
objects.
The disagreement that blob theory has with common sense perception may appear to
be a large issue to pass over, but I do not consider it so, for the following reason. Since
quantum mechanics is entirely unable to explain why humans experience the commonsense
macroscopic world of color patches and solid surfaces as they do, but is taken seriously, this _o , .,, appears to be evidence that fundamental reality is in opposition to commonsensical
X macroscopic reality, and thus in addition to quantum physics, blob theory should also be taken
seriously. Since quantum experimental and empirical evidence leads to doubt about the
correctness of our commonsense macroscopic perception of reality, it appears that if quantum
physics is on track, we should be at ease when our philosophical reasoning leads to the
X position that our ordinary, commonsensical, macroscopic view of reality is incorrect. And it
appears that we should be leery of theories that imply our ordinary, commonsensical,
macroscopic view of reality is correct. This appears to imply that model of reality put forth
X L by blob theory can be trusted more than the ordinary, commonsensical, macroscopic view of
reality, due to the fact that blob theory may be more in line with an anticommonsensical view
of reality that denies ordinary perception and our macroscopic view of reality. Quantum
X
theorists also ubiquitously avoid discussion of how or why ordinary common sense reality
X
appears the way it does in light of the highly counterintuitive quantum theories. - n
i q F m
ԍOne article, by a philosopher, that discusses issues to do with the conflict disagreement quantum theories and
ordinary experience, is Quentin Smith, 1997. Quantum
theories are taken seriously even though they have enormous disagreement with commonsense,
ordinary, macroscopic reality. Since quantum theories show that reality at the fundamental
level might be largely unstructured, then I see no reason why blob theory, which arrives at
X a similar picture of reality, cannot also be taken seriously. + . & i q F k ԍA second reason why I consider my passing over any discussion of the issue that the philosophy of
propertylessness is weakened by its disagreement with common sense reality, or by its inability to explain how
humans have experiences of objects that have properties, is given as follows. Similar to the philosophy of
propertylessness, it is widely discussed in the literature that philosophers of mind (connectionism, dualism, identity
theory, etc.) cannot explain specifically why or how humans have experiences of properties"for example, it is not
known how neural mechanics can give rise to qualia. For this reason, there is no reason to consider the position
that there are no nadic properties as weakened any more or less than any of the philosophies of mind due to this
lack of explanation. The problem that blob theory is apparently unable to describe how or why humans seem to
have experiences of ordinary objects that have properties is no more a problem for any of the philosophies of mind
that it is for blob theory. + If my arguments in this article
are correct, then blob theory is not only more in accord with the most progressive areas of
quantum reality (namely quantum gravity), it is also more coherent than the metaphysical
realist theories since my arguments in sections 2 ! 4 apparently show metaphysical realism
X > to be incoherent.
X a 1.3 Substances that are Bundles, and NonBundle Substances ă
Before moving to my arguments, I will briefly discuss theories of ordinary objects,
according to the metaphysical realist account. In this subsection, I will discuss that according
X to metaphysical realism, objects are substances or bundles: perduring or enduring particulars
X that exemplify properties, or bundles of compresent properties. These are the two available
analytic metaphysical realist theories of ordinary objects that are considered by the
contemporary philosophers who describe and discuss in great detail the nature of property
possession by ordinary objects. (I do not discuss Quinnean nominalism since Quinnean
nominalism involves some sort of property possession " all particulars possess the polyadic
X J property, set membership " and for that reason, the problems I discuss to do with bundle
theory and substance theory apparently also apply to Quinnean nominalism.)
When one looks out into the world, as it is given to phenomenal consciousness, one
X } is presented with attributes, and they come in groupings which the metaphysical realist
typically calls substances
, bundles
, ordinary objects
, or things
. For example, the lion h p . .,, (substance) is a grouping of attributes: felinity, fourleggedness, goldenness, sublimity,
X ferocity, etc. The metaphysical realist's task is to figure out how it is the case that properties
can amass, conglomerate, bundle, or group to coherently (noncontradictorily) give rise to
objects. In other words, the metaphysical realist is very concerned with the question: If we
know that properties exist, then how do properties interrelate, interconnect, hold together, or
tie together, to give rise to objects, where objects in turn give rise to nature? This comprises
a major principle of the metaphysical realist: To understand reality, one must understand how
X a reality is composed of objects, where objects can be considered groupings, collections,
X L assemblages, or sets of attributes that are coherently interrelated and connected either to each
other (this is called the bundle theory, which I discuss below), or to an item that holds the
properties together (this is the nonbundle substance theory, which I discuss below).
The metaphysical realist tells us that the difference between attributes and ordinary
X h objects which are not attributes, is: ordinary objects have (possess) attributes (the lion has
X S ferocity, the electron has structurelessness, / %S i q F ԍThis is the way physicists often refer to the electron or quark, as being structureless, since they have no
evidence for its having any parts, or, as they say, they have no evidence for it having any internal structure. See
Kane, 2000, 22. a galaxy might have the property of being a
X >
quasar); and an object is not had (possessed) by anything else (this will be discussed in more
X ) detail in later sections). ` 0 ) i q F ԍSee Lowe, 2001 for discussion on properties as ways
.`
There are two major theories of objects: bundle theory and substance theory. According
X q to the bundle theory, properties are tied to one another. According to substance theory,
X \ properties are tied not to one another, but rather are tied to an entity that is a nonproperty.
X G Many metaphysical realists use the word substance
to denote either the substances of
X 2 substance theory, or the bundles of the bundle theory of objects. 1 n2X i q F _ ԍFor a good example of a philosopher that makes a point of talking about bundles as substances, see
Glouberman, 1975. Bundles are often called
substances that are bundles of properties
; and substances are often merely called
substances
, but I will call them nonbundle substances
. The nonproperty item that
properties tie to in nonbundle substance theory will also be discussed in detail in sections
X below; for now, it can be called a thin particular, or an internally bare particular. 2 % i q F _ ԍIf the properties of the nonbundle substance were not tied to a nonproperty, then they would be tied to a
property, and the nonbundle substance would be a bundle. So on the nonbundle account, the properties must be
held by a nonproperty.
Armstrong discusses what is meant by internally bare particular
, or thin particular
:
t M " X Here is a problem that has been raised by John Quliter (1985). He calls it the Antinomy of Bare
v M Particulars.
Suppose that particular a instantiates property F. a is F8 a and F are different entities,
one being a particular, the other a universal. The is' we are dealing with is the is' of instantiation
" of the fundamental tie between particular and property. But if the is' is not the is' of identity,
v M ( then it appears that a considered in itself is really a bare particular lacking any properties. But in
X that case a has not got the property F. The property F remains outside a8 ; 3
} q F ) ԍArmstrong, 2001, 7879.;
(I will discuss in section 2 that many metaphysical realists hold that properties of the
X 3 nonbundle substance are not tied to a bare nonproperty (internal bare particular or thin 3 q f 3 .,, particular), but rather are tied to what they call a thick particular
. I will give an argument
showing that this position is apparently incorrect, where my argument is based on the issue
that the definition of thick particular
has not been outlined specifically by metaphysical
realists. Often the issues of bare or thick particulars are not addressed in the literature, and
rather than specifying if the thin or thick particulars are of concern when talking about
property possession, philosophers will merely write that particulars exemplify properties
( x
has F
) (e.g., an electron has charge, Jones is present, God is omnipresent, etc.) without
discussing the issues that those who are specialists in the philosophy of property possession
bring up to do with bare or thick particulars.)
X Properties are held to the thin particular in the case of substance theory, or to one
X another in the case of bundle theory, by special ties, often referred to as predicating ties
.
The tie in substance theory (where properties do not tie to one another, but tie to the internally
X f bare particular) is called the exemplification tie; and in bundle theory (where properties tie to
X Q one another) the tie is called the compresence tie (or it is sometimes called the compresence
X <
relation). These ties are discussed more in sections below.
Bundle theories, and nonbundle substance theories, are the only theories given to us
by metaphysical realists. Loux discusses how there are very few theories of ordinary objects
offered to us by metaphysical realists:
t M X If we follow bundle theorists and substratum theorists [substratum theorists are bare particular
theorists] in holding that any metaphysician who concedes that concrete objects have some sort of
ontological structure must endorse one of the two theories we have so far discussed, we are likely
v M to conclude that few options are genuinely viable8 By any standards, the list of available options
v M is depressingly short; its brevity is especially depressing for the philosopher who has sympathies
X with metaphysical realism8 (Emphasis added.)
t M X !! But not all metaphysicians agree that the substratum theory and the bundle theory are the only
accounts of concrete particulars available to the philosopher who attributes an ontological structure
to familiar objects. According to a very old tradition, ontologists have another option: they can take
concrete particulars themselves, or at least some among them, to be basic or irreducibly
X fundamental entities. 9 4 } q F s ԍLoux, 1998, 117118. 9
What I want to point out about these passages, is how he discusses that there so few
X B options
for theories of property possession by ordinary objects, and that the options only
X - consist of bundles or nonbundle substances. In the above passage, Loux mentions bundle
theory, Aristotelianism, and substratum theory (the latter two are nonbundle substance
theories). In a passage I cite next, Loux mentions the remaining major branch of metaphysical
X realism, which is the platonistic account of property possession by ordinary objects (platonists
X are either bundle theorists n 5 G i q F $ ԍSee Oaklander, 1978 for a paper that discusses bundle platonist theories. n or nonbundle substance theorists).
t M 4" X What are the issues separating the Aristotelian realists from Platonists? 8 Aristotelians typically tell
us that to endorse Platonic realism is to deny that properties, kinds, and relations, need to be
anchored in the spatiotemporal world. As they see it, the Platonist's universals are ontological free
floaters
with the existence conditions that are independent of the concrete world of space and time.
But to adopt this conception of universals, Aristotelians insist, is to embrace a twoworlds
ontology8 On this view, we have a radical bifurcation of reality, with universals and concrete %r 5 .,, particulars occupying separate and unrelated realms8 [T]here [is a] connection between
X spatiotemporal objects and beings completely outside of space and time. 6 % } q F : ԍLoux, 1998, 46. There is one more major branch of analytic metaphysics, which is the trope bundle theory. But
I did not make mention of it above, since I already made mention of bundle theory, and the majority of bundle
theorists are trope theorists.
In sections 2 4, I discuss hitherto unnoticed problems to do with substance and
bundle theories. I do not discuss blob theory in detail. Rather, my goal is only to argue for
it by revealing hitherto unnoticed problems to do with property possession in analytic
metaphysics.
X : a 2. Any NonBundle Substance Involves an Internally Bare Particular ă
In this section I will find that, contrary to what many metaphysicians typically tell us,
X nonbundle substances can only be considered in terms of a bare particular: the properties of
X m a nonbundle substance attach to a nonproperty: an internally bare particular.
Since not all nonbundle substance theorists hold that there are such bare and thin
X particulars, in this section I present an argument for the position that any nonbundle
X substance can only involve an internally bare thin particular. (I will discuss in more detail
X
what is meant by internally bare particular
in section 3. But in section 3, I will not be
primarily concerned with the problems to do with bare particulars that are typically discussed
X a in the literature, such as: How can an entity be (internally) propertyless?) If this reasoning is
correct, I will discuss in the next section that there are hitherto unnoticed problems for the
tying of (exemplifying of) properties to internally bare particulars. But it is a long while until
I arrive at that conclusion; many issues need to be discussed before that.
In showing that any nonbundle substance can only be considered in terms of bare
particular, I will discuss the exemplification tie (subsection 2.1), thin and thick particulars
(subsection 2.2), the nature of properties (subsection 2.2), and property possession (subsection
2.3). As I discuss these issues, my arguments that any nonbundle substance can only be
described in terms of a bare particular will be revealed. After this, in section 3, I will be able
to discuss the specific problems to do with property possession in nonbundle substance
theory.
X R a( 2.1 The Exemplification Tie ă
X In this section I will discuss the nonrelational tie of exemplification that holds a
property and particulars together in the nonbundle substance account of property possession
by ordinary objects. (In this subsection of this section, I will refer to the particular
rather
than the bare particular
, or thin particular
, as the item that exemplifies properties on the
nonbundle account of substances; and only after I establish that only bare particulars can be
the items that exemplify properties on the nonbundle account of substance will I refer to all
particulars as bare particulars
.) On the nonbundle substance account, properties do not
X " directly attach to the particular. Rather, I will discuss in this subsection that metaphysical
X " realists tell us there is a tie, called the exemplification tie, acting as an intermediary between
properties and particulars. In other words, properties do not directly attach to particulars, but
rather the intermediary of the exemplification tie directly attaches to the properties and
particulars. Loux lucidly explains these issues (and he discusses a few other issues to do with
the details involved in the exemplification tie being an intermediary between particular and
properties, which I will discuss after Loux's passage): 's 6 .,, Ԍ v M X !! According to the realist, for a particular, a, to be F, it is required that both the particular, a,
v M and the universal, Fness, exist. But more is required; it is required, in addition, that a exemplify
v M Fness. As we have formulated the realists theory, however, a's exemplifying Fness is a relational
v M I fact. It is a matter of a and Fness entering into the relation of exemplification. But the realist
insists that relations are themselves universals and that a pair of objects can bear a relation to each
other only if they exemplify it by entering into it. The consequence, then, is that if we are to have
v M the result that a is F, we need a new, higherlevel form of exemplification (call it exermplification2)
v M Q whose function it is to insure that a and Fness enter into the exemplification relation.
t M Unfortunately, exemplification2 is itself a further relation, so that we need a still higherlevel form
v M of exemplification (exemplification3) whose role it is to insure that a, Fness, and exemplification
t M are related by exemplifiaction2; and obviously there will be no end to the ascending levels of
exemplification that are required here. So it appears8 that the only way we will ever secure the
v M desired result that a is F is by denying that exemplification is a notion to which the realist's theory
t M applies.
t M X !! The argument just set out is a version of the famous argument developed by F.H. Bradley.
Bradley's argument sought to show that there can be no such things as relations8 Realists claim
that while relations can bind objects together only by the mediating link of exemplification,
exemplification links objects into relational facts without the mediation of any further links. It is,
we are told, an unmediated linker; and this fact is taken to be a primitive categorial feature of the
concept of exemplification. So, whereas we have so far spoken of exemplification as a relation
tying particulars to universals and universals to each other, we more accurately reflect the realist
thinking about the notion if we follow realists and speak of exemplification as a tie' or nexus'
v M where the use of these terms has the force of binging out the nonrelational nature of the linkage
X this notion provides. 8 7 } q F X ԍ Loux, 1998, 3841. 8
(Note that Loux mentions that the exemplification tie is to be considered ontologically
X ' primitive. This issue is not important to the reasoning of this paper since I do not make any
attempt to analyze the exemplification tie (I only inquire in 3.2 as to whether or not the tie
has properties), but it is interesting to note that, at least to my knowledge, specifically why
the tie is to be considered primitive has never been argued for, but rather it has merely been
assumed by many philosophers that the tie is not to be analyzed. This assumption arose after
Bradley's regress revealed problems to do with the theories of property possession. Placing
an unanalyzable tie into the metaphysics of property possession removes the regress, but this,
it could be objected, is merely an ad hoc solution, perhaps covering up deeper problems with
the currently accepted theories of property possession. I discuss such deeper problems in much
more in three recent papers (2003, 2004a, 2004b). But more importantly, my reasoning in this
X C paper about problems with theories of property possession follow whether the exemplification
X . tie is primitive or not.)
X On the nonbundle substance account, properties are considered to not involve a
relational or direct attachment to the particulars that exemplify them. If they did, a Bradleyesque regress would ensue. Rather, properties, and the particulars that exemplify them,
X L" according to nonbundle substance theories, are typically considered to be mediated by a non X 7# relational tie, called the exemplification tie, and for that reason, Bradley's regress is avoided.
There are two types of attaching that metaphysical realists are concerned with when
X % they discuss the property possession of substances: (i) intermediary attachment, which is the
sort of attachment between property and particular by way of the nonrelational intermediary
X U' tie of exemplification, and (ii) unmediated attachment, which is the sort of attachment between
the exemplification tie and property, and the exemplification tie and the particular. The nonrelational exemplification tie stands between property and particular, mediating the particular ))t G 7 .,, and its properties, and for that reason, the attachment between property and particular is an
X intermediary attachment. Unmediated attachment is a kind of attachment that entities are
X involved in where an intermediary is not involved. Let unmediated attachment
denote the
way that the exemplification tie attaches to both the property and particular. Unmediated
attachment is an attachment that is not a relation that the exemplification tie has with the
property and particular. Unmediated attachment does not involve any sort of entity
X z (intermediary) that is between the exemplification tie and the property, or that is between the
exemplification tie and the particular. Rather, properties and particulars each involve an
unmediated attachment to the nonrelational exemplification tie, and the exemplification tie,
in turn, involves an unmediated attachment to the particular, and to the properties that the
particular exemplifies.
X
Ya* 2.2 Properties are Ways ă
X In the next two subsections I discuss properties. Specifically, I will inquire as to what
X properties are, and what, exactly, properties tie to (via the exemplification tie) when they are
exemplified by a particular. In other words, I will be concerned with the issue of specifically
X what it is about a nonbundle substance that the exemplification tie involves an unmediated
attachment with in its unmediated attaching to a particular.
X A property is a way any substance (bundled or nonbundled substance) is: a property
X is what a substance is like. In discussing substances (Heil uses the word objects
instead of
the word substances
), Heil discusses the differences between properties and the objects that
have them:
t M X 8[O]bjects are bearers of properties. When we consider an object we can consider it as a bearer of
properties, itself incapable of being borne as a property, or we can consider its properties8 A
X property is nothing more than an object's being a particular way. 7 8 } q F ԍHeil, 1998, 17778.7 (Emphasis added.)
X Armstrong also discusses how properties are ways objects (substances) are:
v M H X Properties are ways things are. The mass or charge of an electron is a way the electron is8
v M Relations are ways things stand to each other.
X !! If a property is a way that a thing is, then this brings the property into very intimate connection with
X the thing, but without destroying the distinction between them. ; 9 G } q F ! ԍArmstrong, 1989, 9697.; (Emphasis added.)%"
I have italicized the last sentence of Armstrong's citation to stress that, aside from the
X M exemplification tie, there are two distinct entities (in the broadest sense of the word entity
)
X 8 that must be involved when a particular exemplifies a property: there is (1) the property, and
X # there is (2) some other entity that the property is a property of. If it were not the case that
X ! property possession involved two distinct entities " property, and the particular that the
X ! property connects to (via the exemplification tie) " a property would not be a way a
X " particular is. A property not connected to another entity (via the exemplification tie) is an
X # impossible entity: 5 : %# i q F ) ԍHere I am referring to properties as entities
. I mean to use the word entity
in the broadest possible sense,
and in the way that many other realists refer to nadic properties as entities (for example, Moreland (2001, 13), Lowe
(2002, 16), and many others).5 it is a way that is not a way a particular is. ; n# i q F ԍPlatonistic properties can allegedly be unexemplified (unborne). It is unclear to me how they can be unborne
q F G and yet be ways. For there to be properties, #u ; .,, X there must be two entities (in the broadest sense of entity
) that are nonidentical, where one
X is tied to (linked to, < i q F ԍLoux discusses the connection between property and particular using the word link
in Loux 1998, 3841. borne by) the other. This issue, which is integral in my arguments in
this article, has been ignored by philosophers " especially substance theorists who assert that
property and particular are not necessarily distinct.
X Ya 2.3 What Specifically Do Properties Attach to (Via the Exemplification Tie)? ă
On the nonbundle substance account, since, as discussed in the last subsection, there
X e must be two distinct items involved in property possession (a property tied to a particular that
is entirely distinct from the property), there must be a specific entity that the exemplification
X 9 tie is involved in an unmediated attachment with. In this subsection, I inquire as to what
properties of a nonbundle substance are connected to (via the exemplification tie). I consider
X
this a rather simple inquiry to make, for one reason: any of the firstorder =
i q F ԍFirstorder properties are not properties of other properties, but are properties of things. It is the firstorder
properties of nonbundle substances that tie to an internally bare particular in order to constitute a thing (lion, planet,
etc.). And it is the firstorder properties of a bundle that tie to one another to give rise to a substance that is a
bundle. properties of a
X
nonbundled substance must attach to some entity, but if they attached to each other, the
X substance would be a bundle, thus firstorder properties must attach to a nonproperty. (This
issue, which is integral in my theorization in this article, has been ignored by metaphysical
realists.) If a substance is not a bundle, there must be something about the nonbundle
X substance that is not a property, and it is that something
that the first order properties attach
to (via the exemplification tie).
Many metaphysicians who are nonbundle substance theorists tell us that nonbundle
substances do not involve a bare particular. They tell us that we cannot get below the
X concept of a concrete particular
. 4 > i i q F ԍLoux, 1998, 118.4 On the Armstrongian account of nonbundle
X substances, ? n
i q F ԍArmstrongian substances are widely discussed substances involving spatiotemporal, physical universals that
are exemplified by a thick particular. some Aristotelian accounts of nonbundle substances, and platonistic accounts
X of nonbundle substances, properties are widely held to be properties of thick particulars, @ f i q F " ԍA thick particular is the entire complex of exemplified properties that constitute a substance. Armstrong:
Consider now a particular, not a particular considered in abstraction form all its properties, but the particular taken
q F 7$ along with its nonrelational properties. (This is a thick particular8) Let it be a particular that changes over time.
(Armstrong, 1997, 100)
X y as when we say: the lion (thick particular) is sublime (property)
, A Jy. i q F |' ԍLoux writes:
X According to a very old tradition, ontologists have another option: they can take concrete particulars
themselves, or at least some among them, to be basic or irreducibly fundamental entities. On this view,
having complexity of structure is compatible with being a basic or underived entity. The tradition is one that
can be traced back to Aristotle8 [A]t least some concrete particulars, living beings"plants, animals, and
persons"[are considered by this tradition] as fundamental entities that cannot be reduced to more basic + @ .,,ee entities. Philosophers in this Aristotelian tradition reject the constructivist approach to concrete particulars
that underlies both the substratum and bundle theories. As they see thing, the ontologist is not to construct
the concept of a concrete particular from antecedently given materials8 On this view, the ontologist cannot
get below the concept to of a concrete particular, and both the substratum theorist and bundle theorist are
mistaken in thinking that they succeed in doing so. (Loux, 1998, 117118)%"
X since lion
may yv A .,, appear to refer to a complex of properties, and not just to the nonproperty entity that
X properties tie to. ( 8is8
denotes the exemplification of the property.) Loux discusses this:
t M J X 8Aristotelians [and Armstrongians]8 find an important insight in the substratum theory. They agree
that the attributes associated with a concrete particular require a subject, but they take the
v M substratum theorist to be wrong, first, in construing that subject as a constituent of the concrete
particular, and second in characterizing it as bare. Aristotelians insist that it is the concrete
particular itself that is the subject of all the universals associated with it; it is what literally
exemplifies those universals. But8 Aristotelians contend that the concrete particular is, in virtue of
belonging to its kind, a thing with an essence, so they reject the central assumption of the
substratum theorists' account of subjects, that, for any attribute, the thing that exemplifies or
exhibits it is something with an identity independent of that attribute8 We have a subject whose
essence or core being does not include the attribute for which it is the subject8
X !! 8[T]he kinds to which concrete particulars belong represent irreducibly unified ways of being.
The Aristotelian wants to claim that because they do, the particulars that belong to them can be
construced as basic entities. What a concrete particular is,8 is simply an instance of its proper
X kind8 7 B } q F > ԍLoux, 1998, 12021.7 (Emphasis added.)
It is fair to say that most nonbundle substance theorists who are not bare particular
X theorists are not entirely clear on specifically which nonproperty entity it is in nonbundled
substance theory that properties are connected to (via the intermediary tie of exemplification).
X Consider a passage from Armstrong's widely discussed book, A World of States of Affairs:
t M X When we have talked about particulars up to his point, the tacit assumption has usually, though not
always, been that we are talking about the particular in abstraction from its properties8 It is, of
course, a controversial question in metaphysics whether there is such a thing as a particular in this
v M O sense8 If properties are not so much thingy entities, but rather are ways that things are (something
that in no way derogates from their mindindependent reality,8), then we cannot dispense with
v M particular in this sense, with what can be called the thin particular8
X !! The thinness is the trouble. It seems so thin that we think it cannot be what we meant when
we talk about particulars8 [W]e might explain our revulsions from the thin particular as no more
than the mind easily sliding away from it to the fullblooded thick particular. Even if the particular
be but an unimportant stone, we are not in the ordinary way interested in its bare particularity,
X something every particular has, but rather in what sort of thing it is. = C } } q F b# ԍArmstrong, 1997, 123125.=
Armstrong continues over the next several pages without telling us, or even hinting to
X X us, what specifically it is about the thick particular that any firstorder property of a thick
X C particular attaches to (via exemplification). Consequently, no definition of what specifically
the exemplification tie attaches to via unmediated attachment is found, and it appears that
regardless of Armstrong's motivations, the thick particular can only involve a thin particular
and the exemplification tie being involved in an unmediated attachment.
The reasoning above, if correct, indicates that on the nonbundle account of substance,
there must be an entity that is not a property, that has no properties in itself, and which all H"w C .,, the firstorder properties of the nonbundle substance tie to. Nonbundle substance theorists
who are not bare particular theorists however not only fail to tell us what that entity is, but
they also inform us that there is not an internally bare nonproperty involved with any
substance, such as the sort of internally bare particular Loux discussed in an above passage.
Nonbundle substance theorists who are not bare particular theorists, such as Davis and
Armstrong, do not include a bare particular, and thus their account appears to be incorrect, if
my reasoning above is correct.
A synopsis of the argumentation I have given in this section is as follows. The first X order properties of any nonbundle substance must be tied to an internally bare particular. The
internally bare particular cannot be composed of properties, for if it were, the nonbundle
X substance would be a bundle. A property is a way some other entity is, which is to say that
X }
the property must be tied some other entity. In the case of a firstorder property in nonbundle
X h substance theory, the other entity that a firstorder property is linked to apparently can only
be a thin particular that is internally bare. Any nonbundle account of a substance is a bare
particular account. For these reasons, I will hereafter only discuss bare particular accounts of
X % nonbundle substance. D % } q F ԍAnother passage from Loux, where he uses different reasoning than I have used above, shows that a thin
q F U particular account of nonbundled substances must be an internally bare particular account.
X [W]hat kinds of things function as constituents of concrete particulars? We have already mentioned the
attributes"the properties or tropes"that are associated with a concrete particular as its constituents. Is
there anything else that enters into the constitution of a concrete particular? One influential view insists
that among the constituents of any concrete particular there is a quite different sort of thing"something
that is not an attribute, but functions as the literal bearer, possessor, or subject of the attributes associated
with the concrete particular. On this view, then, there are two different kinds of entities that enter into the
constitution of any concrete object: the various attributes associated with the concrete object and
something that functions as the literal bearer or possessor of those attributes8%"
X !! 8[N]either the attributes that actually are nor those that could have been associated with [a concrete
particular, such as, a] ball [call it s,] can figure in the identity of s. Might some other attributes do so? If
they do, they must be attributes related to s in the way that the attributes associated with the ball are
related to the ball, as constituents to wholes. But, then, these new attributes need a subject or bearer; and
just as the ball could not be a subject for these new attributes. What we need, then, is a subject in our
subject, a constituent of s, that will function as literal bearer of the attributes that are supposed to fix s*s
identity. But what attributes will fix the identity of our new subject (s*)? Obviously, not the new attributes
for which it is subject. It looks as though the only way attributes could fix the identify of s' is for s' to be
a further whole made up of still further constituents; and obviously we are off on an infinite regress, a
regress that can be avoided only by conceding that there are subjects for attributes whose identity involves
no attributes whatsoever. And since we must concede that subjects whose being the things they are
involves no attributes make their appearance at some point in our analysis, we are best advised to make
this concession for s itself and thereby eliminate the need for new and intrusive subjects like s* and its
descendants. But if we do, we are committed to the view that each familiar concrete object is a whole
whose constituents include, first, the attributes whose being
or identity involves no attributes.
q F 4# Philosophers have given a special name to this subject; they call it bare substratum8 The points of the
label should be clear. The constituent in question stands under or supports attributes, but its being the
thing it is involves no attributes. (Loux, 1998, 9597)
X a 3. The Unmediated Attachment of Propertyless Entities ă
X In this section I next discuss problems with the unmediated attachment of an internally
bare particular and the exemplification tie. I will discuss that this unmediated attachment is
an unmediated attachment of propertyless entities (in the broadest sense of the word entity
),
which leads to serious problems for nonbundle substance theory. Since the exemplification
tie is considered to be unanalyzable, this unmediated attachment between the internally bare x K D .,, particular and the exemplification tie is typically also considered to be unanalyzable. Moreland
writes that [i]t is a primitive fact that properties are tied to [bare particulars] and this does
X not need to be grounded in some further capacity or property within them.
8 E i q F K ԍMoreland, 2001, 155.8 In writing
about this passage from Moreland, Davis has claimed that [b]y embracing tied to'
predication we have averted both the incoherence and infinite regress objections [that have
X been posed against bare particulars] in one stroke
. 5 F G i q F ԍDavis, 2003, 538.5 But if my arguments in this section are
correct, there is an apparently fatal problem that is not addressed in the literature regarding
the primitive unmediated attachment between the internally bare particular and the
exemplification tie. Regardless of whether or not such an unmediated attachment is primitive
or not, my argumentation will show that the mere issue of there being an unmediated
attachment between the internally bare particular and the exemplification tie lead to apparently
fatal problems for property possession in nonbundled substance theory and the bundle theory
of substance.
I will first discuss what has been called the internal nature
of bare particulars (3.1).
Then I will discuss the whether or not the exemplification tie itself has properties (3.2). Those
issues will enable me to discuss whether or not the exemplification tie can coherently be
involved in an unmediated attachment with the internal nature of the bare particular (3.3). I
will find that it cannot, and I will discuss that for that reason, the nonbundle account of
property possession by ordinary objects is impossible.
X 7 a+ 3.1 Bare Particulars ă
X Bare particular theorists commonly tell us that the bare particular is internally bare, not
X externally bare: it has no properties in itself, but it is tied to properties that are distinct from
X l it. Moreland and Pickavance write:
t M X !! Advocates of bare particulars distinguish two different senses of being bare' along with two
different ways something can have a property. In one sense, an entity is bare if and only if it has
no properties in any sense. There is another sense of bare', however, that is true of bare
particulars. To understand this, consider the way a classic Aristotelian substance has a property, say,
some dog Fido's being brown. On this view, [unlike a bare particular,] Fido is a substance
constituted by an essence which contains a diversity of capacities internal to the being of Fido.
These capacities are potentialities to exemplify properties or to have parts that exemplify
properties8 When a substance has a property, that property is seated within' and, thus, an
expression of the inner nature' of the substance itself8
X !! By contrast, bare particulars are simple and properties are linked or tied to them. This tie is
asymmetrical in that some bare particular x has a property F and F is had by x. A bare particular
is called bare', not because it comes without properties, but in order to distinguish it from other
v M O particulars like substances and to distinguish the way it has a property (F is tie to x) from the way,
v M ! say, a substance has a property (F is rooted within x). Because bare particulars are simples, there
X ! is no internal differentiation within one of them. G G ! } q F ' ԍMoreland and Pichavance, 2003, 34.G (Underlining added.)
X 4# Bare particular theorists discuss an internal and external nature of the bare particular.
The external nature is not propertyless; the internal nature is propertyless. A nonbundle
substance might be considered in terms of the following diagram, where a nonbundle %y G .,, X substance consists of an internally bare particular, IS, which exemplifies the properties A, B,
C, and D:
b H 5 D
) / \
b V $
C Is A
) \ /
b d 5 B
+
#
A, B, C, D are firstorder properties
X - IS is the internal nature of the bare particular
5 a
X g 2 a We can consider (i) the entire nonbundle substance, which is the bare particular along
X R with the properties the bare particular has. This is the internal and external nature of the bare
particular, which is, of course, not a propertyless item. But since the bare particular must be
distinct from the firstorder properties of the nonbundle substance, we can consider (ii) the
X bare particular's internal nature, represented by the encircled IS
at the center of the above
X diagram. IS, in itself, is propertyless (where propertylessness
somehow does not denote the
X property, propertylessness), and is entirely distinct from A, B, C, and D. Armstrong discusses
(i) and (ii):
v M + X The thin particular is a, taken apart from its properties (substratum). It is linked [tied] to its
properties by instantiation, but it is not identical with them. It is not bare because to be bare it
would have to be not instantiating any properties. But though clothed, it is thin8
X !! This is the thick particular. But the thick particular, because it enfolds both thin particulars and
properties, held together by instantiation, can be nothing but a state of affairs.
v M X !! Suppose that a instantiates F, G, H,8 They comprise the totality of a's (nonrelational) properties. Now
v M form the conjunctive property F&G&H. 8 Call this property N, where N is meant to be short for a's nature.
v M d a is N is true, and a's being N is a (rather complex) state of affairs. It is also the thick particular. The thick
v M ' particular is a state of affairs. The properties of a thing are contained within it
because they are
constituents of the state of affairs8%"
X !! Therefore, in one sense a particular is propertyless. That is the thin particular. In another sense
it enfolds properties within itself. In the latter case it is the thick particular and is a state of
X ! affairs. 8 H ! } q F $ ԍArmstrong, 2001, 79.8
X # The internal
nature (IS), to use the word Moreland used in a passage above, is what
I am concerned with. In this section, I am concerned specifically with the issue of the bare
particular's being involved in an unmediated attachment with the exemplification tie.
X 3& IS is the literal possessor of firstorder properties, and IS is tied to firstorder properties
of the nonbundled substance via the exemplification tie. Regardless of whether or not the
exemplification tie has properties, I will discuss that the unmediated attachment of the
X ( exemplification tie with IS will involve an unmediated attachment of propertyless items. It is (z G H .,, this sort of unmediated attachment that I will be concerned with near the end of this section.
I will argue that this unmediated attachment leads to a fatal problem for property possession
in nonbundle substance theory. Before discussing this unmediated attachment, I will discuss
the issue of whether or not the exemplification tie has properties.
X
a 3.2 Does the Exemplification Tie have Properties? ă
X y If the exemplification tie were also propertyless, the unmediated attachment of IS and
the exemplification tie would be an unmediated attachment of propertyless items, which is an
unmediated attachment I will later describe as fatal for metaphysical realism. If this is the
case, it may be better for the metaphysical realist to consider the exemplification tie to have
properties.
X |
If the exemplification tie did have properties, it would be a substance, for the following
X g reasons. The exemplification tie is not a property of a substance (it is not a way that a
X R substance is), and for that reason, it is unexemplified. Entities that are not exemplified, but
X =
which also have properties, are primary substances. If the exemplification tie is a primary
substance, it would have an internally bare particular that its firstorder properties tie to. I will
X call the exemplification tie's internally bare particular, IE. In being involved in an unmediated
X attachment with a property and with IS, for reasons I discuss in the remainder of this
X subsection, it is not the entirety of the exemplification tie (IE and all its nonrelational
X properties) that is involved in an unmediated attachment with IS. Rather, it is merely IE that
X involves an unmediated attachment with IS, rather than the entire exemplification tie substance
X (IE + all nonrelational properties IE has). I will next explain this.
X In being involved in an unmediated attachment with IS, we can ask: What is it about
X