[Foot Note 1_1]

On substructural logics see Dosen/Schröder-Heister (1993). This field is in fact not new, just the name for it. For example Avron (1988) shows that there are some striking resemblances between linear logic and relevant logic.




[Foot Note 1_2]

Da Costa's works were first published in the Comptes Rendus de l'Académie des Sciences de Paris (CRASP, first note (da Costa 1963), the referenes of other notes can be found e. g. in D'Ottavinao (1990)), through Marcel Guillaume (see Guillaume 1996). At this time I wrote to the latter who kindly sent me a joint work of him with da Costa published in Brazil that I was not able to find in France. Later on da Costa started to publish in Notre Dame Journal of Formal Logic (NDJFL) where a lot of papers on paraconsistent logic have appeared along the years.




[Foot Note 1_3]

See Raggio (1968). Raggio was a former student of Bernays who worked on cut-elimination for natural deduction before Prawitz. At the same time of my work, W. A. Carnielli built a tableau system for C1 and proved cut-elimination for it (see Carnielli 1990).




[Foot Note 1_4]

See (B 90a), the main results of it were later published in (B 93a).




[Foot Note 1_5]

This turns into my first published paper (B 89).




[Foot Note 1_6]

According to Curry, this is a special case of an even more general result to be found in MacLane's PhD (cf. MacLane 1934).




[Foot Note 1_7]

I discovered later on that da Costa had also been influenced by this book, in particular to develop the algebaic counterparts of his C-systems, which he called «Curry algebras» (see da Costa 1966).




[Foot Note 1_8]

This work was later published (B 97e) in a joint paper with D. Krause a disciple of da Costa working mainly in Schrödinger logics, i. e. logic for which the principle of identity does not hold in general, the motivation being that according to Schrödinger the micro-objects of quantum physics do not obey this law (see da Costa/Krause 1994). The principle of identity is also one fundamental law of logic whose study and rejection have attracted me over the years (see B 96b).




[Foot Note 1_9]

On this logic see de Souza (1997).




[Foot Note 1_10]

A set of formulas is saturated iff there is a formula not deductible of it but deductible of any extension of it. A saturated set is maximal iff it is saturated for any formula outside of it. Saturated sets are also called relatively maximal sets, especially in the Polish school.




[Foot Note 1_11]

This result was not new in the sense that there is an algebraic version of it which is known for years. However the logical version of this theorem apparently was not known or properly understood (cf. Suszko 1977), nor the consequences of it, for example the fact that intuitonistic logic cannot be characterized by maximal sets. This result shows also that the standard semantics for propositional classical logic is minimal, since it is made of maximal sets and classical logic is absolute.




[Foot Note 1_12]

David W. Miller proved independently the result about implication. He visited da Costa during my stay in São Paulo and turned to be interested in my work, due to the fact that at this time he was working on the question of the quantity of maximal extensions of a set. The expression «absolute logic» was suggested to me by David Makinson.




[Foot Note 1_13]

See (B94c). This paper is the first extensive exposition of da Costa's theory of valuation. A shorter and simpler exposition is to be found in Grana (1990).




[Foot Note 1_14]

This has been developed in more details in my philosophy PhD (B 96a).




[Foot Note 1_15]

An individual study of C1+ was later on presented in (B 95c) and also in my math PhD (B95e).




[Foot Note 1_16]

This result is presented in (B 98h) (B 96a) and (B 95e). The relation with da Costa and Suszko's reduction results is discussed in (B 96c).




[Foot Note 1_17]

Farah proved the equivalence between the axiom of choice and the general distributivity law (see Farah 1955).




[Foot Note 1_18]

See e. g. da Costa/Doria (1991). They in fact mainly use Suppes predicate which is a kind of adaptation of Bourbaki's notion of strcuture (see da Costa/Doria 1994).




[Foot Note 1_19]

I learned a lot about Bourbaki in Brazil but of course I already had heard of him before! In fact I was part of the generation of school boys who have been Bourbakized by ultra-bourbachic pedagogues. But when I went at the University of Paris, the Bourbakian ideology was already widely dismissed. People were making the bill of the alleged disastrous effect of modern mathematics and the high-school programs had been changed in order to come back to 19th century pre-Bourbakian mathematics and get rid of abstract non sense, viewed as anti-democratic (sic). Moreover Bourbaki was not well considered among French logicians who had been persecuted by him. However, as an exception, my first course on set theory was given by M. Eytan and was based on Bourbaki and category theory (this course was considered as a monstrosity and was later on suppressed for «technical» reasons).




[Foot Note 1_20]

Porte's book is quite unknown and had no influence. It was several years ahead of his time. Porte himslef spent most of his career in Algeria. I tried to contact him in Paris but he was already in a senile state.




[Foot Note 1_21]

The interplay between the Brazilian school and the Polish school was in fact limited, for example Kotas/da Costa (1980) is rather a juxtaposition of valuation and matrix than a work of synthesis. The terminology is generally different, with some random similarities. Funny enough, Wójcicki used as a key word «logical calucli» in the title of his books rather than the tyical Polish expression «consequence operator» which shows up only timidly in the subtitle of his 1988's book. On the connection between da Costa and Suszko's results on bivalent semantics see Batens (1987).




[Foot Note 1_22]

The main results of these investigations are to be found in Brown/Suszko (1973) and Bloom/Brown (1973). Suszko was in a sense quite an isolate figure in Poland and his work on «abstract logic» has not been pursued there, neither in the USA, but it was recently revived by the Barcelonas logic group (see Font/Jansana 1996).




[Foot Note 1_23]

P. Février developed a three-valued logic in order to deal with Heisenberg's indeterminacy principle (see Février 1937). This logic has been rightly considered as «quasi-formal» by J.-L. Destouches (see Destouches 1948). Discussion about this can be found in (B 95 g).




[Foot Note 1_24]

For a discussion of these topics, see (B95b). In (B 96a) I proposed to consider the domain of a logic as any kind of structure, results which do not depend on this structure being properly abstract results.




[Foot Note 1_25]

More generally, the metamathematics of Hilbert was replaced by the methodology of deductive science, with different objectives and methods, in particular, by contrast with Hilbert, Tarski allowed himself to use any mathematical tools at the «meta» level.




[Foot Note 1_26]

As it is known, Gentzen originally used the arrow for sequents (under P. Hertz's influence). For this discussion see (B 99b).




[Foot Note 1_27]

It seems that it is also a confusion between permissible and derived rules that Lukasiewicz made in his odd paper about intuitionistic logic (Lukasiewicz 1952), as pointed out by Legris/Molina (200?).




[Foot Note 1_28]

(Birkhoff 1987), Birkhoff explains that he took the expression «universal algebra» from Whitehead (1898) but recalls that the creator of this expression is J. J. Sylvester; Corry (1996) erroneously states that it is Whitehead. Birkhoff also says that it is in (Birkhoff 1940), his famous Lattice theory, that he decided to use this expression to denote a general study of algebras. The first systematic exposition of the subject was (Birkhoff 1946) whose title is simply «universal algebra».




[Foot Note 1_29]

Czelakowski (1980), Blok/Pigozzi (1989) and Font/Jansana (1996). The road leading from the algebra of logic to algebraic logic is an interesting object of study for the historian of modern logic which has yet to be fully examined. Curry stands in the middle of the road, he was the first to use the expression «algebraic logic» in (Curry 1952) and not Halmos as erroneously stated in (Blok/Pigozzi 1991, p. 365), but what he meant by it was still close to algebra of logic. Halmos introduced this expression rather to denote the algebraic treatement of first-order logic, but nowadays the expression «algebraic logic» is used to include both the zero and the first-order levels.




[Foot Note 1_30]

I was therefore jointly presenting two different tendencies, Bourbaki and universal algebra, which historically, for some odd reasons, have been conflicting.




[Foot Note 1_31]

I prefer the terminology «sequent system» than «sequent calculus», because a sequent calculus is not necessarily a calculus, in the algorithmic sense, if it is undecidable. More generally, I think that the word «calculus» in logic is inappropriate. It suggests that logic'algorithm, a thesis dismissed by the fall of Hilbert's program.




[Foot Note 1_32]

Apart of Wójcicki's work on non-monotonic logics, there are some works by G. Malinowski where the axioms for the consequence operator are weakened (Malinowski 1990).




[Foot Note 1_33]

These two papers were presented respectively at the 27th International Symposium on Multiple-Valued Logic (Antigonish, Canada, May,1997) and at the First World Congress on Paraconsistency (Ghent, Belgium, July 1997). My researches on many-valued logic started with a discussion with da Costa and O. A. Bueno (B 96c) about (Malinowski 1993).




[Foot Note 1_34]

This nice title was suggested to us by Michel Paty.




[Foot Note 1_35]

N. C. A. da Costa since more than ten years has started to work on the connections between logic and physics, logic and biology, logic and economy, etc. (see da Costa 1997).




[Foot Note 1_36]

I found this logic by studying the paracomplete dual of C1 and mentioned it in my math PhD (B 95e).




[Foot Note 1_37]

The translation problem was not eschewed by the Polish school, people such as Wójcicki worked on it and Suszko and his collaborators were probably the first to work on a «category of logics». In Brazil, the logic group of Campinas has few years ago taken this subject as a main subject research (see Carnielli/D'Ottaviano 1997).




[Foot Note 1_38]

In order to get an intuitive idea about the paraconsistent negation of C1 I worked with diagrams (B 98d).




[Foot Note 1_39]

confusions have proliferated recently, see e. g. (Dunn/Hardegree 200?). The expression «substructural» was put forward by people working within a proof-theoretical framework with probably very few knowledge of Polish logic in which the expression «structural» is used since many years with a totally different meaning. On the other hand, Gentzen's work was not well-known in Poland and people were no aware that Genzten already used the expression «structural rule» in a different context.




[Foot Note 1_40]

Presented as one chapter in (Gabbay 1994) and fully developed in (Gabbay 1996).




[Foot Note 1_41]

A general abstract semantical approach can already be found in (van Fraasen 1973). A less abstract semantical approach is related to «Abstract model theory» which includes such results as Lindströms theorem, see e. g. (Barwise 1974).




[Foot Note 1_42]

Curry was already using this expression: he wrote a book entitled Foundations of mathematical logic (Curry 1963), which was a kind of augmented version of (Curry 1952) which, as we have seen, also bears a prophetic title.




[Foot Note 1_43]

On this question see (B 93f).




[Foot Note 1_44]

Unfortunately this name is not very used outside Poland.




[Foot Note 1_45]

Suszko liked to say that «abstract mathematics can be a genuine philosophy». His ideas about philosophy of logic are similar to the one defended here, cf. (B 00e). In this paper we show how the mathematical concepts developed by the Polish school of logic can be a basis for a new approach in philosophy of logic, less formal or symbolic in style, but conceptually more mathematical.




[Foot Note 1_46]

Nor a blend of paraconsitency and relevancy, or any other system which will play the role of a «universal» system; see (B 99d).




[Foot Note 1_47]

On Vasiliev see (Bazhanov 1990) and (Arruda 1990).




[Foot Note 1_48]

One could think that it will be a good idea to call «metalogic» what we have called «universal logic», but on the one hand the suffix «meta» has different meanings and has been already used in such expression as «metaphysics» and «metamathematics» with a meaning not corresponding to our intention, on the other hand the expression «metalogic» is already used and has already been used in various different ways. In fact one can find it, even before Vasiliev, but with a similar meaning in Schopenhauer. On this question see (B 92) and (B 93b).




[Foot Note 2_1]

«Time is the dimension of movement in its before-and-afterness, and is continuous (because movement is so)» (Aristotle, Physics 4. 11. 220a25 ff., trans. P. Wicksteed and F. Cornford [Cambridge, Mass.: Harvard University Press; London: W. Heinemann ltd, 1980], vol. I, p. 395).




[Foot Note 2_2]

«Now some attributes or modes are in the very things of which they are said to be attributes or modes, while others are only in our thought. For example, when time is distinguished from duration taken in the general sense and called the measure of movement, it is simply a mode of thought. For the duration which we understand to be involved in movement is certainly no different from the duration involved in things which do not move. This is clear from the fact that if there are two bodies moving for an hour, one slowly and the other quickly, we do not reckon the amount of time to be greater in the latter case than the former, even though the amount of movement may be much greater. But in order to measure the duration of all things, we compare their duration with the duration of the greatest and most regular motions which give rise to years and days, and we call this duration «time». Yet nothing is theeby added to duration, taken in its general sense, except for a mode of thought» (Descartes, Principles of Philosophy 57, in The Philosophical Writings of Descartes, vol. I, trans. J. Cottingham, R. Stoothoff, and D. Murdoch [New York: Cambridge University Press, 1985], p. 212).




[Foot Note 2_3]

For example in his De corpore: «As a body leaves a phantasm of its magnitude in the mind, so also a moved body leaves a phantasm of its motion, namely an idea of that body passing out of one space into another by continual succession. And this idea, or phantasm, is that, which (without receding much from the common opinion, or from Aristotle's definition) I call Time. [...] A complete definition of time is such as this, TIME is the phantasm of before and after in motion; which agrees with the definition of Aristotle, time is the number of motion according to former and later; and time is a phantasm of motion numbered. But that other definition, time is the measure of motion, is not so exact, for we measure time by motion, and not motion by time» (Thomas Hobbes, De corpore [Concerning body], in The English Works of Thomas Hobbes, vol. I, collected and ed. Sir William Molesworth in 1839 [London: Scientia Aalen, 1962],II. vii. 3, p. 95).




[Foot Note 2_4]

«In reality the decrees could not have been separated from God: he is not prior to them or distinct from them, nor could he have existed without them. So it is clear enough how God accomplishes all things in a single act» (Descartes, Conversations with Burman 50, trans. and intro. John Cottingham [Oxford: Clarendon Press, 1976]). «In God, willing, understanding and creating are all the same thing without one being prior to the other even conceptually» (Descartes, Letter to Mersenne, 27 May 1630, in The Philosophical Writings of Descartes vol. III, trans. J. Cottingham, R. Stoothoff, and D. Murdoch [Cambridge, UK: Cambridge University Press, 1985]). «I do not see why God should not have been able to create something from eternity. Since God possessed his power from all etrnity, I do not see any reason why he should not have been able to exercise it from all eternity» (Descartes' Conversation with Burman 23,p. 15. The reasons why Descartes eliminates the temporal distance between God and his creation (i. e. the priority in time of the former) seems to be analogous to the elimination of that distance in the case of God's creation of himself. Arguing for the latter, Descartes writes: «I think it is necessary to show that, in between «efficient cause» in the strict sense and «no cause at all», there is a third possibility, namely «the positive essence of a thing», to which the concept of an efficient cause can be extended. [...] I thought I explained this in the best way available to me when I said that in this context the meaning of «efficient cause» must not be restricted t o causes which are prior in time to their effectsor different from them. For, first, this would make the question trivial, since everyone knows that something cannot be prior to, or distinct from, itself; and secondly, the restriction «prior in time» can be deleted from the concept while leaving the notion of an efficient cause intact.» (Fourth Set of Replies 239-40, in Philosophical Writings II pp. 167-68). And also: «The answer to the question why God exists should be given not in terms of an efficient cause in the strict sense, but simply in terms of the essence or formal cause of the thing. And precisely because in the case of God there is no distinction between existence and essence, the formal cause will be strongly analogous to an efficient cause, and hence can be called something close to an efficient cause. (Fourth set of Replies 243, in Philosophical Writings II, p. 169).




[Foot Note 2_5]

Leibniz, New Essays on Human Understanding, trans. and ed. Peter Remnant and Jonathan Bennett (New York: Cambridge University Press, 1996), p. 216.




[Foot Note 2_6]

Leibniz's third letter to Clarke 17, in H. G. Alexander, ed., The Leibniz-Clarke Correspondence (Manchester: University of Manchester Press, 1956), p. 30. I must confess that I do not understand the use of certain punctuation signs in this translation, in particular the use of commas, semi-colons and the colons. If we follow them strictly the reading of the book, which by itself is not specially difficult, sometimes becomes unintelligible. I will not, however, modify the punctuation given in Alexander's edition.




[Foot Note 2_7]

Leibniz's fourth paper to Clarke 44, in ibid., p. 43.




[Foot Note 2_8]

Ibid.




[Foot Note 2_9]

«I certainly grant you can imagine that the world is eternal. However, since you assume only a succession of states, and since no reason for the world can be found in any one of them whatsoever (indeed, assuming as many of them as you like won't in any way help you to find a reason), it is obvious that the reason must be found elsewhere. For in eternal things, even if there is no cause, we must still understand there to be a reason. In things that persist, the reason is the nature or essence itself, and in a series of changeable things (if, a priori, we imagine it to be eternal), the reason would be the superior strength of certain inclinations, as we shall soon see, where the reasons don't necessitate (with absolute or metaphysical necessity, where the contrary implies a contradiction) but incline. From this it follows that even if we assume the eternity of the world, we cannot escape the ultimate and extramundane reason for things, God» (Leibniz, «On the Ultimate Origination of Things,» op. cit., p. 150).




[Foot Note 2_10]

Leibniz, «On the Radical Origination of Things,» in Philosophical Papers and Letters, vol. II, selec., trans., and ed. Leroy E. Loemker (Chicago: University of Chicago Press, 1956), p. 792.




[Foot Note 2_11]

For example, in Leibniz, Monadology 87 and 89, trans. N. Rescher (Pittsburgh: University of Pittsburgh Press, 1991), p. 29.




[Foot Note 2_12]

«Once having assumed that being involves,ore perfection than nonbeing, or that there is a reason why something should come to exist rather than nothing, or that a transition from possibility to actuality must take place, it follows that, even if there is no further determining principle, there does exist the greatest amount possible in proportion to the given capacity of time and space (or the possible order of existence), in much the same way as tiles are laid so that as many as possible are contained in a given space» (Leibniz, «On the Radical Origination of Things,» p. 792).




[Foot Note 2_13]

«We can now understand in a wonderful way how a kind of divine mathematics or metaphysical mechanism is used in the origin of things and how the determination of the maximum takes place. So the right angle is the determined one of all angles in geometry, and so liquids placed in a different medium compose themselves in the most spacious figure, a sphere. But besr of all is the example in ordinarymechanics itself; when many heavy bodies pull upon each other, the resulting motion is such that the maximum possible total descent is secured. For just as all possibilities tend with equal right to existence in proportion to their reality, so all heavy objects tend to descend with equal right in proportion to their weight. And just as, in the latter case, that motion is produced which involves the greatest possible descent of these weights, so in the former a world is produced in which a maximum production of possible things takes place» (ibid., p. 792).




[Foot Note 2_14]

«[...] the world is not only the most perfect naturally or, if you prefer, metaphysically --in other words, that that series of things has been produced which actually presents the greatest amount of reality --but also that it is the most perfect morally, because moral perfection is truly natural in minds themselves. Hence the world not only is the most wonderful mechanism but is also, insofar as it consists of minds, the best commonwealth, through which there is conferred on minds as much felicity or joy as possible; it is in this that their natural perfection consists» (ibid., pp. 794-795). «As we have already established a perfect harmony between two natural realms, the one of efficient and the other of final causes, we must here also recognize a further harmony between the physical realm of nature and the moral realm of grace, that is, between God considered an architect of the mechanism of the universe, and God considered a monarch of the divine city of spirits» (Monadology 87, op. cit., p. 28).




[Foot Note 2_15]

H. G. Alexander, ed., The Leibniz-Clarke Correspondence (Manchester: University of Manchester Press, 1956), 55-59 & 106. I must confess that I do not understand the use of certain punctuation signs in this translation, in particular the use of the commas, the semi-colons and the colons. If we follow them strictly, the reading of the book, which by itself is not specially difficult, sometimes becomes unintelligible.




[Foot Note 2_16]

Ibid., p. 75.




[Foot Note 2_17]

Ibid., p. 75.




[Foot Note 2_18]

Ibid., p. 75.




[Foot Note 2_19]

«Supposing any one should ask, why God did not create every thing a year sooner; and the same person should infer from thence, that God has done something, concerning which «tis not possible there should be a reason, why he did it so, and not otherwise: the answer is, that his inference would be right, if time was anything distinct from things existing in time. For it would be impossible there should be any reason, why things should be applied to such particular instants, rather than to others, their succession continuing the same» (Leibniz's third paper to Clarke 6, in Leibniz-Clarke Correspondence, op. cit., pp. 26-27).




[Foot Note 2_20]

Leibniz's fifth paper 50, in op. cit., p. 73: «If the reality of space and time, is necessary to the immensity and eternity of God; if God must be in space; if being in space, is a property of God; he will in some measure, depend upon time and space, and stand in need of them.




[Foot Note 2_21]

Ibid. 36, p. 66.




[Foot Note 2_22]

Leibniz's fifth paper to Clarke 49, in op. cit., p. 72.




[Foot Note 2_23]

Ibid. 26, p. 62. And a clarification must be added: «When I deny that there are two drops of water perfectly alike, or any two other bodies indiscernible from each other; I don't say, «tis absolutely impossible to suppose them; but that «tis a thing contrary to the divine wisdom, and which consequently does not exist» (ibid. 25, p. 62).




[Foot Note 2_24]

Ibid. 60, p. 77.




[Foot Note 2_25]

«One cannot say [...] that the wisdom of God may have good reasons to create this world at such or such a particular time: that particular time, considered without the things, being an impossible fiction; and good reasons for a choice, being not to be found, where everything is indiscernible» (Leibniz's fifth paper to Clarke 58, pp. 76-77).




[Foot Note 2_26]

Leibniz's fourth paper to Clarke 41, in ibid., p. 42.




[Foot Note 2_27]

Ibid. 27, p. 63.




[Foot Note 2_28]

Leibniz, «On the Radical Origination of Things,» op. cit., p. 791.




[Foot Note 2_29]

«For it must be that, if there is a reality in essences or possibilities, or indeed in eternal truths, this reality be founded in something existent and actual, and consequently in the existence of the Necessary Being, in whom essence includes existence, or in whom being possible suffices for being actual» (Monadology 44, op. cit., p. 22).




[Foot Note 2_30]

Leibniz's fifth paper to Clarke 57, p. 76.




[Foot Note 2_31]

Ibid. 33, op. cit., p. 64.




[Foot Note 2_32]

Ibid., p. 27.




[Foot Note 2_33]

Leibniz, Letter to Louis Bourguet, August 5, 1715, in Philosophical Papers and Letters, vol II, op. cit., p. 1079.




[Foot Note 2_34]

Ibid., p. 1080.




[Foot Note 2_35]

Leibniz's fifth paper to Clarke, p. 76.




[Foot Note 2_36]

Ibid., pp. 76-77.




[Foot Note 2_37]

Ibid., p. 61.




[Foot Note 2_38]

Ibid., p. 96. This is the whole fragment in defense of the principle of sufficient reason and, in particular, against Newton's absolute time, which Leibniz places at the end of the Fifth Paper:

He pretended, that I have been guilty of a petitio principii. But, of what principle, I beseech you? Would to God, less clear principles had never been laid down. The principle in question, is the principle of the want of sufficient reason; in order to any thing's existing, in order to any event's happening, in order to any truth's taking place. Is this a principle, that wants to be proved? [...] -- I dare say, that without this great principle, one cannot prove the existence of God, nor account for many other important truths. -- Has not everybody made use of this principle, upon a thousand occasions? 'Tis true, it has been neglected, out of carelessness, on many occasions: but that neglect, has been the true cause of chimeras; such as are (for instance,) an absolute real time or space, a vacuum, atoms, attraction in the scholastic sense, a physical influence of the soul over the body, [and of the body over the soul] and a thousand other fictions, either derived from erroneous opinions of the ancients, or lately invented by modern philosophers. [...] -- To deny this great principle, is likewise to do as Epicurus did; who was reduced to deny that other great principle, viz. the principle of contradiction; which is, that every intelligible enunciation must be either true, or false. (The Correspondence..., pp. 95-96).




[Foot Note 2_39]

Ibid., 77.




[Foot Note 2_40]

That God made a decision is clear, for example, in the following passage: «Since the designs of God which concern this whole universe are all interrelated in conformity with his sovereign wisdom, he made no decision about Adam without taking into consideration everything which has any connection with him. It is therefore not because of the decision made about Adam but because of the decision made at the same time about everything else (to which the decision made about Adam involves a perfect relation) that God makes up his mind about all human events. There seems to be no fatal necessity in thiis, or anything contrary to God's freedom, any more than in the generally accepted hypothetical necessity to which God himself is subject, of carrying out what he has resolved» (Letterr to Arnauld, July 14, 1686, in Leibniz, Philosophical Papers and Letters, 2nd ed., sel., trans., and intro. Leroy E. Loemker [Dordrecht, Holland: D. Reidel Publishing Company, 1956, 1970], pp. 331-32).




[Foot Note 2_41]

Leibniz, «On the Radical Origination of Things,» in Philosophical Papers and Letters, op. cit., p. 794. (My italics)




[Foot Note 2_42]

Leibniz's fifth paper to Clarke 50, in op. cit., p. 73: «If the reality of space and time, is necessary to the immensity and eternity of God; if God must be in space; if being in space, is a property of God; he will in some measure, depend upon time and space, and stand in need of them.




[Foot Note 2_43]

«Beyond the world, that is, beyond the collection of finite things, there is some One Being who rules, not only as the soul is the ruler in me, or, better, as the shelf is the ruler in my body, but also in a much higher sense. For the One Being who rules the universe not only rules the world, but also fashions or creates it; he is above the world, and, so to speak, extramundane, and therefore he is the ultimate reason for things» (Leibniz, «On the Ultimate Origination of Things,» 23 November 1697, in Philosophical Essays, ed. and trans. R. Ariew and D. Garber [Indianapolis, IN: Hackett, 1989], p. 149). [I have used two different editions of «The Ultimate Origination» for the only reason that I discovered some pages missing in the copy I was using.]




[Foot Note 2_44]

Leibniz's third paper 15, op. cit., p. 29.




[Foot Note 2_45]

This becomes quite clear, for example, a few lines earlier in the same letter to Clarke and it is not very different, at least in its general formulation, from the common view which defends the presrence of God in the world: «God is not present to things by situation, but by essence: his presence is manifested by his immediate operation» (Leibniz's third paper to Clarke 12, op. cit., p. 28). And later some specification of that «presence»: «I never gave any occasion to doubt, but that God's conservation is an actual preservation and continuation of the beings, powers, orders, dispositions, and motions of all things [...]» (ibid. 16, p. 29).




[Foot Note 2_46]

Ibid., 90.




[Foot Note 2_47]

Leibniz's fifth paper 49, pp. 72-73.




[Foot Note 2_48]

Leibniz's fourth paper to Clarke 41, p. 42.




[Foot Note 2_49]

Augustine deals with the problem of a possible earlier creation and God's decision, mainly, in three books: The City of God (XI,5,6); Two Books on Genesis Against the Manichees (I,2) and Confessions (XI).




[Foot Note 2_50]

The City of God, trans. M. Dods (New York: Random House, 1950), p. 349.




[Foot Note 2_51]

Ibid., p. 349.




[Foot Note 2_52]

Ibid., p. 349. Agustine continues: «For here the question is with those who, with ourselves, believe that God is spiritual, and the creator of all existences but Himself. As for others, it is a condescension to dispute with them on a religious question, for they have acquired a reputation only among men who pay divine honours to a number of gods, and have become conspicuous among the other philosophers for no other reason than that, though they are still far from the truth, they are near it in comparison with the rest. While these, then, neither confine in any place, nor limit, nor distribute the divine substance, but, as is worthy of God, own it to be wholly though spiritually present everywhere, will they perchance say that this substance is absent from such immense spaces outside the world and is occupied in one only, (and that a very little one compared with the infinity beyond,) the one, namely, in which is the world? I think they will not proceed to this absurdity» (ibid., 349).




[Foot Note 2_53]

Ibid., p. 350.




[Foot Note 2_54]

Ibid. The «lack of merit of the chosen spot» was precisely the reason Leibniz found to deny even that God could have a preference for a particular point in time to place the world at.




[Foot Note 2_55]

Ibid.




[Foot Note 2_56]

Ibid., p. 350.




[Foot Note 2_57]

Leibniz saw in the indiscernibility of two moments in time before the world was created the argument to deny the existence of a sufficient reason in God to create the world at a particular moment.




[Foot Note 2_58]

Ibid.




[Foot Note 2_59]

This reference to the Scriptures here is a little bit surprising. Instead of it, it is expected his claim that time is dependent on movement (as we have just read). By referring to the sacred text, Augustine 1)could be telling that the Scriptures have more authority than his thought (when both show the same truth) or 2) he could be expressing problems in making compatible his «time does not exist without some movement» and his idea of time as distentio animi. By doing the latter he would be, at the same time, providing one more of the many texts where scholars have found difficulties in reaching an agreement on the «degree of dependence" of time on movement.




[Foot Note 2_60]

Ibid., p. 350.




[Foot Note 2_61]

«Two Books on Genesis Against the Manichees» in Saint Augustine on Genesis, trans. Roland J. Teske (Washington, D. C.: The Catholic University of America Press, 1991): pp. 45-141, p. 49.




[Foot Note 2_62]

Ibid., p. 49.




[Foot Note 2_63]

Augustine, Confessions, XI, 14, trans. E. B. Pusey (New York: Dutton; London: Dent, c1907, 1966), p. 262.




[Foot Note 2_64]

Augustine, Two Books on Genesis Against the Manichees, I,2,3, op. cit., p. 50.




[Foot Note 2_65]

John F. Callahan, in his Four Views of Time in Ancient Philosophy (Cambridge,Mass.: Harvard University Press, 1948) is more interested in the concept of time itself and in this regard, like an introduction, he simply paraphrases Augustine without any further reflection: «When there was no time, therefore, St. Augustine insists, God had not made anything, because He is the author of time itself. No expanse of time is co-eternal with God, because He persists in eternity, while time by its very nature cannot persist in the present» (p. 150).




[Foot Note 2_66]

«Thou art the same, and Thy years fail not. Thy years neither come nor go; whereas ours both come and go [...] Thy years are one day; and Thy day is not daily, but To-day, seeing Thy To-day gives not place unto to-morrow» (Confessions, I,13, pp. 261-62).




[Foot Note 2_67]

All these problems are enclosed in Augustine's words «God, in whose eternity is no change at all, is the Creator and Ordainer of time» (The City of God XI,6, trans. Marcus Dods; Intr. Thomas Merton [New York: The Modern Library, 1950], p. 350).




[Foot Note 2_68]

Against the Manichees, p. 50.




[Foot Note 2_69]

An even clearer expression of this contradiction can be found in Two Books on Genesis Against the Manichees (Bk. I,2): «God who is the maker of time is before time» (op. cit., p. 50).




[Foot Note 2_70]

Ibid., p. 50.




[Foot Note 2_71]

Ibid., pp. 51-52.




[Foot Note 2_72]

Ibid., p. 51.




[Foot Note 2_73]

Augustine, Confessions, I. 13, trans. E. B. Pusey (New York: Dutton; London: Dent, 1966), pp. 261-62.




[Foot Note 2_74]

Confessions 1,14, p. 262.




[Foot Note 2_75]

Cf. Aristotle, Physics (IV, x-xiv). The Greek philosopher presented exactly the same doubts as part of the «opinion of those who do not believe in the existence of time.» It could have perfectly been a parallel thought since Augustine may not have known of Physics. This book was not known in the West until the first half of twelfth century through different Latin translations.




[Foot Note 2_76]

This is the main difference between time as «the number of motion» (Aristotle) and Augustine's «distention of the man's soul».




[Foot Note 2_77]

See De Genesi, 5,5,12; Confessions 12,8; and Civitas Dei, 11,6.




[Foot Note 2_78]

Augustine, Confessions, 11,14-30. See John F. Callaghan, Four Views of Time in Ancient Philosophy (Cambridge: Harvard University Press, 1948), 149-87; and Étienne Gilson, The Christian Philosophy of Saint Augustine, trans. L. E. m. Lynch (New York: Random House, 1960), 193-95.




[Foot Note 2_79]

Augustine, Confessions, p. 273.




[Foot Note 2_80]

Bertrand Russell, A History of Western Philosophy (New York: Simon and Schuster, 1945), p. 353.




[Foot Note 2_81]

Wetzel is one witness of this disagreement: «Interpreters disagree over whether Augustine offered a definitive answer to the question of time's nature and even more so over whether any putative answer of his stands up to scrutiny» (James Wetzel, «Time After Augustine,» Religious Studies 31, 3 [September 1995]: 341). Wetzel himself thinks that «It is misleading at best, but more likely mistaken, to think of Augustinian inwardness as a retreat to subjectivity. If time is in the mind, as Augustine seems to conclude, it does not follow that time must be an item in a mental inventory, or an artifact of immaterial creativity. / Just what does follow, on the other hand, is not easy to determine» (ibid., p. 341). In that article he opposes Russell's view of Augustine's thought on time to Wittgenstein's. The latter's view, in Wetzel's words, can be summarized like this: «The answer he supposes Augustine to suggest to the philosophical question of time is not a piece of speculation, a subjective theory of time, but a life returned to time, secured from its original estrangement» (Ibid., p. 343).




[Foot Note 3_1]

Here I want to ignore any worries concerning what would count as a subject's intending to play one language game rather than another.




[Foot Note 3_2]

Actually, it is Davidson's theory of mental states that is a token/token identity theory, not his theory of meaning. Davidson is something of a nihilist about meaning given that he holds, with Quine, that what a speaker means by an utterance will be ineluctably indeterminate. See Davidson, 1984.




[Foot Note 3_3]

Here, I want to count such things as fields and energy as physical whether or not they count as physical substances.




[Foot Note 3_4]

It is easy to see how this would go with the meaning of a mathematical symbol such as `+', much harder with ordinary material object terms such as `cat', but, so as not to beg the question against meaning functionalism, I will suppose that there is no in principle impossibility in providing such a reduction.




[Foot Note 3_5]

Actually, if physicalism is not a necessary condition for naturalism, I am not entirely sure why this view should not count as a form of meaning naturalism.




[Foot Note 4_1]

Frankfurt, Harry G., The Importance of What we Care About, (New York: Cambridge University Press, 1988), 95-103. Cited in text hereafter as «(Frankfurt, x)».




[Foot Note 4_2]

Van Inwagen, Peter, «Ability and Responsibility», Philosophical Review, 87 (1978), 201-224.




[Foot Note 5_1]

Or `plus'.




[Foot Note 5_2]

See also Kripke 1982, pp. 38-9, where it is explicitly stated that meaning skepticism is not merely epistemological. I take this to mean that the skeptical problem about meaning and intention is also a constitutive one.




[Foot Note 5_3]

For example, Paul Boghossian (1989, p. 515), Ruth Millikan (1990) and Barry Smith (1998).




[Foot Note 5_4]

In Kripke 1982, pp. 41-51.




[Foot Note 5_5]

To say that self-knowledge of meaning an intention is not objective is simply to claim that the subject lacks the objective criteria for attribution of such a knowledge to himself.




[Foot Note 5_6]

Kripke 1982, pp. 34-5, 37.




[Foot Note 5_7]

Soames notes that the counterfactual should be spelled out without the use of any intentional notions if we are to avoid smuggling into the disposition any problematic concept involving content, and therefore opening ourselves to a similar skeptical challenge concerning the new concept. I take it that the above counterfactual is free from this objection, but even if it was not entirely satisfactory, we could always reformulate it so as to bring it into line with Soames' desideratum.




[Foot Note 5_8]

Kripke 1982, p. 37.




[Foot Note 5_9]

Kripke 1982, p. 28.




[Foot Note 5_10]

Concerning this, see also note 7.




[Foot Note 5_11]

Soames 1997, p. 216: n. 6. The reason for this constraint is given in note 7.




[Foot Note 5_12]

This is the kind of non-intentional meaning-determining fact suggested by Soames. We will be elaborating on these facts in section 6.




[Foot Note 5_13]

Soames 1997, p. 232. The passage is slightly modified for stylistic reasons but its content remains unchanged.




[Foot Note 5_14]

Soames 1997, pp. 223-4.




[Foot Note 5_15]

Soames 1997, pp. 220-1.




[Foot Note 5_16]

Soames 1997, pp. 230-1.




[Foot Note 5_17]

Soames 1997, p. 227.




[Foot Note 5_18]

In section 6, I will discuss the plausibility of the thesis that the supervenience of one kind of fact upon another implies the relation of necessary consequence between a description of the latter fact and a description of the former.




[Foot Note 5_19]

Here Soames appeals to Kripke himself, part of whose seminal work was dedicated precisely to show that necessary and a priori are not equivalent notions (Kripke 1980). Applied to the case in question, this implies that there may be necessary consequences of a proposition p (which is made true by the dispositional fact cited in P1) which are not a priori consequences of p (these consequences are made true by the meaning fact corresponding to the dispositional fact in question).




[Foot Note 5_20]

See section 3.




[Foot Note 5_21]

See the last paragraph of section 3 and note 7.




[Foot Note 5_22]

See section 3.




[Foot Note 5_23]

Soames' overall position is actually more complex than I have been portraying it so far.

Besides thinking that properly construed dispositions are apt to determine what S means by `+', Soames also believes that S's beliefs and intentions towards `+' are adequately explanatory of his meaning (Soames 1998, pp. 335-38). However, as he does not take the skeptic to be demanding an explanation for our linguistic competence but only asking for the non-intentional basis of meaning and content in general, the part about the explanatory role of beliefs and intentions for mental states like meaning is irrelevant for his strategy to deal with the skeptical problem.




[Foot Note 5_24]

The same point comes expressed in different ways in various passages of Wittgenstein's later remarks. For example, concerning the question of whether our access to the linguistic rules we follow is mediated by an intuition (a kind of inner voice), he says: «how can we tell how a rule which has been used for fourteen steps applies at the fifteenth? (...) We might as well say that we need, not an intuition at each step, but a decision. -- Actually there is neither. You don't make a decision: you do a certain thing. It is a question of a certain practice» (Diamond 1975, p. 237). See also Wittgenstein 1953, par. 217.




[Foot Note 5_25]

I discuss Wittgenstein's position about self-knowledge of meaning and linguistic intention in chapter 3 of my PhD dissertation (Pinto 1998).




[Foot Note 5_26]

The topic of the interpreter, or foreign explorer, who goes to a foreign land and there finds a tribe speaking a language he knows nothing about is recurring in Wittgenstein's later writings. See, for example, Wittgenstein 1953, pars. 205-207. According to him, such an observer may come to a stage where he will be able to interpret the members of the tribe as speaking a certain language if some conditions are satisfied. This means that the interpreter will be able to justify their use of words by appeal to the norms of their language which he will have eventually discovered if the method of interpretation succeeds. The topic is too vast to be discussed here.




[Foot Note 5_27]

For example, in Davidson 1984a and Davidson 1987.




[Foot Note 5_28]

Elsewhere, Soames seems to agree with this pre-theoretic intuition and with Wittgenstein and Davidson (see, for example, Soames 1997, footnote 25, p. 241). If this represented what he really thinks, then Soames should be agreeing with me on the inadequacy of NE to represent first-person knowledge of meaning. Yet, I suspect he would prefer to regard such inadequacy as another reason to reject the framework of the skeptical problem rather than to abandon his parsing of the problem as inappropriate.




[Foot Note 5_29]

Soames 1997, p. 229.




[Foot Note 5_30]

Henceforth ED-fact, for enhanced dispositional fact.




[Foot Note 5_31]

M-facts, for short.




[Foot Note 5_32]

This is the characterisation of supervenience that one can extract, for example, from Davidson 1970, p. 214; Davidson 1973, pp. 253-4; Fodor 1987, pp. 30-33; Kim 1978, pp. 152-154; Stich 1978, pp. 346-348; as well as from many other authors.




[Foot Note 5_33]

See, for example, Davidson 1970; Davidson 1973; Davidson 1980a and Davidson 1995.




[Foot Note 5_34]

Davidson 1984b.




[Foot Note 5_35]

This is explicit, for example, in Davidson 1995, pp. 4-6.




[Foot Note 5_36]

This result is proved in Kim 1978, pp. 152-54. It is also mentioned in Davidson 1973, p. 249.