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SORITES, ISSN 1135-1349

Issue #04. February 1996. Pp. 21-35.

«Logic and Necessary Being»

Copyright (C) by SORITES and Matthew McKeon
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Logic and Necessary Being

<27>Foot note 2_1

Matthew McKeon

Sect.1. Introduction

In a recent publication,<28>Foot note 2_2 Yuval Steinitz argues that since (1) it is logically possible that there are logically necessary beings, it follows that (2) there is at least one logically necessary being. Steinitz switches the Leibnitzean ontological argument's concern from perfect beings to logically necessary beings because he thinks that it is easier to establish the logical possibility of a logically necessary being than the logical possibility of a perfect being (at least as traditionally understood as omnipotent, omniscient, perfectly good, etc...), and, therefore, easier to establish the soundness of his argument than the soundness of a Leibnitzean ontological argument. Steinitz's justification of the soundness of his argument (A) is based on Richard Swinburne's account of logical necessity presented in, The Coherence of Theism.<29>Foot note 2_3

This paper has two primary aims. First, I seek to highlight what I view as the nature of the justification for thinking that (A) is valid. I do not offer a complete treatment of the logic of logical necessity and possibility in first-order logic, but I hope to show that Steinitz's quick treatment is insufficient to establish that (A) is valid. In particular, it is not clear that Swinburne's account of logical necessity grounds the validity of (A). Secondly, I attempt to show that the correct account of logical possibility makes (A)'s premise false. In pursuing both aims, I expose the inadequacies of Swinburne's account of the nature of logical necessity.

Sect.2. The Validity of Steinitz's Argument

In order to assess the validity of (A), it is standard to identify its logical form by translating (1) and (2) into sentences of a well-defined formal language. Since these sentences contain modal terms, we appeal to a first-order modal language which is a standard first-order language supplemented with the operators `M' and `L' for `possibly' and `necessarily' respectively. Consider the following plausible construal of the logical structure of (A), (A'): (1') M(Ex)L(Ey)(x=y) |- (2') (Ex)L(Ey)(x=y).

It is standard to interpret necessary truth as truth at all possible worlds, and possible truth as truth at some possible world. In this way, the modal operators are understood as quantifiers over possible worlds. The truth or falsity of a modal claim on some interpretation of it is understood as its truth or falsity at some possible world. Furthermore, `necessarily' and `possibly' are defined in terms of the relative possibility (or accessibility) relation as follows: `Mp' is true at some world if and only if (iff) there exists at least one world w' such that w' is possible relative to w and `p' is true at w'; and `Lp' is true at w iff for every world w', if w' is possible relative to w, then `p' is true at w'.<30>Foot note 2_4

Different senses of necessity warrant different restrictions on the relative possibility relation, because the possible worlds relevant to assessing the truth value of `Lp' and `Mp' will vary on different senses of `L' and `M'. For example, a historian of religion claims, «Biologically speaking, Mary can't be a mother while a virgin at the same time.»<31>Foot note 2_5 Presumably, she means that Mary can't be a mother while a virgin, given the laws of biology. Parsing this in a modal language we get: `~(Mary is a mother and a virgin)' is true at all worlds at which the laws of biology hold. Worlds at which these laws fail are irrelevant to assessing the truth value of the claim that Mary can't be a mother while a virgin. So the relative possibility relation serves to restrict the range of `L' in `L~(Mary is a mother and a virgin)' to those worlds at which the laws of biology hold. It is this subset of the totality of possible worlds that is relevant to assessing the truth value of the historian's claim (i.e., her claim is false if it is true that Mary is a mother while a virgin at one of these worlds). In general, the relative possibility relation serves to restrict the domain of the modal operators to those worlds relevant to assessing the truth value of the sentences within their scope.

If the relative possibility relation has a certain structure, then it can happen that an object necessarily exists from the point of view of one world, but not from others. In order for, say, `It is possible that Al Gore not exist' to be true, there must exist a world w possible relative to this one at which `Al Gore exists' is false. `It is necessary that Al Gore exists' is true iff `Al Gore exists' is true at all worlds possible relative to the real world. Obviously, it can't be true at a world that Gore exists contingently (i.e., it is not necessary that Gore exists) and that it is necessary (in the same sense) that he exist. However, this doesn't imply that `Al Gore exists contingently' and `Al Gore necessarily exists' can't each be true at distinct worlds that are not possible relative to one another.

For example, suppose that the species of possibility at work in the above sentences is metaphysical possibility. So, in evaluating the truth of sentences at a world w, we consider only those worlds that are metaphysically possible from the point of view of w. Consider a Leibnitzean view of metaphysical possibility. That is, suppose a strong form of metaphysical determinism is true, so that for any constellation of objects and initial conditions there is just one way things can go. Still, different choices of objects and conditions are, we imagine, possible for God. The universe of metaphysically possible worlds is, then, a set of worlds partitioned into equivalence classes -- each class is made up of a single world, and from the point of view of any one world w, the only world metaphysically possible relative to w, is w itself. Al Gore necessarily exists from the point of view of our world w, but there is another world -- impossible (in a metaphysical sense) from the point of view of w -- in which Al Gore's non-existence is metaphysically necessary, because Gore doesn't exist there (God could have not created Gore). On this view, the fact that an object necessarily exists from the point of view of one world, does not entail that it exists at each world.

Since the modal operators in (A') are understood to be operators for logical possibility and logical necessity, the validity of (A') turns on the nature of these modal notions. Steinitz's argues that (A) is valid because if we, «... assume that these necessary beings might not exist, that is, that their absence is only contingent, ,,, then it logically follows that they could also, in principle, contingently exist.»<32>Foot note 2_6 But it isn't clear that this does follow, because the content of the assumption «these necessary beings might not exist» is unclear. If we assume that an individual doesn't exist at our world w, and necessarily exists at a world w', then this makes 's existence at w' contingent only if w is possible relative to w'. But perhaps our world is impossible from the point of view of a world at which there exists a necessary being.

In fact, the validity of (A') turns on the rationale for thinking that the structure of the relative possibility relation on the totality of logically possible worlds is both symmetric and transitive. For if `M(Ex)L(Ey)(x=y)' is true at a world w, then `(Ex)L(Ey)(x=y)' is true at some w' possible relative to w, and so `L(Ey)(b=y)' is true at w' for some b in w'. Then, given that the relative possibility relation is symmetric, `(Ey)(b=y)' is true at w, so b exists in w. If `(Ex)L(Ey)(x=y)' were false at w, `(x)M~(Ey)(x=y)' would be true at w, and so `M~(Ey)(b=y)' would be true at w, so `~(Ey)(b=y)' would be true at some w» possible relative to from w. Given that the relative possibility relation is transitive, `(Ey)(b=y)' would be true at w», which is ridiculous.

This semantic proof establishes that it is logically impossible for (A')'s premise to be true while its conclusion is false on a conditional basis: if the structure of the relative possibility relation on the collection of logically possible worlds is both transitive and symmetric, then it is logically impossible for (A')'s premise to be true while its conclusion is false. Hence, this proof needs to be underwritten by some account of the collection of logically possible worlds in order to establish the validity of (A').<33>Foot note 2_7

Intuitively, logical truth is a species of necessary truth, i.e., if a sentence p is logically true, then it is impossible for p to be false. Two traditional ways of unpacking the modal notion are: (i) if a sentence p is logically true, then p remains true (at the real world) on all possible meaning assignments to the non-logical terms occurring in p; and (ii) if a sentence p is logically true, then there is no way the world could be which would make p false. In what follows, I first show that (A') is invalid in the totality of worlds generated by (i), and secondly, I show why it is unclear that the totality of possible worlds generated by (ii) makes (A') valid.

(i) is derived from an approach which defines logical truth in first-order logic as follows: a sentence p is logically true iff it remains true on all meaning assignments to the non-logical terms occurring in it. This approach understands a logically possible situation (or a logically possible world) as a meaning assignment in the (real) world.<34>Foot note 2_8 Meaning assignments to predicate letters, variables, and constants correlate sentences to subsets of the totality of individuals in the world in such a way as to make sentences true or false. We can establish that, say, `Bill and Hillary are married' is not logically true by imagining that `Bill' and `Hillary' refer to 2, and `married' to the less than relation. Following Lehmann, I will call it the possible meaning (PM) approach.<35>Foot note 2_9 Given that the domain of the world is denumerably infinite, this approach secures all of standard first-order logic.<36>Foot note 2_10 However, it has been pointed out that one consequence of the PM-treatment of logical truth is that if the world were finite, then more sentences would be logically true.<37>Foot note 2_11

For example, if Parmenides were correct and the world w contained exactly one thing, then (Ex)(y)(x=y) would be logically true, since there would be no meaning assignment which falsifies this sentence at w (such a meaning assignment requires a domain greater than one, and in the Parmenidean world there is only «The One»). On the PM approach , a logical truth at a world w is a sentence p for which there is no meaning assignment which falsifies it at w. So, what is logically true at a possible world w, turns on the cardinality of w's domain. Judgments about what is logically true are a posteriori (in sofar as the determination of the cardinality of the world's contents is a posteriori), as well as revisable (in sofar as this determination is revisable).

The PM approach generates the following totality of logically possible worlds.[38] Let's suppose that the totality of individuals in the real world is denumerably (countably) infinite. (1) A logically possible world is any sub-collection of individuals, properties and relations from the real world. (2) A logical law at a world is a first-order logical truth at that world; a sentence p is a logical truth at a world w iff p is true under all meaning assignments at w. A meaning assignment to p at a world w is a function that assigns to each individual constant and variable occurring in p an element of the domain of w; to each n-place predicate letter a set of n-tuples from that domain. The truth rules for the logical constants determine the truth values of logically compound sentences at a world w given the truth values of their atomic parts at w. (3) The relative possibility relation is defined in terms of (1) and (2): w' is possible relative to w iff each logical law at w is a logical law at w'. Then (4) the relative possibility relation is non-symmetric. For any two worlds w, w', if the domain of w is larger than the domain of w', there will be a sentence p which is a logical law at w' but not at w, and so w will not be possible relative to w'. But since all the laws of w are laws of w', w' will be possible relative to w. (5) The truth or falsity of modal claims at worlds is unpacked by (1) and (4) (e.g., `Mp' is true at w iff `p' is true at a world (given by (1)) which is possible relative to w (given by (4)). For example, `M~(Ex)(Ey)~(x=y)' is true at our world (reading 'M' as 'it is logically possible that'), because `~(Ex)(Ey)~(x=y)' is true at the parmenidean world w which contains just one thing; w is possible relative to our world because all logical laws at our world are logical laws at w. But since ~(Ex)(Ey)~(x=y) is a logical law at w and not at our world, the latter is not possible relative to the former.

(A') doesn't turn out to be valid on this view of logical truth, because the relative possibility relation is non-symmetric. Here is a countermodel for the argument. Suppose that a world w contains exactly one object a, and another world w' contains exactly two objects, each distinct from a. Then w is possible relative to w', but the only world possible relative to w is w itself. Then (1') is true at w' because `(Ex)L(Ey)(x=y)' is true at w. But (2') is false at w' since no member of its domain exists at w.

On the PM approach, sentences whose denials can only be true in infinite domains (e.g., ((x)~Txx&(x)(y)(z)((Txy&Tyz)->Txz))->(Ex)~(Ey)Txy ) turn out to be logically true at all finite worlds. Nevertheless, those sentences logically true at our world are necessary in a very strong sense: they are true at each world, since all worlds are possible relative to ours (assuming that there does exist a denumerably infinite totality). But for each world w/i/ containing less than the real world, there will be sentences logically true at w/i/ which are not logically true at each world.<39>Foot note 2_13

The main criticism of the PM approach to logical truth is that it generates a notion of logical necessity that is too weak.<40>Foot note 2_14 What is logically necessary should be true regardless of the empirical makeup of the world, and so what is logically true at a world w should not turn on what exists at w. An adequate account of logical necessity must reflect that (ii) if a sentence is logically true, then there is no way the world could be which would make p false. But the significance of this criticism rests on a clarification of the modal notion in (ii). How are we to understand the appeal to ways the world could be in a way that grounds the validity of (A')?

In his paper, Steinitz relies on Swinburne's account of logical truth.<41>Foot note 2_15 According to Swinburne, (iii) a statement p is logically true iff ~p is incoherent, where the meaning of the words occurring in p is fixed.<42>Foot note 2_16 On his view, to say that p is logically possible means that p is coherent.<43>Foot note 2_17 The notion of coherence is unpacked as follows. (iv) «A coherent statement is, I suggest, one for which it makes sense to suppose is true; one such that we can conceive of or suppose it and any other statement entailed by it being true; one such that we can understand what it would be like for it and any statement entailed by it to be true.»<44>Foot note 2_18 Swinburne offers `All bachelors are unmarried' and `1=/=3' as examples of logical truths.

The validity of (A') requires that we understand logical truth in such a way so that p is a logical truth at each possible world if it is one at any one world. By (iii), this amounts to appealing to a sense of coherence which makes the coherence or incoherence of a sentence invariant from one world to another.<45>Foot note 2_19 However, Swinburne's conflation of logical possibility with coherence results in psychologizing logic by making the logical necessity of a sentence p consist of the fact that p must be thought of as true. Critics of the psychologistic interpretation of logical necessity will argue that this misrepresents the modal notion in (ii). The fact that, say, human x is boy and at the same time not a boy, must be false is not due to the fact that the human mind is so made that it cannot understand the conditions required for the truth of this claim, but rather is due to mind independent facts about the world.

At any rate, the arguments in the literature levied against the reliability of conceivability -- on any of its senses -- as a guide to possibility, make dubious the identification of a way the world could be with a coherent (in Swinburne's sense) state of affairs.<46>Foot note 2_20 For example, as Tidman points out, what is conceivable (in Swinburne's sense) seems to depend on what is possible. «Whether we can really conceive of, say, having a headache without being in a particular brain state, depends upon what is possible, in particular, on whether these are two essential aspects of one thing, a question that cannot be resolved by what we can conceive of.»<47>Foot note 2_21

Whether a seeming conceivability is truly conceivable depends on what is possible. We don't want to say that it was conceivable to ancient astronomers that (a) the morning star exist without the evening star existing. Rather (a) only seemed conceivable to them, for their understanding of the truth conditions of (a) was based on an ignorance of the fact that the existence of the one entails the existence of the other. Knowing that it is conceivable that p, requires a knowledge of what is possible in order to know what is entailed by p. Tidman concludes that, this «...removes from our grasp any direct ability to make judgments about possibility based on conceivability.»<48>Foot note 2_22 More relevant to the concern here is the fact that the invariance of what is coherent from world to world obtains only if what is possible is invariant from world to world. So, we return to the original problem of unpacking the nature of the modal notion at work in (ii) in order to ground the latter.

One suggestion is that this modal notion is metaphysical<49>Foot note 2_23: if a sentence p is a logical truth, then there is no way the world could metaphysically be which would make p false. This motivates the following definition of logical truth: p is logically true iff it is not metaphysically possible for p to be true on any meaning assignment to the non-logical terms occurring in p. On this approach, a logically possible situation is a meaning assignment in a metaphysically possible world. ((x)~Txx&(x)(y)(z)((Txy&Tyz)->Txz))->(Ex)~(Ey)Txy is a logical truth on this approach only if it is metaphysically impossible for there to be a denumerable infinite totality of things. One might object that even if the existence of such a totality is metaphysically impossible it may nevertheless be logically possible. However, this approach is here being pursued precisely to get at the latter notion, and it is not clear what objection there can be if it turns out that such a totality exists in no metaphysically possible world.

The appeal to meaning assignments makes logical possibility weaker than metaphysical possibility. For example, a Kripkean can hold that `Saddam Hussein is a dog' could logically be true on the basis that, say, `is a dog' could have meant, is an Iraqi. The validity of (A') then would turn on the structure of the relative possibility relation on the totality of metaphysically possible worlds. It must be at least both symmetric and transitive. While there are many who believe that what is metaphysically necessary does not vary from one possible world to another (and subscribe to an S5 modal semantics as the correct semantic representation of the logic of metaphysical possibility), this view is not universally held. I subscribe to the modal situationalism<50>Foot note 2_24 illustrated by the above Leibnitzean view of metaphysical possibility. That is, on my view the laws of modal metaphysics may vary from world to world. In particular, I believe that it is metaphysically possible that metaphysical determinism be true of worlds whose domains are finite. In such worlds, it is not metaphysically possible that there be more individuals. However, the actual world is not, on my view, deterministic. The deterministic worlds are (metaphysically) possible relative to the actual world, but not vice versa. This view will not support the validity of (A'). So, on this understanding of the modal notion in (ii), a defense of the validity of (A') must consist of, in part, an argument against modal situationalism.

There are other ways of understanding the modal notion in (ii).<51>Foot note 2_25 I do not claim to have taken the matter far here. I merely wish to point out that the challenge for the proponent of the validity of (A) is to unpack the modal notion in the ordinary concept of logical truth in a way which will ground the invariance of logical truth from world to world. People like Quine believe that the notion ways the world could be is deeply mysterious, and opt for weaker notions of logical necessity (e.g., the one embodied in the PM notion of logical truth). I don't believe that the invariance of logical truth from world to world is self-evident or obvious, and therefore it seems to me that it needs to be defended by argument. Steinitz does not provide one. What justifies his confidence that (A) is valid?

Sect.3. Is it logically possible that a logically necessary being exist?

Although Steinitz claims that there can be no conclusive demonstration for the coherence of any concept,<52>Foot note 2_26 he thinks that there is reasonable justification for the coherence of the existence of a logically necessary being.

Quine emphasizes that every self-contradictory concept forms a necessary non-being, i.e., in no possible world does there exist a barber who shaves all and only all those who don't shave themselves. Necessary non-being forms a coherent concept, why shouldn't necessary being as well? For if the combination of logical/analytical necessity with negative existential propositions can be coherent, it means that there is no essential opposition between modality and ontology. This seems to remove the only difficulty ... from which the internal inconsistency of necessary beings was alleged to emerge.<53>Foot note 2_27

I am not sure what Steinitz has in mind by internal inconsistency. Perhaps a concept is internally inconsistent if it pictures that something is both the case and not the case. But Swinburne claims that a sentence is also incoherent if it conflicts with another coherent sentence. Hume, of course, believed that for each object, it is conceivable that it not exist, and so would argue that a necessary existent is incoherent. I don't see why the coherence of a thing whose non-existence is necessary is a reason for maintaining the coherence of an object whose existence is necessary. The condition required to establish the necessary non-existence of, say, a barber who shaves all and only all those who do not shave themselves is clear in sofar as it is clear that this claim is internally inconsistent. But Steinitz must show that not only is the concept of necessary existence internally consistent, but also that it does not conflict with other coherent claims.

At any rate, it seems to me that this is besides the point because Swinburne's approach to logical possibility is unmotivated, and so it is unclear that the sentences it makes logically impossible are really logically impossible. Logical possibility is a logical property and all logical properties are, on my view, properties of sentences. So, the concept of the existence of a logically necessary being is internally consistent only if the claim that it exists is logically possible. Recall that on the Swinburne's approach, p is logically possible iff p is coherent, i.e., the conditions required for the truth of p are understandable, where the meaning of the words occurring in p are fixed. In what follows, I question the motivation for the latter constraint

On the standard approach to logical possibility in first-order logic, a sentence p could logically be true iff there exists an interpretation which makes p true. An interpretation of a first-order sentence p consists of two components: a domain and a meaning assignment, which (as indicated above) is a function that assigns to each individual constant and variable an element of the domain; to each n-place predicate letter a set of n-tuples from the domain. A sentence is a logical truth iff there is no interpretation which makes it false. On this approach, in order to ascertain whether or not the concept of the existence of a necessary being is consistent, we need to identify the logical structure of the claim that an individual necessarily exists. I have construed it as (2') (Ex)L(Ey)(x=y). So, the logical possibility of (2') boils down to whether or not there exists a meaning assignment to (2') which makes it true in some domain (i.e., at some possible world).

The appeal to meaning assignments in the standard approach to logical possibility in first-order logic reflects the fact that possible uses for variables, individual constants, and predicates are elements of possible situations to be countenanced in fixing the extension of logical possibility. In fact, this approach constrains the possible uses or meanings of variables, names, predicates, and primitive sentences only by the type of expressions they are (e.g., properties to predicates, first-order particulars to names, etc). To elaborate, consider the treatment of the existential quantifier in classical semantics. There the quantifier is attached to a variable which may be used to range over various collections of individuals. The actual use of variables is given by the kind terms in the quantifiers.

For example, the logical structure of `There exists at least two natural numbers' can be represented as, `(Ex)(Ey)~(x=y)', where `x' and `y' range over the collection of natural numbers. In actual use, these quantifiers might point to all sorts of different collections of objects. This suggests that a possible use of variables, and therewith the quantifiers, is given by specifying some collection over which they could be used to range. By moving to a possible use of variables, we can make existential quantifications false. For example, we can make the above existential quantification false by using `x' and `y' to range over the offspring of Bill and Hillary Clinton.

So, if a first-order sentence p is true on a possible use of the non-logical terms occurring in it, then this establishes that it is logically possible for p to be true. In other words: a sentence is logically necessary at a world w only if it remains true at w on all possible uses to its non-logical terms (whether this is both necessary and sufficient for logical truth at a world, as is maintained by the PM approach, is a point of contention). To establish that `Bill Clinton is a Democrat' could logically be false at the real world w, we need not consider a world in which Clinton has a different party affiliation, but simply consider a re-interpretation of the atomic sentence so that it says something false about w, perhaps that Bill Clinton is a female.

I believe that the classical requirement that the ranges of variables be non-empty is unmotivated. It represents a qualification of the idea that the possible meaning of variables are to range as widely as possible. Since failure of reference is a possible use for a term,<54>Foot note 2_28 the empty world represents the use of terms in which they fail to refer (e.g., one possible use of a variable is to range over the empty set). Since all existential quantifications are false at the empty world, none are logically necessary. The objections that allowing failure of reference generates a semantics that misses some logical truths, e.g., (Ex)(x=x), is circular. The judgment that this sentence is a logical truth presupposes some theory in which names must have referents, domains must be non-empty, etc..., but the latter is what is at issue.

Note that to imagine that, say, `~(Ex)(x=x)' is true is to imagine that `x' could be used so that it fails to refer. It is not required that we imagine an alternative course of evolution such that the individuals of the world fail to exist. Clearly, the possibility of such a use for `x' is independent of considerations about whether the universe could have evolved so that nothing exists. Hence, even if there exists an individual whose existence is, say, metaphysically or mathematically necessary, this is no reason to think that failure of reference is not a possible use for a term. To say that it is logically possible that there be nothing is misleading because, on my view, the appeal to the empty world in determining what is logically possible is grounded on the notion that failure of reference is a possible use for terms, and is not grounded on some claim that the universe could have been empty.

By keeping the meaning of all terms occurring in a sentence p fixed in determining whether p could logically be true, Swinburne's approach makes the evaluation of the logical possibility of a sentence p consist of inspecting different possible (conceivable) worlds in which the extensions of the terms occurring in p are changed. This makes the logical possibility of a quantification p turns on the actual use of the quantifiers occurring in p, which results in fixing the domain of quantifiers in terms of their actual use, and not subject to change from one interpretation to the next. For example, suppose a platonist, who believes that each possible world contains all the arithmetical entities, uses `(Ex)(Ey)~(x=y)', to assert that there exists at least two natural numbers. On this use, the variables range over the collection of natural numbers. On Swinburne's approach, to consider whether this sentence could logically be false is to consider whether it is coherent to suppose that the cardinality of the set of natural numbers be less than two. Since the platonist finds the latter incoherent, she is committed to the logical truth of her assertion.

However, by relativizing logical truth to the actual use of the variables, different views about the nature of mathematical objects can give us different answers to questions about what is logically true.<55>Foot note 2_29 Moreover, on this account, logical form is not decisive to what turns out as logically true for logical truth will vary on distinct uses for terms. For example, the Platonist could use the above formal sentence to assert that Shannon has two marbles in her pocket (on such a use, `x' and `y' range over the marbles in Shannon's pocket). Surely the platonist is not committed to the incoherence of the denial of this assertion.

On the standard approach to logical possibility, we can establish that the above sentence could logically be false by appealing to the fact that the variables could be used to range over, say, the set of even natural numbers that are prime. By appealing to possible uses/meanings we make what is logically true a matter of form and thereby reduce the need to do metaphysics in order to do logic. On Swinburne's approach in considering whether or not the theist's assertion, `there exists a necessary being' -- (Ex)L(Ey)(x=y) - - is logically true the range of `x' and `y' is fixed in terms of one object b, and not subject to change from one interpretation to the next. But then the issue of whether this sentence is logically possible turns on whether there {could} {conceivably }exist such an object. But why make logic hostage to the resolution of issues in modal metaphysics? By restricting the possible uses of variables we make logical truth turn on things other than logical form, and this results in decreasing the epistemic transparency of judgments about what is logically true. Since we have more to say about what meanings are possible than about ways the world could conceivably be, it is better to base our assessments of logical possibility on the former in order to secure the strongest possible epistemological foundations for our logical judgments.

Since the theist believes that God is metaphysically necessary, the theist is committed to believing that the above sentence is true (reading `L' as the metaphysical necessity operator). Reading `L' as the logical necessity operator, is the theist committed to regarding the sentence as true? Not on the standard approach. For, if this is true, then there exists an object b such that `(Ey)(b=y)' is true on all possible uses for the variable `y'. But there is no such object; we can use `y' to range over an object a such that (a=/=b) (assuming that there exists at least one object distinct from God to call on as the value of `y'). On such a use, `(Ey)(b=y)' is false.

So, if (1') `M(Ex)L(Ey)(x=y)' is true (reading `M' and `L' as the logical possibility and logical necessity operators), then there exists a referent b for `x' such that `(Ey)(b=y)' is true on all possible uses for `y'. But there is no such b for there as many uses for `y' which will falsify `(Ey)(b=y)' as there are collections of objects which exclude b. For example, if `y' is used to either range over the empty set or the set consisting of just one object a (a=/=b), then `(Ey)(b=y)' is false. Hence, (1') is not true. In sum, the difference highlighted here between Swinburne's approach to logical possibility and what I have been calling the standard approach is that on the former one determines whether it is logically possible that a given sentence p is true by looking to other possible (i.e., coherent -- in Swinburne's sense) extensions of the terms occurring in p, while on the latter one can look to the actual world with its actual extensions in substituting new terms for the non-logical terms occurring in p. I don't see the motivation for adopting an account of logical possibility which diminishes the capacity of logic as a tool for figuring out what is true by decreasing the reliability of the perception of what is logically possible in some cases, and in other cases leaving it an open question whether a sentence is logically possible or not. <56>Foot note 2_30 Placing logic on a more solid epistemological footing by not grounding intuitions about what is logically possible on any one view of metaphysical or mathematical reality underwrites the uses of logic. By using the resources of logic, we can determine the truth values of a number of sentences without having to investigate that part of the world they are about. If we base the determination of logical truth on strong claims in modal metaphysics, then we obviously minimize the value of logic in helping to figure out what is true. Moreover, we use logic to clarify and frame issues in metaphysics. It is not going to have this use if it embodies one point of view. This motivates allowing the range of the possible uses of variables to be as wide as possible to insure that logic is not encumbered with issues in metaphysics. So, if it is logical possibility and necessity at work in (1'), then (1') is false because possible uses of variables should count as elements of possible situations to be countenanced in fixing the extension of logical necessity and possibility. Of course, (1') may be true on a different reading of the modal operators.

Sect.4. Conclusion

Steinitz's argument for the validity of (A) is a reductio from the assumption that the argument is invalid.<57>Foot note 2_31 If (i) it is logically possible for a logically necessary being to be merely logically possible and not actually exist, then (ii) it is logically possible that such a being exist contingently. But (ii) is impossible because a being that exists contingently is not necessary. But (ii) follows from (i) only if what is logically necessary is invariant from possible world to possible world. So there is no reason to take Steinitz's reduction from the assumption that (A) is invalid seriously unless there is reason to think that what is logically necessary does not vary from world to world. I have tried to make explicit the challenge of clarifying the notion of logical necessity in a way which grounds the validity of (A). The fact that this challenge is substantial motivates interest in weaker notions of logical necessity like the one captured by the PM approach. This approach does not make (A) valid.

Furthermore, the correct approach to logical necessity must account for those possible situations in which the meanings of some of the terms in our language might have been different. On such an approach, the premise of Steinitz's argument is false. This suggests that arguments for the possibility of a necessary existent which do not make logical possibility the operative modal notion are more promising than those that do.<58>Foot note 2_32

Matthew McKeon

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