Comments on Truth at A World for Modal Propositions

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1 Comments on Truth at A World for Modal Propositions Christopher Menzel Texas A&M University March 16, 2008 Since Arthur Prior first made us aware of the issue, a lot of philosophical thought has gone into sorting out the relationship between quantified modal logic (QML), actualism, property actualism (a.k.a. serious actualism), and modal existentialism roughly, the view that so-called singular propositions are ontologically dependent upon the individuals they are about. Many actualists find modal existentialism intuitively appealing: If Socrates, say, hadn t existed, then, since there would have been no such thing as Socrates, there simply would have been no propositions about him, no information about him in any sense. The view, however, seems to have grave consequences for any QML based upon the popular propositional modal logic S5. Prior himself argued that the first QMLcasualty of the view is the interdefinability of the modal operators necessarily and possibly ; in particular, Necessarily, ϕ is not equivalent to It is not the case that, possibly, not-ϕ. To see this, let us first accept that, intuitively, a sentence of the form Possibly, ϕ is true just in case the proposition that-ϕ [ϕ], for short is true with respect to some possible world w. The core of Prior s reasoning behind this Draconian conclusion is more or less the simple argument that Franklin cites from Turner [6] (p. 192). The argument (itself simply a variant of Plantinga s antiexistentialist argument in [4]) can be laid out in just a bit more detail as follows: 1. Possibly, Socrates does not exist. (Assumption, for RAA) 2. For any sentence ϕ, possibly, ϕ if and only if the proposition [ϕ] is true with respect to some possible world. (Standard possible world semantics) 3. Hence, the proposition [Socrates does not exist] is true with respect to some possible world. (From 1 and 2) 4. For any world w, if [Socrates does not exist] is true with respect to w, then [Socrates does not exist] exists in w. (By property actualism)

2 5. For any world w, if [Socrates does not exist] is true with respect to w, then Socrates does not exist in w. (By the logic of truth) 6. Hence, for some possible world w, [Socrates does not exist] exists in w even though Socrates does not exist in w, contradicting modal existentialism. (From 3, 4, and 5) 7. Hence, it is not the case that, possibly, Socrates does not exist. (From 6, by RAA) Prior boldly embraced this conclusion, arguing that our apparently incompatible intuitions about Socrates contingency can be accommodated simply by denying that his existence is necessary a move rendered compatible with line 7 by denying the interdefinability of necessarily and possibly. Socrates contingency is thus represented, not by virtue of [Socrates does not exist] being true with respect to some world, but rather by virtue of [Socrates exists] failing to be true with respect to all worlds. To capture these ideas formally, Prior developed his rather awkward and unfortunate quantified modal logic Q (see [5], Ch. XIII). Q features a classical quantification theory with identity but only pale shadows of the usual S5 axioms and theorems notably, instead of the characteristic S5 axiom ϕ ϕ, in Q we can prove only ϕ ϕ. In Q, ϕ is defined as Sϕ ϕ, where Sϕ expresses that ϕ is necessarily statable, i.e., in effect that it expresses a proposition that itself exists necessarily. In Q, then, the closest we get is to the characteristic S5 axiom is the theorem Sϕ ( ϕ ϕ). Even Modus Ponens requires modification to avoid the theorem x x = x, i.e., the theorem that it is not possible that nothing exist, whose truth or falsity Prior (rightly) considers a purely philosophy matter and hence inappropriate as a theorem of formal logic. Awkward and unfortunate though it may be, Prior evidently believed Q to be worth the cost as Franklin himself unflichingly asserts in the face of similarly dramatic restrictions that he believes to be required by his own philosophical commitments: Better to have a complicated modal logic that accurately represents our ontology than to have a simple modal logic that fails to represent our ontology. Now, as Franklin points out, there is in fact a very plausible response to the argument above, one indeed suggested by Prior himself in his realization that there are worlds with respect to which [Socrates exists] fails to be true. Specifically, there are two ways of understanding what it is for a proposition to be true with respect to to characterize, for short a possible world. The distinction is perhaps best phrased in terms of perspective. One the one hand, one can evaluate whether a proposition p characterizes a world w internally. On this approach, we adopt the standpoint of an agent existing within w hence able to avail himself only of the objects, properties, and relations of w and ask whether the agent could, in principle at least, 2

3 grasp p and recognize it as true. From the internal perspective, then, a proposition characterizes w only if would have existed had w been actual. If p is characterizes w internally, p is said to be true in w. On the other hand, we can also evaluate whether p characterizes w externally. In this case, we ask whether p characterizes w from our standpoint in the actual world. The external perspective encompasses the internal: a proposition that is true in a world w stills obviously characterizes w from our perspective in the actual world; we see the truth of [Socrates exists], say, at a world in which Socrates does in fact exist as much as any agent existing within w. But from our vantage point in the actual world, we also see more. In particular, as Prior himself seemed intuitively to grasp, we can see that a negative proposition [ ϕ] can characterize a world w simply in virtue of its unnegated counterpart [ϕ] failing to characterize w notably, in the case of propositions like [Socrates does not exist], in virtue of its unnegated counterpart failing to be true in w. A proposition that characterizes a world w externally, whether existing in w or not, is said to be true at w. Armed with this distinction it is easy to see how the modal existentialist can avoid the conclusion of the argument above: The argument is sound only if truth with respect a world is taken to mean truth in a world. But (as Prior seemed not to recognize, despite having a glimmer of the internal/external distinction) this interpretation is not forced. If truth with respect to a world w is understood consistently as truth at w, Premise 4 is clearly false; the truth of [Socrates does not exist] at w does not entail its existence at w, only the failure of [Socrates exists] to be true in (hence also at) w. So understood, the argument fails and the possibility of Socrates nonexistence returns, and with it in a logic based semantically on truth-at rather than truth-in the interdefinability of necessarily and possibly. Now, how does all of this play out with respect to the evaluation of modal propositions the proposition [Possibly, Socrates exists], say with respect to other worlds? Jason Turner claims that, if we evaluate such propositions consistently from the external standpoint, the following principle emerges for determining the truth values of modal propositions at other possible worlds: (C4) Modal proposition we are true simpliciter are true at all possible worlds. 1 As Turner nicely puts it, from the external standpoint: [w]e are standing outside of a world, looking into it, and using the propositions, objects, properties, and relations of our own world to describe what we see. It makes sense to think that which predications of 1 This of course isn t exactly principle (C4) but it s a bit more intuitive and it s all we need for purposes here. 3

4 [an individual] a are true at a world is determined solely by things going on in that world how could facts from other worlds ever get into the picture? But we tend to think that modal truths are not made true solely by what is going on in any one world but by what goes on in the entire space of possible worlds. Furthermore, on the model of standing outside of a world looking into it, it is not implausible to think that we should be able to see the entire space of possible worlds. We can say, of a non-a world W, that at that world it is possible that a is P, precisely because, standing outside of W, we can see other worlds worlds where a exists and is P. 2 Unfortunately, Franklin argues, principle (C4) is not compatible with modal existentialism. The logical consequences are grave: As in Prior s Q, the characteristic axiom schema for S5 falls by the wayside, as do the characteristic schemas for the systems B (ϕ ϕ) and S4 ( ϕ ϕ). A logic that is little better than Q appears to be in the offing. Franklin argues for the invalidity of (C4) by pointing out two fatal shortcomings in the principle. First, (C4) fails to capture that the set of the possibilities there are is not identical with the set of the possibilities there could have been (p. 5). According to modal existentialism, if Socrates hadn t existed there would have been no propositions about him, hence no possibilities about him. Thus, if w is a Socratesfree world, the proposition [Possibly, Socrates exists] does not exist in w. According to Franklin, [t]he most natural way of expressing this is by maintaining that [Possibly, Socrates exists] is false at w and its negation true (p. 6). Hence, where s is Socrates and E! is the property of existence, we have the truth of E!s E!s, i.e., ( E!s E!s), i.e, the S5 schema is not generally valid. Franklin s argument for this view is that [Possibly, Socrates exists] simply does not properly characterize a Socrates-free world w. Notably, he argues, it does not properly characterize the fact that, were w actual, it would be possible that Socrates exists, since this would contradict existentialism (p. 6). But I cannot agree. It seems to me that the reason that Franklin himself believes this to be the case is that his semantical approach is a bit schizophrenic. When it comes to the evaluation of negative singular propositions like [Socrates does not exist] with respect to other worlds, he (rightly) adopts the external perspective. But when it comes to the evaluation of modal singular propositions like [Possibly, Socrates exists] at other worlds, he adopts the internal perspective [Possibly, Socrates exists] must be false at a Socrates-free world w because there is no such proposition in w; there would be no such proposition if w were actual. True enough. But exactly the same thing is true of [Socrates does not exist]. There is no such proposition as [Socrates does not exist] in w. But, from our perspective 2 [6], p

5 in the actual world, we see that the proposition characterizes w simply in virtue of the failure of [Socrates exists] to characterize w. By the same token it seems to me that, while there is no such proposition as [Possibly, Socrates exists] in w, from our perspective in the actual world we see that that proposition characterizes w simply in virtue of the fact that [Socrates exists] characterizes some world, in particular, the actual world. In both cases, we evaluate the propositions in question from our vantage point in the actual world, making full use of our ability to describe worlds in light of everything we know from that perspective. So, as far as I can see, the most we have here are two competing semantic options for determining the truth values of propositions with respect to possible worlds. The only thing approaching an argument that I could find from Franklin for his hybrid approach is that the truth of a proposition in or at a world W must be grounded in the ontological fabric of W. I must say I m not at all sure what particular piece of the fabric of a Socrates-free world W grounds the truth of [Socrates does not exist]. But I do admit that, in some sense, we take the proposition to be true of such a W in virtue of something about W itself, even if what that is cannot be described in terms of what exists in W. But it is far from obvious that the same intuitive criterion should be what determines which semantic option we choose for evaluating the truth values of modal propositions at worlds. As Turner says, we tend to think that modal truths are not made true solely by what is going on in any one world but by what goes on in the entire space of possible worlds. From the external perspective, it is the space of all worlds, viewed from our vantage point, that determines the truth values of modal propositions at any world. Franklin, it seems, simply declares that that isn t enough: When assessing the truth of non-qualitative modal claims at W, we must...first consider the ontological furniture of W and only then is the entire space of possible worlds relevant. If Socrates fails to exist in W, there should be no true predicative modal propositions directly about him at W. If Socrates exists in W, then, and only then, do we proceed to examine other possible worlds, here we include the actual world, in order to discern whether certain modal claims about Socrates are true. There is simply no argument here, just a prescription. And, Franklin s assertions to the contrary, there is certainly nothing here that demonstrates any incompatibility between modal existentialism and Turner s externalist semantic option. So, with regard to the first alleged shortcoming that Franklin finds in (C4), at best he shows only that there is a way of determining truth values one that involves shifting between internal and external perspectives with respect to which (C4) is incorrect. But this semantic option is no more forced than is the consistently internalist approach that underlies Prior s Q. If we adopt Turner s consistently externalist approach for determining truth values, (C4) appears to be exactly right. 5

6 The second shortcoming that Franklin finds in (C4) (p. 7) is that it [precludes] any way of representing the fact that if a Socrates-less world were actual, there would be no possibilities concerning Socrates...The most obvious way to make this point clear is to have a logic in which propositions such as [Possibly, Socrates exists] are false at any world in which Socrates does not exist. Here, however, Franklin simply mistakes lack of expressivity for a shortcoming in principle (C4). It is an expressive shortcoming in the usual language of QML that one cannot refer directly to propositions; one can only express them via sentences (and even that is not typically represented explicitly in the semantics of QML). In this language, there is no obvious way to represent the fact that there are no possibilities about Socrates in Socrates-free worlds, as there is no way to represent that fact at all. To do so, one must have terms that denote propositions. I ve been using such a language informally in these comments. But (leaving semantic details aside for purposes here 3 ) suppose we in fact introduce a grammatical rule into the language of QML to the effect that, if ϕ is a formula, then [ϕ] is a term intuitively, one that denotes the proposition expressed by ϕ. Then the way to represent the absence of any possibilities or more generally, any singular propositions at all about Socrates in Socrates-free worlds is to have a QML, so augmented, in which all instances of the following are valid: (1) ( E!s E![ϕ]), where s occurs in ϕ. We can then, in particular, express quite nicely the truth of both [Socrates does not exist] and [Possibly, Socrates exists] at Socrates-free worlds together with the modal existentialist s commitment to their nonexistence as follows: (2) ( E!s ( E![ E!s] E!s E![ E!s])) In sum, then, it seems to me that Franklin has exposed no shortcomings at all in (C4). The road is thus clear for an S5-based QML that is fully compatible with the commitments of actualism, property actualism, and modal existentialism. 4 References [1] George Bealer. Quality and Concept. Oxford University Press, See, e.g., Bealer [1] and Menzel [3] for more details. 4 For which, of course, see Menzel [2]! 6

7 [2] Christopher Menzel. The true modal logic. Journal of Philosophical Logic, 20(4): , [3] Christopher Menzel. Singular propositions and modal logic. Philosophical Topics, 21(2): , [4] Alvin Plantinga. On existentialism. Philosophical Studies, 44:1 20, [5] Arthur Prior. Papers on Time and Tense. Oxford University Press, Oxford, [6] Jason Turner. Strong and weak possibility. Philosophical Studies, 125: ,

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