Ways Modality Could Be

Cross-posted at Scholardarity: Click Here

In this post I want to introduce the idea of a higher-order modal logic—not a modal logic for higher-order predicate logic, but rather a logic of higher-order modalities. “What is a higher-order modality?”, you might be wondering. Well, if a first-order modality is a way that some entity could have been—whether it is a mereological atom, or a mereological complex, or the universe as a whole—a higher-order modality is a way that a first-order modality could have been. First-order modality is modeled in term of a space of possible worlds—a set of worlds structured by an accessibility relation, i.e., a relation of relative possibility—each world representing a way that the entire universe could have been. A second-order modality would be modeled in terms of a space of spaces of (first-order) possible worlds, each space representing a way that the entire space of (first-order) possible worlds could have been. And just as there is a unique actual world which represents the way things really are, there is a unique actual space which represents the way that first-order modality actually is.

Why, though, should we adopt a framework like this? To motivate it, consider the fact that people have mutually conflicting intuitions about what the space of all (first-order) possible worlds is like. Does God exist in all, none, or only some worlds? Or consider the famous dispute between Platonists and nominalists concerning predication. Platonists think that at least some predications can be true only if objects exemplify properties, and nominalists deny this. They think that there are no properties, but that predications can still be true. For the one party, some predications essentially involve properties, and for the other none do. Platonism, if true, is necessarily true, and if false, is necessarily false. The same goes for nominalism. Either some predications essentially involve properties or none do. On the face of it, this is problematic for the view that conceivability implies possibility: Platonism and nominalism have both been believed, and by many very able philosophers at that. What is believed is conceivable in some sense, otherwise such “beliefs” would have no content. So both positions are conceivable, but only one is possible. Either way, conceivability doesn't imply possibility.

But maybe that's not quite true. Perhaps, though only one of these positions is actually true, and hence first-order possible, both views are second-order possible. So maybe conceivability does imply possibility—at some order or other. Related considerations might apply to semantic content and possibility: If we can coherently mean something, it can be the case—at some order or other.

And what is the accessibility relation itself like? Presumably it is reflexive, but is it also symmetric, or transitive? And whichever of these properties it may or may not have, could that itself have been different? Could at least some rival modal logics represent ways that first-order modality could have been?

To be clear, the claim is not just that some things which are possible or necessary might not have been so, but rather that the nature or structure of actual modality could have been different. Even if the accessibility relation is actually both symmetric and transitive, maybe it could have (second-order)  been otherwise: There is a (second-order) possible space of worlds in which it is different, where it fails to be symmetric, or transitive. We must, therefore, introduce the notion of a higher-order accessibility relation, one that in this case relates spaces of first-order worlds. The question then arises as to whether that relation is symmetric, or transitive. We can then consider third-order modalities, spaces of spaces of spaces of possible worlds, where the second-order accessibility relation differs from how it actually is. I can see no reason why there should be a limit to this hierarchy of higher-order modalities, any more than I can see a reason why there should be a limit to the hierarchy of higher-order properties.

The accessibility relation is not the only thing that might be thought to vary between spaces of worlds: Perhaps the contents of the spaces can vary as well. While I presume that the contents of the worlds themselves remain constant—it makes doubtful sense to suppose that in one space an object o exists in w_1 and in another space o doesn't exist in w_1—we may suppose that the spaces differ as to which worlds they contain. Thus we might have a higher-order analogue of a variable-domain modal logic.

I do not expect this kind of framework to settle the issue of how modality at any order actually is—no more than I expect ordinary first-order modal logic to settle (aside from first-order necessary truths) what is actually the case. What goes for the actual world goes for the actual space of worlds, and for all higher-order spaces of spaces. What I do hope for is that it will, if it proves to be coherent, help to clarify the terms of the debate about the way modality is—to help us to state the issues, and to see their interrelations, as clearly as we can.

I think that's enough for this time. I'll leave the further development of such a framework for another occasion--or occasions—provided that you, my readers, think it merits further development.

Conceivability, Consciousness and the Content of Belief

[Cross-Posted at Scholardarity's DiaBlog, here]

What follows is the text of my final paper for a philosophy of mind course that I took in the spring semester of this year. Due to constraints on length, the paper is much shorter than it needs to be. I would very much like to expand it to deal with David Chalmers' nuanced views on conceivability, among other things, and so I would be very grateful for any feedback that could help me to do that.  

Conceivability, Consciousness and the Content of Belief

1. Introduction

            In his article “Consciousness and Its Place in Nature”, David Chalmers presents an argument against materialism—the view that truths about consciousness and indeed mental phenomena in general are in some sense fixed by truths about physical entities—which is based on conceivability. If the argument is sound, the fact that one can conceive materialism to be false entails that materialism actually is false. In this paper I will argue that the argument is unsound, and I will do so by giving a parallel argument that is clearly unsound.

2. Sketching Chalmers’ Argument

            Let us say that something is metaphysically possible if it really could obtain, even if it actually does not. Let us also say that something is metaphysically necessary if it really has to obtain no matter what. In other words, if we say that a “possible world” is a way that absolutely everything—the entire universe—really could turn out to be, then something is metaphysically necessary if it obtains in every possible world. Now, Type-A materialists think that phenomenal truths—truths about “what it is like” to have experiences of various kinds—can be derived from physical truths via a priori reasoning. Type-B materialists, on the other hand, think that phenomenal truths are entailed by physical truths even though one cannot know a priori either that this is so or which physical truths entail which phenomenal truths. However, on both views phenomenal truths are fixed by physical truths as a matter of metaphysical necessity: There is no possible world in which the physical truths are as they actually are while the phenomenal truths are different.

            Chalmers offers an argument against Type-B materialism based on our ability to conceive that the physical truths about the world could be just as they are while some phenomenal truths are different. Let P be some sentence expressing the complete truth about all things physical, and Q be some particular truth about someone’s phenomenology. Chalmers’ argument would then be:

1. It is conceivable that P ⋀ ¬ Q.

2. If it is conceivable that P ⋀ ¬ Q, then it is metaphysically possible that P ⋀ ¬ Q.

3. If it is metaphysically possible that P ⋀ ¬ Q, then materialism is false.

4. Materialism is false.

(Philosophy of Mind, p. 249). So if it is conceivable that P holds while at least one truth about someone’s phenomenology fails to hold, materialism, and hence Type-B materialism, is false. This argument is clearly valid, but I think it is unsound, and I propose to show this by giving a parallel argument that I take to be clearly unsound.

3. Presenting the Parallel

            Consider Platonism and nominalism about predication. Platonists think that the truth of sentences of the form “x is F” (or some restricted class of such sentences) require the further truth of sentences of the form “x exemplifies F-ness”, where ‘F-ness’ refers to the property expressed by ‘F’. Nominalists deny this. They would say that sentences of the form “x exemplifies F-ness” are (necessarily) false.[1] Nevertheless, for nominalists sentences of the form “x is F” are perfectly fine as they are.

            There are at least two different possible forms of Platonism. Type-A Platonists hold that sentences of the form “x exemplifies F-ness” can be derived from sentences of the form “x is F” via a priori reasoning. Type-B Platonists hold that sentences of the first form cannot be derived via a priori reasoning from sentences of the second form, although they do follow from them as a matter of metaphysical necessity. Nominalism has been upheld by many able philosophers over a long span of time, and since it is unlikely that this would be so if Platonism could be easily established a priori from the platitude that there are true predications, I think Type-B Platonism is more plausible than Type-A Platonism.

            Now for the parallel argument. Let R be some sentence expressing the conjunction of all sentences of the form “x is F”, and S be some particular sentence of the form “x exemplifies F-ness.” The argument would then be:

1. It is conceivable that R ⋀ ¬ S.

2. If it is conceivable that R ⋀ ¬ S, then it is metaphysically possible that R ⋀ ¬ S.

3. If it is metaphysically possible that R ⋀ ¬ S, then Platonism is false.

4. Platonism is false.

So if it is conceivable that R holds while at least one particular sentence of the form “x exemplifies F-ness” is false, it follows that Platonism, and hence Type-B Platonism, is false. I take it to be clear that Type-B Platonism cannot be refuted so easily. Something has gone wrong, but what?

4. Two Kinds of Conceivability

The second argument, like the first, is clearly valid. Thus, it must be unsound. Type-B materialists are committed to accepting premise (1) of first argument, and Type-B Platonists are committed to accepting premise (1) of the second. Also, in both arguments premise (3) appears to be necessarily true. This casts suspicion on premise (2) of each argument, though I will focus on premise (2) of the first argument.

I think premise (2) is questionable for two related reasons. The first reason is that I think the term ‘conceivable’ is ambiguous, and has two senses. Taking ‘thinkable’ and ‘comprehensible’ as technical terms, I will say that something is thinkable if one can understand it, and that something is comprehensible if, in virtue of understanding it, one can tell that it is possible.[2] So everything which is comprehensible is thinkable, but I think the converse is not true. Some expressions—nonsense strings like “!#?@”, and ungrammatical “sentences” like “Is and Caesar two” are not conceivable in either sense. On the other hand, sentences expressing logical or metaphysical impossibilities—e.g., “It’s true that Socrates was brave and it’s not true that Socrates was brave,” “Tyler is a married bachelor”—are, in my opinion, thinkable but not comprehensible. Some would take such sentences to be strictly meaningless, but I think that view is mistaken. For it seems that we can understand necessarily false sentences, because there are many cases where people have believed things which just can’t be true. Consider those who thought they could prove Euclid’s parallel postulate before the discovery of non-Euclidean geometries, or those who thought, before Gödel, that one could derive all the truths of mathematics within a single formal system. Once we grant that people have believed such things, we must also grant that necessarily false sentences are meaningful, and can thus be understood, for if they could not be understood they could not be the content of someone’s belief.

5. The Perilous Parallel

We now have the resources to see why my parallel to Chalmers’ argument is significant, and what its significance is. Regarding predication, either Platonism or nominalism is true. Now, either the truth of predications (or some restricted class of predications) requires that objects exemplify properties, or it does not. So either Platonism or nominalism is necessarily true, and if Platonism is necessarily true nominalism is necessarily false, and vice versa. Either way, one side thinks, but does not comprehend, something that is metaphysically impossible.

This brings us, at last, to the second reason why premise (2) is problematic. Let’s say that a world satisfies some sentence (or whatever one takes the ultimate bearers of truth to be) if that sentence is true at that world; or, alternatively, that that sentence would be true if that world were actual. If something is comprehensible then it is indeed satisfied by some possible world.  But either nominalists or Platonists think something which is metaphysically impossible, and which cannot be satisfied by any possible world. Despite that, both of these alternatives are epistemic “possibilities” in the sense that we cannot rule them out a priori given our current knowledge and limited inferential abilities. Thus some epistemic possibilities are not satisfied by any metaphysically possible worlds. If we nevertheless wish to count Platonism and nominalism as epistemic possibilities in the above sense, we will have to hold that some things are satisfied by impossible worlds—worlds in which some necessary truths may fail to hold. One could not say that such worlds are really misdescribed possible worlds or scenarios, as when a world in which water is purportedly XYZ is really a world in which the watery stuff is XYZ while water is still H₂O. A “world” in which predication essentially involves properties is not really a misdescribed world in which predication doesn’t essentially involve properties. Some work in semantics has shown that one can make sense of impossible worlds[3], but this is no help to Chalmers’ argument because the argument only works if worlds where P ⋀ ¬ Q holds are metaphysically possible. P ⋀ ¬ Q may be epistemically possible in the above sense, but that does nothing to show that it is metaphysically possible.

6. Objection and Reply

            Chalmers could try to question the idea that one could believe impossible things. If one cannot, my distinction between thinkability and comprehensibility threatens to collapse.

            I think, however, that the view that one cannot believe the impossible cannot be sustained. Consider the content of that very belief: If it is impossible to believe impossible things, it is impossible to believe that one can believe impossible things. What then could Chalmers make of his opponents’ position? He could not say that someone believes that someone can believe impossible things. Are sentences like “Wyman believes that P”, where ‘P’ is an arbitrary impossible sentence, meaningless? If so, “Wyman believes that someone believes impossible things” is meaningless. Could Chalmers maintain that his opponents are not mistaken, but simply confused? In that case he should not deny what they say, holding it to be false. He should instead claim that his opponents have an illusion of belief, and on pain of incoherence this illusion must not itself involve a false belief about the semantic status of certain of their apparent beliefs.  Vindicating that claim is no easy task, and I doubt it can be done.

7. Conclusion

We can now see why Chalmers’ argument against Type-B materialism fails. It fails because it does not distinguish two different senses of ‘conceivable’, namely thinkable and comprehensible, and while the falsity of materialism is thinkable we have no real evidence that its falsity is comprehensible. So we can conceive that materialism is false in one sense, but this does not entail that it is really possible that it is false. Maybe materialism is false anyway, but Chalmers’ argument does not show that it is.

References

Chalmers, David J. “Consciousness and Its Place in Nature,” Philosophy of Mind: Classical and Contemporary Readings, ed. David J. Chalmers. New York, Oxford University Press, 2002.

Chalmers, David J. Philosophy of Mind: Classical and Contemporary Readings. New York, Oxford University Press, 2002.

Restall, Greg. “Ways Things Can't Be”, Notre Dame Journal of Formal Logic, 38: 583–96. 1997.

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[1] There is one exception: “Trope” theorists do believe in properties, but for them properties are tropes.  These are particular properties like Socrates’ wisdom, not Platonic properties like wisdom in general which are thought to exist outside of space and time. In the main text I intend ‘property’ to be understood as referring to Platonic properties, not tropes.

[2] I think ‘conceivable’ is really an operator, not a predicate. Instead of saying, e.g., “Nominalism is conceivable,” one should say, “It is conceivable that nominalism is true.” The same goes for ‘thinkable’ and ‘comprehensible’.

[3] See for example Restall (1997)

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