Ways Modality Could Be: Revised and Expanded

My article "Ways Modality Could Be: Revised and Expanded" is now up on Scholardarity.

Here's an excerpt:

1. Introduction

            In this paper I 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 terms 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 (first-order) possible worlds could have been. And just as there is a unique actual world which represents the way that things actually are, there is a unique actual space which represents the way that first-order modality actually is.

            One might wonder what the accessibility relation itself is like. Presumably, if it is logical or metaphysical modality that is being dealt with, it is reflexive; but is it also symmetric, or transitive? Especially in the case of metaphysical modality, the answer is not clear. 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 different ways that first-order modality could have been?

            To be clear, the idea behind my proposal is not just that some things which are possible or necessary might not have been so at the first order, as determined by the actual accessibility relation, but also that the actual accessibility relation, and hence the nature or structure of actual modality, could have been different at some higher order of modality. Even if the accessibility relation is actually both symmetric and transitive, perhaps it could (second-order) have 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. There will thus be an infinity of orders, one for each positive integer, and each order will have an accessibility relation of its own. To keep things as clear as possible, a space of first-order points (i.e., of possible worlds) shall be called a galaxy, a space of second-order points, a universe, and a space of any higher order, a cosmos. However, to keep things as simple as possible, in what follows I will deal with but a single cosmos at a time, and hence will not deal with modalities higher than the third order.

            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 some entity e exists in a world w and in another space e doesn’t exist in that same world w—we may suppose that different spaces may differ as to which worlds they contain, just as different worlds may differ as to which objects they contain. Thus we might have a higher-order analogue of a variable-domain modal logic. There seem, then, to be three ways in which spaces can differ: First, as to the properties of the accessibility relation; second, as to which worlds the relation relates; and third, as to which worlds or spaces are parts of their domains.

            The paper will be structured as follows. In Section 2 I provide some reasons why one might want to pursue this kind of project in the first place. In Section 3 I outline the syntax and semantics of my proposed logic. Section 4 covers semantic tableaux for this system; and after giving the rules for their construction, I construct a few of them myself to establish some logical consequences of the system and give the reader a feel for how it works. In Section 5 I outline a potential application of my framework to the metalogic of modal logics. In Sections 6, 7 and 8 I explore some of  its potential philosophical implications for areas besides logic, namely the philosophy of language; metaphysics, including the metaphysics of modality, the philosophy of time, and laws of nature; and finally the philosophy of religion, before concluding the paper in Section 9.

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.

Deflating Debates over Essential Properties

Suppose we have a debate as to the essential properties of something, or over whether some x is really an F. For example, let’s say there's a dispute between an epistemic internalist and an epistemic externalist as to what knowledge or justification essentially is. It seems to me that we can avoid debates such as this in the following way: Instead of arguing over whether knowledge requires accessibility or not, or whether a belief’s being the product of reliable cognitive faculties is sufficient to justify it or not, we could simply coin terms such as “knowledge_e” and “knowledge_i”, or “justification_e” and “justification_i”. Then we could say that knowledge_i requires accessibility but knowledge_e does not. And we could say, similarly, that being the product of reliable cognitive faculties is sufficient for justification_e but not for justification_i. So long as each of these notions is consistent, there is no a priori obstacle to their all having instances. We might, of course, be able to find evidence or devise arguments to show that, as a matter of fact, either knowledge_e or knowledge_i or both do not exist. And then again, we might not. What I want to know is why we should think there is some other thing, knowledge simpliciter, concerning which we are unsure of its essential properties. If there is no reason for supposing there is such a thing, we risk only the loss of much fruitless debate if we eliminate it from our ontology.

Does every proposition have a negation?

Some philosophers have held—as Wittgenstein seems to in the Tractatus—that if some (apparent) statement is meaningless then so is its “negation”, and conversely that if the negation of a statement is meaningful then the negated statement must be meaningful too. Thus some have held that, since statements such as “It is not the case that Jones is identical to himself” are (so they think) obviously meaningless, then “Jones is identical to himself” is likewise meaningless. On the other hand, some have supposed that since statements such as “Jones is identical to himself” are (so they think) obviously meaningful, then so is “It is not the case that Jones is identical to himself”; it’s just that the latter is necessarily false. What has never been questioned, so far as I know, is that every meaningful statement has a negation. I think one might reasonably maintain something like the following: Suppose you think the meaning of a declarative sentence is the proposition it expresses. In that case you could say that while a given declarative sentence, say “Jones is identical to himself”, expresses a proposition, the sentence which results from prefixing a negation operator to it, say “It is not the case that Jones is identical to himself”, expresses no proposition. One could thus maintain that there are necessary truths but no necessary falsehoods. I think many would see this as beneficial, since if we could grasp the meaning of a necessarily false statement—that is, if we could understand full well what things would be like if it were true—what would we mean by calling it necessarily false or impossible? On the other hand, since we would still believe in necessarily true propositions, we could (at least potentially) accept the existence of a priori knowledge. We could also avoid a major pitfall of theories which reject (apparently) necessary truths as pseudo-propositions, namely, that on such theories a statement like “Every genuine statement has its truth value contingently” would seem not have its truth value contingently.

Such, I think, are the merits of this view. But what do you think? Is this view tenable, or does it suffer from problems comparable to those of the rival views discussed above? I have my suspicions, but for now I’m just interested in your own opinion.