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.