Topology and Modality: The Topological Interpretation of First-Order Modal Logic

Abstract

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

Identity and intensionality in Univalent Foundations and philosophy

“…159-185) dressed up in topological clothing-and it is quite possible that there is a limit to how far it can be taken. 24 It should perhaps be noted that the topological semantics given here is quite different from the one of Awodey and Kishida (2008), which is a more direct generalisation of McKinsey and Tarski (1944), and which, in distinction to our semantics, validates the Kripkean inference s = t s = t. We also note that HoTT furthermore contains its own notion of modality (Univalent Foundations Program 2013, pp. 245-249), but this concept is again rather different from the one presented here.…”

Categories for Dynamic Epistemic Logic

“…3 We thank an anonymous reviewer for their suggestion of this convention. 4 Categories with the following structures are studied e.g. in [33], where they are called "ordered categories with involution".…”

General Topos Semantics for Higher-Order Modal Logic

“…In the general case, the operator is of course interpreted by ∆ A Γ A , which always satisfies the axioms for an S4 modality, since ∆ A Γ A is a left exact comonad. The specialist will note that both ∆ A and Γ A are natural in A, in a suitable sense, so that this interpretation will satisfy the Beck-Chevalley condition required for it to behave well with respect to substitution, interpreted as pullback (see [1]).…”

“…This is the logic that the current paper investigates. The first step of our approach is to observe that, because higher-order logic includes a type of "propositions", interpreted by a subobject classifier Ω, the natural operations on the various subobject lattices in (1) can be internalized as operations on Ω. Moreover, the relevant part of the geometric morphism f : F → E, giving rise to the modal operator, can also be internalized, so that one really just needs the topos E and a certain algebraic structure on its subobject classifier Ω E .…”

“…induced by the continuous inclusion |X| ֒→ X gives rise to sheaf models for classical first-(and higher-) order modal logic, studied in [1]. The exact structure of these examples will be discussed in more detail in section 4 below.…”