David Gabelaia

Some results on modal axiomatization and definability for topological spaces

We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the six classes of spaces considered in the paper are pairwise distinct, while the C-logics of some of them coincide.

Modal languages for topology: Expressivity and definability

In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt-Thomason definability theorem in terms of the well established first-order topological language

Bitopological duality for distributive lattices and heyting algebras

L_t

Topological completeness of the provability logic GLP

.

Topological Interpretations of Provability Logic

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakias duality.

Spectral and T 0 -spaces in d-semantics

Abstract Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.

THE MODAL LOGIC OF STONE SPACES: DIAMOND AS DERIVATIVE

Provability logic concerns the study of modality \(\Box \) as provability in formal systems such as Peano Arithmetic. A natural, albeit quite surprising, topological interpretation of provability logic has been found in the 1970s by Harold Simmons and Leo Esakia. They have observed that the dual \(\Diamond \) modality, corresponding to consistency in the context of formal arithmetic, has all the basic properties of the topological derivative operator acting on a scattered space. The topic has become a long-term project for the Georgian school of logic led by Esakia, with occasional contributions from elsewhere. More recently, a new impetus came from the study of polymodal provability logic \(\mathbf {GLP}\) that was known to be Kripke incomplete and, in general, to have a more complicated behavior than its unimodal counterpart. Topological semantics provided a better alternative to Kripke models in the sense that \(\mathbf {GLP}\) was shown to be topologically complete. At the same time, new fascinating connections with set theory and large cardinals have emerged. We give a survey of the results on topological semantics of provability logic starting from first contributions by Esakia. However, a special emphasis is put on the recent work on topological models of polymodal provability logic. We also include a few results that have not been published so far, most notably the results of Sect. 10.4 (due to the second author) and Sects. 10.7, 10.8 (due to the first author).

Modal logics of metric spaces

In [6] it is shown that if we interpret modal diamond as the derived set operator of a topological space (the so-called d-semantics), then the modal logic of all topological spaces is wK4--weak K4--which is obtained by adding the weak version ⋄⋄p → p ∨ ⋄p of the K4-axiom ⋄⋄p → ⋄p to the basic modal logic K. In this paper we show that the T0 separation axiom is definable in d-semantics. We prove that the corresponding modal logic of T0-spaces, which is strictly in between wK4 and K4, has the finite model property and is the modal logic of all spectral spaces--an important class of spaces, which serve as duals of bounded distributive lattices. We also give a detailed proof that wK4 has the finite model property and is the modal logic of all topological spaces.

Connected modal logics

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces. §

Admissible Bases Via Stable Canonical Rules

It is a classic result (McKinsey & Tarski, 1944 ; Rasiowa & Sikorski, 1963 ) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3: