TOPOLOGICAL COMPLETENESS OF LOGICS ABOVES4

Abstract

It is a celebrated result of McKinsey and Tarski [28] that S4 is the logic of the closure algebra X + over any dense-in-itself separable metrizable space. In particular, S4 is the logic of the closure algebra over the reals R, the rationals Q, or the Cantor space C. By [5], each logic above S4 that has the finite model property is the logic of a subalgebra of Q + , as well as the logic of a subalgebra of C + . This is no longer true for R, and the main result of [5] states that each connected logic above S4 with the finite model property is the logic of a subalgebra of the closure algebra R + .In this paper we extend these results to all logics above S4. Namely, for a normal modal logic L, we prove that the following conditions are equivalent: (i) L is above S4, (ii) L is the logic of a subalgebra of Q + , (iii) L is the logic of a subalgebra of C + . We introduce the concept of a well-connected logic above S4 and prove that the following conditions are equivalent: (i) L is a well-connected logic, (ii) L is the logic of a subalgebra of the closure algebra T + 2 over the infinite binary tree, (iii) L is the logic of a subalgebra of the closure algebra L + 2 over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logic L above S4 is connected iff L is the logic of a subalgebra of R + , and transfer our results to the setting of intermediate logics.Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics for S4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames for S4 and descriptive spaces are isomorphic. It follows that every logic above S4 is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of both C and R. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients of L 2 is an interior image of R. Contents2000 Mathematics Subject Classification. 03B45; 03B55.

LOGICS ABOVES4 AND THE LEBESGUE MEASURE ALGEBRA

“…Our present interest in the infinite binary tree stems from the following lemma, proved in Bezhanishvili et al, (2015). What follows is a sketch of the proof given there; for full details, the reader should consult Bezhanishvili et al, (2015), Lemma 5.4. LEMMA 4.2.…”

“…Although for any logic L above S4, each nontheorem of L can be refuted in some general frame over the infinite binary tree 2 <ω , it is not the case that each logic above S4 is the logic of a subalgebra of B(2 <ω ). For further discussion, see Bezhanishvili et al, (2015). Proof.…”

“…One of the difficulties with the real line, as these authors point out, is that the interior algebra B(R) is connected-the only clopen (open and closed) elements in the algebra are the top and bottom elements, 1 and 0. Following Bezhanishvili & Gabelaia (2011) and Bezhanishvili et al, (2015), we can say that a logic L is connected if it is the logic of a connected interior algebra. One can show that not every logic above S4 is a connected logic; but of course every subalgebra of B(R) is connected.…”

“…So not every logic above S4 is the logic of a subalgebra of B(R). (It is proved in Bezhanishvili et al, (2015) that every connected logic above S4 is the logic of a subalgebra of B (R). )…”

“…We follow Bezhanishvili et al, (2015) in referring to normal extensions of S4 as logics above S4. We refer to the formulas in a normal modal logic L as the theorems of L.…”