Strong Completeness of S4 for any Dense-in-Itself Metric Space

In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space. In the topological semantics for modal logic (McKinsey, 1941; McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963), S4 is well-known to be complete for the class of all topological spaces, as well as for a number of particular topological spaces, notably the rational line, Q, the real line, R, and Cantor space, C. The results for Q, R, and C are special cases of S4’s completeness for any dense-in-itself metric space: see Rasiowa & Sikorski (1963, Theorem XI, 9.1), which is derived from McKinsey (1941) and McKinsey & Tarski (1944). In Kripke semantics, it is customary to strengthen the completeness of a logic, L, for a Kripke frame (a class of frames) to strong completeness, that is, the claim that any L-consistent set of formulas is satisfiable at some point in the frame (in some frame in the class): the simplest completeness arguments, using canonical frames and models, tend to deliver this stronger result.1 In topological semantics, as long as the language is countable, the construction used to prove the completeness of S4 for Q can be slightly amended to show the strong completeness of S4 for Q. But no similarly easy amendment is available for R or for C, and the question of the strong completeness of S4 for these spaces has until now remained open. In the current paper, we prove that S4 is strongly complete for any dense-in-itself metric space, including R and C. It should be noted that the question of strong completeness in topological semantics has not been widely addressed. That said, given the well-known identification of reflexive transitive Kripke frames with Alexandroff spaces (see p. 549, below), and given the wellknown strong completeness of S4 for the class of countable reflexive transitive Kripke frames (Makinson, 1966), S4 is strongly complete for the class of countable Alexandroff spaces and therefore for the class of all topological spaces. Also, Theorem XI, 10.2 (v) in Rasiowa & Sikorski (1963) immediately entails a kind of restricted strong completeness of Received: February 26, 2013. 1 Such arguments were first published in Makinson (1966). There are, however, modal logics L that are complete for some class of Kripke frames for L, but strongly complete for no class of Kripke frames for L. Standard examples are GL and Grz: see Zakharyaschev et al. (1997), especially Section 1.5, “Stronger forms of Kripke completeness”. c © Association for Symbolic Logic, 2013 545 doi:10.1017/S1755020313000087