Quantified Modal Logic on the Rational Line

In the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line. In the topological semantics for propositional modal logic (McKinsey, 1941; McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963), it is well known that (S4all) S4 is complete for the class of all topological spaces, (S4Q) S4 is complete for the rational line, (S4C) S4 is complete for Cantor space, and (S4R) S4 is complete for the real line.1 Rasiowa & Sikorski (1963) extend the topological semantics to quantified modal logic. Let QS4 be classical first-order logic, without identity, enriched with a modal operator satisfying the axioms of S4. The four above results suggest four conjectures: (QS4all) QS4 is complete for the class of all topological spaces, (QS4Q) QS4 is complete for the rational line, (QS4C) QS4 is complete for Cantor space, and (QS4R) QS4 is complete for the real line. Rasiowa & Sikorski (1963) prove (QS4all) but leave (QS4Q), (QS4C), and (QS4R) open. (QS4R) fails, since QS4 is not complete for any locally connected space (Theorem 3.4). The main result of the current paper is a strong version of (QS4Q): QS4 is complete, indeed strongly complete, for Q with a constant countably infinite domain (Theorem 2.5). This result follows from a more general strong completeness theorem, Theorem 6.1.2 (QS4C) remains open. Received: March 3, 2013. 1 See Rasiowa & Sikorski (1963), Theorem XI, 9.1, which is derived from McKinsey (1941) and McKinsey & Tarski (1944) . (S4Q), (S4C), and (S4R) are special cases of Rasiowa & Sikorski (1963), Theorem XI, 9.1, (vii), which states that S4 is complete for any dense-in-itself metric space. Kremer (2013) strengthens this: S4 is strongly complete for any dense-in-itself metric space. 2 The proof here grew out of my work on two-dimensional propositional modal logic: after reading that work, Valentin Shehtman drew my attention to its connection to topologically interpreted quantified modal logic. In particular, he alerted me to the result in Rasiowa & Sikorski (1963) that QS4 is sound and complete for all topological spaces with a constant domain, and claimed without proof that, in a language without function symbols, QS4 is complete for Q with a constant domain. The proof in the current paper is an application of the technique I had originally c © Association for Symbolic Logic, 2014 439 doi:10.1017/S1755020314000021