Frame definability in second-order arithmetic

Abstract

We study the reverse-mathematical strength of frame definability in modal logic. The central principle is the Valuation Extension Lemma (VEL), which asserts that every assignment of propositional variables on a frame extends to a full valuation. We show that, over $\mathrm{RCA}_0$, VEL is equivalent to $\mathrm{ACA}^{+}_0$, and as are frame-definability principles for Geach axioms and for $\mathbf{GL}$. We also obtain analogous $\mathrm{ACA}^{+}_0$-equivalences for the Barcan and Converse Barcan formulas in modal predicate logic. Finally, we examine variants of VEL for $\mathbf{CTL}$ and $\mathbf{LTL}$ and locate their strengths between familiar subsystems of second-order arithmetic.