Mirroring Theorems in Free Logic

Abstract

Linnebo and Shapiro have recently given an analysis of potential infinity using modal logic. A key technical component of their account is to show that under a suitable translation ${}^{◊}$ of nonmodal language into modal language, nonmodal sentences ${\varphi }_1,\dots ,{\varphi }_n$ entail $\psi$ just in case ${\varphi }_1^{◊},\dots ,{\varphi }_n^{◊}$ entail ${\psi }^{◊}$ in the modal logic S4.2. Linnebo and Shapiro establish this result in nonfree logic. In this note I argue that their analysis of potential infinity should be carried out in a free logic. I then extend their key theorems to the setting of negative free logic.