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 ϕ1,…,ϕn entail ψ just in case ϕ◊1,…,ϕ◊n entail ψ◊ 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.
Bibliographical noteThanks to Øystein Linnebo for discussion of these issues and to Stewart Shapiro and an anonymous referee for helpful comments on a draft of this paper.
- potential infinity
- modal logic
- free logic
- mirroring theorem