Counterfactual Logic and the Necessity of Mathematics

Abstract

This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne (2018), who seek to establish that mathematics is committed to its own necessity. I demonstrate that their assumptions collapse the counterfactual conditional into the material conditional. This collapse entails the success of counterfactual strengthening (the inference from 'If A were true, then C would be true' to 'If A and B were true, then C would be true'), which is controversial within counterfactual logic, and which has counterexamples within pure and applied mathematics. I close by discussing the dispensability of counterfactual conditionals within the language of mathematics.

Counterpossibles

“…As we discussed earlier, some nonvacuists explicitly deny this (Berto et al., 2018, p. 704). Furthermore, as Elgin (2021, Section 4) observes, the indispensability claim is in tension with the logic of counterfactuals Yli‐Vakkuri and Hawthorne adopt, on which counterfactuals are (informally provably) equivalent to the material conditional. Of course, that's only for counterfactuals within the language of pure mathematics, but that's precisely what's at issue when justifying the necessity of mathematics.…”