Invariance conditions for truthmaker models

Konstantinos Konstantinou, Konstantinos Konstantinou

When dealing with a particular sort of semantics, logicians are accustomed to thinking about invariance conditions for the relevant models. That is, they examine how two prima facie distinct models must be related to one another in order to “count as the same,” i.e., to satisfy the same formulas of the interpreted language. However, (to my knowledge) we still lack such invariance criteria when it comes to truthmaker models. How must two truthmaker models be related to one another so as to verify and falsify the same formulas of some language L? This talk suggests some directions for moving toward an answer. My entry point is the translation proposed by Van Benthem (2019), and elaborated by Knudstorp (2023), between truthmaker semantics and the models for the so-called “modal information logic.” The satisfaction of modal formulas is known to be invariant under a family of bisimulation relations. In virtue of Van Benthem’s translation, it therefore makes sense to ask whether truthmaker models also exhibit invariance under some variant of bisimulation. Of course, invariance conditions for modal information logic and truthmaker semantics must differ somehow, as the latter (1) rests on non-classical negation and (2) does not by default interpret languages including modal operators. But after some manoeuvres, variants of bisimulation suggest promising invariance results for different kinds of truthmaker semantics. None of these results is global (yet), partly because the candidate proof techniques are sensitive to one’s assumptions about how the mereology of states interacts with verification and falsification. Still, beginning to think in terms of invariances can allow us to discover how much truthmaker models can “see” in a language L, and thus a principled way of arguing about the indefinability of properties. Moreover, introducing invariance conditions is bound to enrich the philosophical applications of truthmaker semantics (e.g., subject matter), especially in light of the realization that the framework is not limited by a unique carving of the world into states and their parts.