Towards a truthmaker semantics for modal logics

Pietro Berardi Gili (Universitá della Svizzera Italiana)

In this talk, I present an exact truthmaker semantics for the language of propositional modal logic that is powerful enough to interpret Kripke's possible worlds semantics and more general than the other truthmaker semantics approaches to modal logic extant in the literature (such as those of Kim (2024) and Litland (2024)). To do so, I build upon an idea already introduced by Kit Fine and Dongwoo Kim, that of augmenting standard modalized state spaces with two states-to-propositions relations that behave as modal counterparts of exact verification and falsification: exact exclusion and exact allowance. After having developed a semantics based on this idea, I show that the weakest normal modal logic K is both sound and complete with respect to a particular class of truthmaker models. Additionally, I establish that the semantics developed interprets standard possible worlds semantics, in the sense that given any class of Kripke frames C there is a corresponding class of modal state spaces C' such that logical consequence in C coincides with loose consequence in C'. Finally, I present an axiomatization of the logic of single-premise exact entailment in the class of all modal state spaces, and I establish the relevant soundness and completeness results. After having presented these metalogical results, I compare my framework to one of the most developed truthmaker semantics approaches to modal logic, that of Kim (2024). Specifically, I argue that my framework is preferable to Kim's since (i) unlike his, it allows for state spaces in which there are no possible worlds (which are needed to represent metaphysical views according to which modal reality is essentially open and incompletable), and (ii) it extends naturally to the semantics of non-normal and non-classical modal logics. I conclude by presenting an overview of the main results and briefly sketching some directions for future work.