Johannes Korbmacher: Exact Truthmaker Semantics for Propositional Modal Logic(s)

I develop an exact truthmaker semantics for propositional modal logic. The idea of exact truthmaker semantics is that we can give the semantic content of a statement by saying what precisely it is in the world that makes the statement true: by giving its exact truthmakers. Intuitively here, an exact truthmaker of a statement is a state (of affairs) such that whenever the state obtains, it is directly and wholly responsible for the truth of the statement.

In particular, an exact truthmaker of a statement will not contain as a part any other state that is not wholly relevant to the truth of the statement. This idea traces back to a paper by Bas van Fraassen (1969), who uses it to give adequate truth-conditions for Anderson and Belnap's tautological entailments (Anderson and Belnap 1962).  A central result of van Fraassen’s paper is a characterization of FDE-consequence (the consequence relation of the logic of first-degree entailment) in terms of exact truthmaking. Let’s say that a state is an inexact truthmaker of a statement iff it contains an exact truthmaker as a part. Then, van Fraassen’s result is that preservation of inexact truthmakers in all models coincides with FDE-consequence. In this paper, I shall prove the corresponding result for (propositional) modal languages: preservation of inexact truthmakers in all modal exact truthmaker models coincides with the consequence relation of K[FDE]: the consequence relation of the smallest normal modal logic extending FDE (Priest 2008). Given natural conditions on these exact truthmaker models, this result also holds for stronger normal modal logics extending FDE, such as T[FDE], S4[FDE], and S5[FDE]. I shall then briefly discuss to what extent this result can be transferred to other background-logics in the vicinity of FDE, such as the logic of paradox LP, strong Kleene logic K3, and classical logic. Finally, I shall illustrate the applications of the semantics by looking at the concepts of partial content and analytic equivalence for modal statements. In particular, I shall extend Kit Fine’s semantics for these concepts (Fine 2015) and axiomatize the resulting modal logic of partial content and analytic equivalence.

When & where?

Thu 31 March, 3.15-5 pm
Room Gamma, Faculty of Philosophy, University of Groningen, Oude Boteringestraat 52