Semantic Investigations of De Re Modal Logics of Terms

Abstract In this paper, by a de re modal logic of terms, we mean a set of tautologies determined by a given semantics. We present a comprehensive study of several such semantics. First, we analyse semantics with a natural interpretation of de re modal propositions. We use the Johnson–Thomason models, but reject their four artificial conditions. Second, we discuss Johnson–Thomason semantics, which—with the unnatural interpretation of some modal propositions—is adequate for McCall’s axiomatic system L-X-M of Aristotle’s assertoric and apodeictic syllogistic. Finally, by weakening Johnson’s two controversial conditions for models and adopting the natural interpretation of modal propositions, we achieve an effect similar to that obtained by Johnson and Thomason. The phrases ‘is necessarily’ and ‘is possibly’ used in de re modal propositions can be interpreted in two ways: either literally, or as ‘is essentially’ and ‘is potentially’, respectively. The first interpretation Thomason uses in [23, 24], and the second one Johnson uses in [4]. With the second interpretation, following Thom’s lead, we consider the so-called Principle of Predication regarding properties. In the paper, we refer to the works of Johnson, Thom, and Thomason. The final part of the paper is devoted to providing suitable axiomatic systems for all previously given semantics. Proven adequacy theorems confirm that these systems express and derive all valid formulas under a given semantics.