Completeness and Decidability Results for Some Propositional Modal Logics Containing “Actually” Operators

Abstract

The addition of "actually" operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing "actually" operators has concentrated entirely upon extensions of KT5 and has employed a particular modeltheoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing "actually" operators, the weakest of which are conservative extensions of K, using a novel generalisation of the standard semantics.

The Meaning of ‘Actually’

“…A simple way to achieve this is to introduce, in each model, a distinguished index and adopt (C). The same effect can be achieved in other ways, such as that in Gregory (2001, 64–65). Thus, at least within the standard model‐theoretic framework for modal logic, validating ‘@ p ⊃ □@ p ’ requires a treatment of ‘@’ that will either incorporate (C) or, at any rate, somehow guarantee that if ‘@ p ’ is true at an index i , it is also true at all indices accessible from i .…”

Actuality, Necessity, and Logical Truth

“…10 Gomez-Torrente (2000) argues persuasively that the notion of an interpretation that Tarski had in mind when he wrote his 1936 paper on logical consequence and logical truth is broader than the set-based notion of an interpretation that has become standard. 11 This approach is taken by Gregory (2001). 12 (R) is closely related to Chalmers' (1996, pp.…”

The Semantics of Actuality Terms: Indexical vs. Descriptive Theories

“…Gregory (: 62) derives ‘Ap ⊃ □Ap’ from ‘◊(Aφ ⊃ ψ) ⊃ (Aφ ⊃ ◊ψ),’ which he takes as an axiom. He consequently rejects ‘□(p ≡ Ap).’ Now the general form ‘◊(p ⊃ q) ⊃ (p ⊃ ◊q)’ is obviously not valid: let ‘q’ be necessarily false and ‘p’ contingently true.…”