The Logic of Free Choice Axiomatizations of State-based Modal Logics
Aleksi Anttila
Abstract:
We examine modal logics employing state-based semantics. In this type of semantics, formulas are interpreted with respect to sets of possible worlds.
The logics studied extend classical modal logic with a special non-emptiness atom ne and with the inquisitive disjunction. We make use of two distinct state-based notions of modality which are equivalent when applied to classical formulas but which come apart in our non-classical setting.
We obtain sound and complete natural deduction systems for three state-based modal logics, and show that each of the logics is expressively complete for the set of state properties invariant under state k-bisimulation for some finite k.
One of the logics studied extends Aloni’s bilateral state-based modal logic (BSML) with the inquisitive disjunction. This logic is bilateral: in addition to the positive support relation between states and formulas, a negative anti-support relation is used. The logic can be used to account for free choice (FC) inferences as Aloni does using BSML. The non-emptiness atom ne allows for the representation of a “pragmatic enrichment” of formulas by the principle “avoid stating a contradiction”. Narrow-scope FC inferences are derived as entailments involving pragmatically enriched formulas. The bilateralism is associated with a negation which tracks the anti-support clauses; this is used to model the interactions between natural language negation and FC inferences. Wide-scope FC inferences and epistemic contradictions are captured in states possessing specific properties; we define these properties using inference rules.