🤖 AI Summary
This study addresses the distinction between logical contradiction and metaphysical impossibility, aiming to enable non-trivial reasoning about counterfactuals with impossible antecedents. To this end, it proposes a stratified possible worlds semantics that partitions worlds into logically normal and anti-logical ones, and introduces a metaphysical accessibility relation within the normal worlds. Coupled with a non-vacuous choice function mechanism, this framework avoids the vacuity problem commonly arising in counterfactual conditionals. Building on this semantics, the formal systems SCP and SCP1 are developed, for which soundness, completeness, and decidability are established. The resulting framework not only provides a fine-grained analysis of distinct kinds of impossibility but also demonstrates promising potential for extension to deontic and epistemic logics.
📝 Abstract
This paper presents two logic systems, SCP and its extension SCP1, to distinguish between different types of impossibility. The semantics use a stratified structure that partitions worlds into logic-normal worlds (N) and anti-logic worlds (I). By defining a metaphysical accessibility relation within N, the systems separate metaphysical impossibility from logical contradiction. This allows for logical reasoning to be maintained even when dealing with metaphysical impossibilities. To address the challenge of vacuism, SCP1 implements a non-empty constraint on the selection function, ensuring that counterpossibles with impossible antecedents are not trivially true but depend on the connection between the antecedent and the consequent. We provide proofs for the soundness, completeness, and decidability of both systems. Finally, we indicate the possibility of applying this stratified approach to other modal domains, such as deontic or epistemic logic.