Restricted Interpolation and Lack Thereof in Stit Logic

TL;DR

This paper investigates the restricted interpolation property in propositional stit logic with multiple agents and finds that it holds only under specific conditions related to the number of agents and how action operators are treated.

Contribution

It establishes the precise conditions under which restricted interpolation holds or fails in stit logic, clarifying the logical properties of this framework.

Findings

Restricted interpolation holds iff there are at most three agents when action operators are logical symbols.

Restricted interpolation fails for any number of agents greater than one if action operators are non-logical symbols.

Unrestricted Craig interpolation generally fails in almost all versions of stit logic.

Abstract

We consider the propositional logic equipped with Chellas stit operators for a finite set of individual agents plus the historical necessity modality. We settle the question of whether such a logic enjoys restricted interpolation property, which requires the existence of an interpolant only in cases where the consequence contains no Chellas stit operators occurring in the premise. We show that if action operators count as logical symbols, then such a logic has restricted interpolation property iff the number of agents does not exceed three. On the other hand, if action operators are considered to be non-logical symbols, the restricted interpolation fails for any number of agents exceeding one. It follows that unrestricted Craig interpolation also fails for almost all versions of stit logic.