Propositional Team Logics

TL;DR

This paper explores propositional team semantics, defining a hierarchy of logics with varying expressive power, and provides axiomatizations for key cases, advancing understanding of dependence and independence in propositional logic.

Contribution

It introduces the full propositional team logic and characterizes a hierarchy of intermediate logics, including syntactic and semantic descriptions and complete axiomatizations.

Findings

Defined the full propositional team logic.

Characterized intermediate logics in the hierarchy.

Provided complete axiomatizations for key logics.

Abstract

We consider team semantics for propositional logic, continuing our previous work (Yang & V\"a\"an\"anen 2016). In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We define an expressively maximal propositional team logic, called full propositional team logic. This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. classical propositional logic, and the full propositional team logic. We characterize these different logics in several ways: first syntactically by their logical operations, and then semantically by the kind of sets of teams they are capable of defining. In several important cases we are able to find complete axiomatizations for these logics.