Dialetheism, Game Theoretic Semantics, and Paraconsistent Team Semantics

TL;DR

This paper introduces a paraconsistent variant of Dependence Logic based on game semantics, capable of expressing its own truth and validity, and establishes its relation to First Order and Second Order Logics.

Contribution

It develops a novel paraconsistent Dependence Logic with game semantics, showing its expressiveness, conservativity, and relation to classical logics.

Findings

Logic is a conservative extension of First Order Logic.

Validity problem reduces to that of First Order Logic.

Expressively equivalent to Universal Second Order Logic.

Abstract

We introduce a variant of Dependence Logic in which truth is defined not in terms of existence of winning strategies for the Proponent (Eloise) in a semantic game, but in terms of lack of winning strategies for the Opponent (Abelard). We show that this language is a conservative but paraconsistent extension of First Order Logic, that its validity problem can be reduced to that of First Order Logic, that it capable of expressing its own truth and validity predicates, and that it is expressively equivalent to Universal Second Order Logic. Furthermore, we prove that a Paraconsistent Non-dependence Logic formula is consistent if and only if it is equivalent to some First Order Logic sentence; and we show that, on the other hand, all Paraconsistent Dependence Logic sentences are equivalent to some First Order sentence with respect to truth (but not necessarily with respect to falsity).