Cyclic Negations and Four-valuedness

TL;DR

This paper explores four-valued logic inspired by quantum computations, focusing on cyclic negations and the formal axiomatization of related logical systems with completeness proofs.

Contribution

It introduces a novel four-valued semantics with cyclic negation inspired by quantum concepts and provides axiomatizations with correctness and completeness results.

Findings

Defined two variants of logical matrices with different truth value orders

Axiomatized the logics of these matrices as systems of binary consequence

Proved correctness and completeness theorems for the deductive systems

Abstract

We consider an example of four valued semantics partially inspired by quantum computations and negation-like operations occurred therein. In particular we consider a representation of so called square root of negation within this four valued semantics as an operation which acts like a cycling negation. We define two variants of logical matrices performing different orders over the set of truth values. Purely formal logical result of our study consists in axiomatizing the logics of defined matrices as the systems of binary consequence relation and proving correctness and completeness theorems for these deductive systems.