# Tarski's relevance logic; Version 2

## TL;DR

This paper defines Tarski's relevance logic, showing it includes many formulas and inference rules, and explores its properties and differences from other relevance logics, especially regarding completeness and axioms.

## Contribution

It introduces a new formulation of Tarski's relevance logic, highlighting its unique features and its divergence from the Anderson-Belnap system R.

## Key findings

- Contains many formulas and derived rules of inference.
- Excludes certain axioms like Contraposition.
- Provides a counterexample to a completeness theorem.

## Abstract

Tarski's relevance logic is defined and shown to contain many formulas and derived rules of inference. The definition arises from Tarski's work on first-order logic restricted to finitely many variables. It is a relevance logic because it contains the Basic Logic of Routley-Plumwood-Meyer-Brady, has Belnap's variable-sharing property, and avoids the paradoxes of implication. It does not include several formulas used as axioms in the Anderson-Belnap system $R$. For example, the Axiom of Contraposition is not in Tarski's relevance logic. On the other hand, the Rules of Contraposition and Disjunctive Syllogism are derived rules of inference in Tarski's relevance logic. It also contains a formula (not previously known or considered as an axiom for any relevance logic) that provides a counterexample to a completeness theorem of T. Kowalski (that the system $R$ is complete with respect to the class of dense commutative relation algebras).