On Tarski's axiomatic foundations of the calculus of relations

TL;DR

This paper analyzes Tarski's axioms for the calculus of relations, demonstrating their independence, redundancy, and equivalence under various modifications, thereby clarifying the foundational structure of the system.

Contribution

It shows the independence and redundancy of Tarski's axioms and establishes their equivalence under certain modifications, enhancing understanding of the calculus of relations.

Findings

Tarski's ten axioms are independent.

Replacing one axiom yields an equivalent but redundant set.

Including both distributive laws makes some axioms redundant.

Abstract

It is shown that Tarski's set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski's axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski's other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski's axiom system.