An axiomatic version of positive semilattice relevance logic

Abstract

In [8] and [9] Urquhart presented the semilattice semantics for a relevance logic with conjunction and disjunction. These semantics combined the simplest known semantics for relevant implication, with minimal modification of the truth conditions for conjunction and disjunction. A slightly different approach leads to the distinct positive relevance logic of [7] and [5]. Sound and complete axiomatic logics for the semilattice semantics were readily available, so long as disjunction was left out.The purpose of this paper is to report the remedy of this lack. In [1] the author developed work of Fine ([3], announced in [4]) and Prawitz [6] to show soundness and completeness of an axiomatic logic and two natural deduction logics, with respect to Urquhart's semantics. Thus Prawitz' positive relevance logic is equivalent to the semilattice logic, rather than the logic of [5]. We present here the axiomatic logic and show its equivalence to the semilattice semantics.In §1 the languages and logic are given, along with a few proof-theoretic remarks. In §2 the semantics are defined, and the soundness proof is sketched. In §3, the completeness proof is given.