A content semantics for quantified relevant logics. I

We present an algebraic-style of semantics, which we call a content semantics, for quantified relevant logics based on the weak system BBQ. We show soundness and completeness for all quantificational logics extending BBQ and also treat reduced modelling for all systems containing BB ^d Q. The key idea of content semantics is that true entailments AB are represented under interpretation I as content containments, i.e. I(A)I(B) (or, the content of A contains that of B). This is opposed to the truth-functional way which represents true entailments as truth-preservations over all set-ups (or worlds), i.e. (VaK) (if I(A, a) = T then I(B, a)= T).This paper was presented to the Australasian Association for Logic Conference, held at the University of Auckland from 9–12th July, 1986. I wish to thank those present for some helpful comments. I also wish to thank Kit Fine for some useful discussion on some topics of this paper.