Abstract

We add to our relational semantics for system r of relevant implication an s4-Type semantics for necessity, Thus furnishing a semantics for system nr (believed to coincide with e). In an nr-Structure m=(o,K,R,S, ), O is in k, Is an operation on k, And r and s are relations on set k which are restrained by reflexivity, Transitivity and monotonicity requirements. Interpretations on m, Which are restricted by an ordering requirement, Are distinguished from modal evaluations by adequate rules for implication and negation. The main work of the paper goes into proving semantical completeness of nr. The semantics is applied to show that rule gamma holds for nr, And that r and nr are reasonable in the sense of hallden and hence have normal characteristic matrices