Hierarchical semantics for relevant logics (Q1206806)

It is well known that a weak characterization of the relevance of a logic \(L\) is: whenever \(\lvdash_ L A\to B,A\) and \(B\) share a variable. In an earlier paper the author introduced a stronger ``depth relevance condition'' that requires that such a shared variable appear at the same depth in \(A\) as in \(B\). This depth counts the depth of \(\to s\) and not that of any other connective. In this paper the author introduces a ``hierarchical'' semantics based on this new condition. This is applicable to those logics to which the Routley-Meyer semantics could be applied, as well as to several weaker logics. Soundness and Completeness are proved. The author comments that there is little prospect of applying other styles of hierarchical semantics to logics that contain contraction or hypothetical syllogism principles.