Abstract:
This paper will deal with combinators, relations and
relevance. It builds on Tarski's investigations into binary relations,
together with Jonsson and Tarski on Boolean algebras with
operators. These suggest the "Kripke-style" worlds semantics for
relevant logics developed in my work with Routley, on which binary
particles like relevant -> are naturally linked to ternary relations
of accessibility.
The general 3-ary relation is the mother of all n-ary relations, under
relational composition. (By contrast, composing binary relations only
leads to more 2-ary ones.) I recall how the 5 graduate logic courses
that I took years ago at Pitt all set out from the truth-functional
logic 2. This is a fine place to start. But I always hankered to move
on to relations.
Working on the semantics of entailment has been a way of moving
on. For the postulates appropriate to various substructural logics are
neatly expressed via the composition of 3-ary relations. When so
expressed they have a Combinatory Logic (CL) flavor. The postulate for
the B axiom LOOKS LIKE "R2abcd IMPLIES R2a(bc)d"; for the W axiom,
LIKE "Rabc IMPLIES R2abbc". People who know CL (or LAMBDA) will NOT be
surprised.
As Brady, Dunn, Belnap, Fine, Urquhart and others have stressed in
their relevant work, there are various perspectives about what it all
means. One perspective is algebraic. Another is relational
semantical. At root, these are the SAME perspective, via an
ALTERNATION OF GENERATIONS. The marvelous coincidence of CL and
relational insights is also NOT so coincidental; as, depending on the
type-theoretic insights of Coppo, Dezani, Venneri, Barendregt et al.,
I shall argue.