On purely relevant logics.
Arnon Avron · Notre Dame Journal of Formal Logic · 1986
ARNON AVRON / IntroductionThe system RMI~ (which consists of the implication-negation axioms of RM) was investigated in [3] and shown there to be an optimal relevance logic in its language.We note there, however, that one cannot add to it an i?-style extensional conjunction Λ, with A κB-+ A, A AB-+ B as axioms and the adjunction rule of inference (A, B \-A A B), without losing its relevance character (see [1], 29.5, and [3], III.8).This state of affairs is not altogether surprising.Anderson and Belnap faced a similar problem when they came to add to R^ (or E^) extensional connectives.In R~, e.g., the meaning of -> is given by the "relevant deduction theorem", according to which a sentence of the form A x -• (A 2 -•...-> (A n -+ B)...) is provable in R~ iff there is a proof in R~ of B from the assumptions A u ... ,A n which uses all theAfs.(Here the meaning of "proof is the usual one, while the meaning of "use" is to be understood according to the relevantist's analysis of this term (see [1], Chapter 1). ) Accordingly, if one wishes to add to R^ an extensional conjunction such that A ΛB \-A, A ΛB \-B and A, B h A A B are all valid modes of inference, then he must recognize A A B -> A, A Λ B -+ B and A -• (B -> A A B)as valid sentences.However, it is well known that by adding these schemes to R^ we get classical logic.Anderson and Belnap's first step in order to solve this difficulty was to give up A -+ (B -> A Λ B) as a valid sentence and to introduce instead adjunction as a new, primitive rule of inference (besides M.P. for ->).A second, unavoidable step was to propose some new concepts of "proof" relative to which some version of the deduction theorem does hold.(In [1] and [5] three competing definitions can be found of what a "proof" in R or E is.ι This is an obvious evidence that the relevantists have no clear intuition at this point.)These concepts of proofs all seem ad hoc and entail many absurdities.Consider an example: A Λ (B -+ B) can be inferred, according to them, if we assume both A and B -> B but not if we assume A alone, although B -> B is a logical truth of the system and so it would be ridicuous to pretend assuming it.