The implicational fragment RI of the relevance logic R can be axiomatized, with substitution and detachment as rules, by the following four axioms:

          CCpqCCqrCpr / (p
q)((qr)(pr))            B' / Syl
          CCpCqrCqCpr / (p
(qr))(q(pr))            C / Com
          Cpp                      p
p                                                 I  / Id
          CCpCpqCpq     / (p
(pq))(pq)                    W/ Hilbert

 
The more standard axiomatization of RI uses B / Syl* = CCpqCCrpCrq / (pq)((rp)(rq)) in place of B' / Syl (whence the common alternate name "BCIW" for RI).   Of course B' and B are interdeducible in the presence of C, but in my experience B' has seemed to be more generally useful in obtaining proofs of other results than B in a variety of settings.
          Anderson and Belnap ask [Entailment: the logic of relevance and necessity, Princeton University Press, Princeton, 1975] if there exists a single axiom for RI, and
the question was answered affirmatively by Rezus [On a theorem of Tarski, Libertas mathematica, vol. 2 (1982), pp. 63-9], who showed how to construct (but did not actually display) such an axiom.  Written out in full, the Rezus axiom developed from the base shown above is of type <93, 23>, that is, is 93 symbols in length and contains occurrences of 23 distinct sentence letters:

CCCpCCCqqCCrrCCssCCttCpuuCCCCCvwCCwxCvxCCCCCyCyzCyzCCCdeCCefCdfgghhCCiCCCjjCCkkCCllCCmmCinnoo.

 

   

NEW RESULT
Th
e following theorem [proof here] of RI of type <35, 8> gives B', C, I, and W [proof here] and, so, is a single axiom for RI:

                    CCCCCpqCrpCCpqCrqCCssCtCuvCCwtCuCwv.

OPEN QUESTION
Is there a shorter single axiom?  (The author expects that the answer is "yes".)

 

Dolph Ulrich, 2007


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