, Volume 13, Issue 2, pp 127-133

Intuitionistic mathematics does not needex falso quodlibet

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Abstract

We define a system IR of first-order intuitionistic relevant logic. We show that intuitionistic mathematics (on the assumption that it is consistent) can be relevantized, by virtue of the following metatheorem: any intuitionistic proof of A from a setX of premisses can be converted into a proof in IR of eitherA or absurdity from some subset ofX. Thus IR establishes the same inconsistencies and theorems as intuitionistic logic, and allows one to prove every intuitionistic consequence of any consistent set of premisses.

This paper grew out of discussion of a survey talk, on earlier work, that I gave to the 5th A.N.U. Paraconsistency Conference in January 1988. I am greatly indebted to the suggestion by Michael MacRobbie on that occasion that I investigate the so-called “non-Ketonen” form of the sequent rule for ⊃ on the right. That suggestion inspired the correspondingly modified rule of ⊃ Introduction in the system of natural deduction given above.