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Logic and Computation Seminar

Monday, February 14, 2022 - 3:30pm

Melvin Fitting



University of Pennsylvania

online - link available upon request

Strict/tolerant logic, ST, evaluates the premises and the conclusions of its consequence relation differently, with the premises held to stricter standards while conclusions are treated more tolerantly.  More specifically, ST is a three-valued logic with left sides of sequents understood as if in Kleene's Strong Three Valued Logic, and right sides as if in Priest's Logic of Paradox.  Surprisingly, this hybrid validates the same sequents that classical logic does, though it differs from classical logic at the metaconsequence level.  A version of this result has been extended to meta, metameta , etc. consequence levels, creating a very interesting hierarchy of logics.  All this raises the question of what it means for logics to be the same.  Here we have agreement on consequence relations, but this does not seem to be enough.

In four recent papers I showed that the original ideas behind ST are, in fact, much more general than first appeared, and an infinite family of many valued logics have Strict/Tolerant counterparts.  Besides classical logic, this family includes both Kleene's and Priest's logic themselves, as well as first degree entailment.   For instance, for both the Kleene and the Priest logic, the corresponding strict/tolerant logic is six-valued, but with differing sets of strictly and tolerantly designated truth values.  There is a reverse notion, of Tolerant/Strict logics, which exist for the same structures.  And the hierarchy going through meta, metameta, \ldots consequence levels exists for the same infinite family of many valued logics.

I will present a sketch of the basic generalizations, of Strict/Tolerant and Tolerant/Strict, but I will not have time to discuss the hierarchies of such logics, nor will I have time to give proofs, beyond a sketch of the ideas involved.  Throughout, my aim is not the philosophical applications of the Strict/Tolerant idea, but the determination of how general a phenomenon it is.