A Family of Strict/Tolerant Logics
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A Family of Strict/Tolerant Logics Melvin Fitting1 Received: 26 December 2019 / Accepted: 27 July 2020 / © Springer Nature B.V. 2020
Abstract Strict/tolerant logic, ST, evaluates the premises and the consequences of its consequence relation differently, with the premises held to stricter standards while consequences 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. A version of this result has been extended to meta, metameta, . . . consequence levels in Barrio et al. (2019). In my earlier paper Fitting (2019) 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. This family includes both Kleene’s and Priest’s logic individually, 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. The present paper extends that generalization in two directions. We examine a reverse notion, of Tolerant/Strict logics, which exist for the same structures that were investigated in Fitting (2019). And we show that the generalization extends through the meta, metameta, . . . consequence levels for the same infinite family of many valued logics. Finally we close with remarks on the status of cut and related rules, which can actually be rather nuanced. Throughout, the aim is not the philosophical applications of the Strict/Tolerant idea, but the determination of how general a phenomenon it is. Keywords Strict/Tolerant · Bilattice · Many valued logic · Kleene logic · Logic of paradox · First degree entailment
Melvin Fitting
[email protected] 1
Departments of Philosophy, Computer Science, Mathematics (all emeritus), The Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY 10016, USA
M. Fitting
1 Introduction Sequents are used with a wide variety of logics, ⇒ where and are finite sets of formulas, but the basic idea is generally the same: if all members of are true then some member of is true. Or what is usually equivalent, if the conjunction of is true then the disjunction of is true. Of course the meaning of “true” is logic dependent, but whatever its meaning is, it applies on both sides of the ⇒ symbol. A case has been made, in [6] for instance, that the antecedents (formulas on the left) of a sequent should be held to higher standards than the consequents (formulas on the right). Consider physics as an example. We have some theory, relativity, quantum, Newtonian, whatever, and in deducing things from such a theory we adopt as premises the ‘laws’ of the theory, which are understood as simply true. We derive consequences, and in testing these we examine the univ
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