On All Strong Kleene Generalizations of Classical Logic

Abstract

By using the notions of exact truth ('true and not false') and exact falsity ('false and not true'), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the SK P and SK N calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the SK P and the SK N calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the SK P and the SK N calculus, we also hint at its philosophical significance.

From many-valued consequence to many-valued connectives

“…In many-valued logics, more notions of consequence become available as the set of truth values expands, and similarly the space of binary operators quickly increases. Perhaps as a result of the greater freedom in the choice of those parameters -though sometimes due to independent desiderata -various popular systems of many-valued logics rest on the choice of a conditional operator that fails one or the other direction of the deduction theorem, and thereby fails to internalize logical consequence adequately in the object-language (see Avron 1991;Cobreros et al 2015;Wintein 2016). For instance, a popular logic such as Strong Kleene's K3, in which v(A → B) = v(¬A ∨ B) = max(1 − v(A), v(B)), and where consequence is defined as the preservation of the value 1, loses conditional introduction (A A, but A → A).…”

“…A recent exception isWintein (2016) looking at 3-valued and 4-valued mixed consequence relations, but not at intersective mixed relations.3 Computer-aided investigations of this kind still seem quite rare, which is striking considering that some pioneers such asFoxley (1962) had bravely started deploying them for very related tasks, when much more ingenuity was needed to compensate for the lower power of computers.…”

OUP accepted manuscript

“…Some additional criteria might be considered for a consequence relation to count as logical (setting aside Tarski's conditions, which would preclude our admission of mixed consequence relations). One such constraint is what we call the constraint of representability, namely for a consequence relation to be associated with a binary conditional operator satisfying the deduction theorem (see [31] for similar considerations). Some of our respectable consequence relations are not representable when the associated set of truth values is not well-ordered.…”

“…The question we propose to investigate in this paper is exactly this: is there a sense in which the three definitions of logical consequence we mentioned form a natural class? Our proposal is to identify plausible desiderata on a consequence relation in many-valued logic, allowing us to do an exhaustive search of candidates and to narrow down the set of possible consequence relations to a distinguished subset (see [31] for a related project in the case of 4-valued logic). The way we proceed is as follows: we start out by examining the three schemes mentioned above, namely pure consequence, mixed consequence, and order-theoretic consequence, to see what they have in common (section 2).…”

Interpolation Methods for Dunn Logics and Their Extensions

“…[24]), a tableau calculus that gives us a uniform syntactic approach to the Dunn logics. The Dunn calculus is a signed tableau calculus with signs coding for truth (1), falsity (0), non-truth (1) and non-falsity (0).…”