Reassurance for the Logic of Paradox

Abstract

Counterexamples to reassurance relative to a "less inconsistent" relation between models of the logic of paradox are provided. Another relation, designed to fix the problem in logic without equality, is introduced and discussed in connection with the issue of classical recapture."it has . . . a certain consonance with common sense which makes it inherently credible. This, however, is not a merit upon which much stress should be laid; for common sense is far more fallible than it likes to believe." (B. Russell) §1. Truth and falsehood. The logic of paradox is the natural paraconsistent logic arising from classical logic by simply dropping the principle of noncontradiction. 1 Thus, an LP-model A, with nonempty universe |A|, is exactly like an ordinary model, except that n-ary relation symbols are interpreted by ordered pairs of their respective extension and antiextension r + A , r − A , such that the exhaustiveness requirement (excluded middle) r + A ∪ r − A = |A| n is met. Constants and function symbols are interpreted, as usual, by objects and functions. Likewise, a valuation is still a function of the set of the variables and the valuation v to |A| extends canonically to an interpretation v A of the terms. The truth and falsehood in a model with respect to a valuation are defined inductively, as follows:

An alternative approach for quasi-truth

“…The semantics for the first-order version of LP, proposed by Priest, (see, for instance, [15] and [6]) is given by means of LP-models A, which are usual tarskian structures, except that any n-ary relation symbol P is interpreted as an ordered pair P The truth and falsehood of sentences in a structure A are defined inductively in the following way:…”

Negation as Cancellation, Connexive Logic, and qLPm

“…5 In that paper, Sylvan (i.e., Routley) introduces a binary lattice-ordered content containment relation on the set of formulas satisfying the condition that A contains B whenever every atomic formula that occurs in B also occurs in A. Sylvan then semantically defines a conditional by assuming the truthconditions of implications in the ternary Routley-Meyer semantics for relevance logics and by demanding, in addition, containment of the consequent in the antecedent of the conditional. 6 Another option for relating containment and entailment is given by defining containment in terms of a notion of entailment and subject-matter inclusion. Such a proposal can be found in Yablo's [33, p. 3…”

What If? The Exploration of an Idea

“…There are also versions of the Curry paradox involving a validity predicate directly. 16 These may also be solved by Strategy 1.…”

“…The rules are exactly what one would expect if A B is simply an object-language way of expressing what a metalanguage deduction from A to B expresses. 16 See, e.g., Beall and Murzi (2013).…”