Non-exhaustive and non-exclusive structural solutions to the Validity Paradox

PAILOS, FEDERICO MATÍAS

Buenos Aires

Congreso; 17th edition of the International Congress on Logic, Methodology and Philosophy of Science and Technology (CLMPST 2023); 2023

There are many kinds of semantic paradoxes. One thing that all of them have in common is that, in order to derive the undesirable result—i.e., triviality, the empty sequent or what not—it seems necessary, at some point, to apply an inferential principle associated with a given logical constant, i.e., some kind of introduction or elimination rule for the given connective.Recently, Beall and Murzi presented in [1] what seems like a version of Curry Paradox. Nevertheless, what seems more relevant from this paradox is that no inferential principle involving any logical connective—i.e., no operational principle is involved in the derivation of the undesirable result. The only inferential rules used are (i) principles related to a naive validity predicate—i.e., the Validity Paradox rule—the rule for introducing the predicate on the right side of the consequence sign—and the Validity Detachment rule (i.e. the rule forintroducing the predicate on the left of the consequence sign), and (ii) structural principles like Cut and Contraction not related in a specific way to any specific logical constant. To solve the paradox it is necessary to give up at least one of these principles. And operational approaches—i.e., the ones that validate every classically valid structural principle—seems to have no choice but to give up one of the principles that regulate the naive use of the validity predicate. Thus, operational approaches cannot be the basis of a theory of naıve validity—or at least cannot be the basis of one that validate both na¨ıve principles.In this presentation, we will present what seems like an unteanable position: an operational solution to the Validity Paradox that retains both naıve principles while not giving up no structural rule. The main idea behind these solutions is to break up the link between satisfaction of an inference and not being a counterexample to that sequent—or, similarly, to being a counterexample to an inference and not satisfying it. A philosophical consequence of this technical move is that, in some of these theories, VD will be accepted, while at the same time rejected, while in others, VD won’t be rejected, though it will not be accepted. Anyway, it will not be possible to have the problematic instances of VD as axioms of the derivations that lead to the paradoxical result.References[1] J. Beall and J. Murzi. Two flavors of curry paradox. Journal of Philosophy, 110:143–65, 2013.