Validity in a paraconsistent framework

PAILOS, FEDERICO MATÍAS

Buenos Aires

Simposio; 16st Latin American Symposium on Mathematical Logic XVI SLALM; 2014

Universidad de Buenos Aires

The so called Validity Paradox can be used to argue that validity is an inconsistent notion that must be characterized in a paraconsistent framework. But, can paraconsistent positions express the concept of validity? Beall 2009 and Beall and Murzi 2012 claim that paraconsistent positions can not do that. We will show how it is possible to do so (both in dialetheist and non-dialetheist frameworks). We will adopt Toby Meadows general strategy in order to give a (paracomplete, in Meadows system) predicate of validity. In Meadows unpublished, he adopts Kripkes fixed point construction for a validity predicate. Its interpretation has a set of pairs of formulae as an extension, and another one as its anti-extension. There will be no formulae that belong to the intersection of both, but there will be pairs of formulae that are neither in the extension nor in the anti-extension. Meadows use a jump operator that adopts a tolerant-strict clause to determine the extension, and a strict-tolerant (or classical) clause to determine the anti-extension. The final interpretation of the validity predicate will be the minimal fixed point of the construction. But it will have some flaws. Meadows define the consequence relation in terms of the validity predicate. And as the interpretation of the validity predicate wont be reflexive, nor monotonic nor transitive, neither will be its consequence relation. We can import Meadows strategy to a paraconsistent framework. We will present versions of it in Ripleys Strong-Tolerant logic, and in LP. The validity predicate will be define in exactly the same way as Meadows does, and so the construction will be non-decreasing, and will eventually reach a fixed point. As the consequence relation of both systems is defined in an independent manner, the validity predicate wont represent exactly the consequence relation of neither of both logics. But both consequences relations will be reflexive and monotonic. Each of them has advantages and disadvantages. LPs consequence relation is also transitive, but ST (expanded with a Meadows-like validity predicate) can the consistently expanded with a truth predicate.