Glutty theories and the logic of antinomies

The logic of paradox (LP) [3, 13] is very well known in philosophy as a logic that accommodates ‘non-trivial glutty theories’ (theories that are negation inconsistent but don’t contain all sentences). We highlight a lesser known logic, namely, the logic of antinomies (LA) from Asenjo and Tamburino’s 1975 article [5], and compare it with LP. We argue that the differences between the logics result from LP’s having a formal account of consequence, and LA a material account of consequence. Further, we show that, like many attempts to extend LP with a detachable conditional, Asenjo and Tamburino’s attempt to add a detachable conditional to the logic fails due to Curry’s paradox. Suppose that you held a view on language in which some predicates are essentially non-classical (i.e., cannot be treated along classical-logic lines). Suppose too that instead of such non-classical predicates yielding ‘gaps’ (where, of some object, the predicate is neither true nor false), you take them to instead deliver the dual of gaps, namely, ‘gluts’ – predicates true and false of objects. Your view might be tied to familiar paradoxical predicates, holding that predicates like ‘is true’, ‘is a member of’, or ‘exemplifies’ are essentially glutty predicates: they cannot be (properly) interpreted in a way that avoids there being objects of which these predicates are both true and false. But you mightn’t think all such predicates are essentially glutty. You might think that some – in fact, the remaining (and majority of) – predicates of the language are essentially classical. One might think that natural languages are much as above (some essentially glutty predicates, but the remaining predicates essentially classical), or one might merely take it that natural languages could be as such. Either way, the question arises: how should the logic of such a language be understood? We know that classical logic cannot provide us with a way to formally model such a language. A paraconsistent approach needs to be considered,