The Difficulties in Using Weak Relevant Logics for Naive Set Theory

The Liberation Argument for Inconsistent Mathematics

“…4 The field has been growing ever since, in many different directions, and by 1995 it was possible to provide a book-length collection of technical results from various areas of mathematics [Mor95]. Nowadays, the list includes so-called paraconsistent set theories trying to either capture the naive conception of set [Res92] [Ist17], allow for the inclusion of particular inconsistent sets [CC13], or ground inconsistent analysis [dC00]; inconsistent nonstandard first-order models of arithmetic [MM84] [PS08]; inconsistent theories of so-called "impossible pictures" [Mor10]; mathematical foundations for an inconsistent metaphysics [Web21a]; and more. 5 More often than not, inconsistent mathematics comes with some nonclassical logic to underlie its theories.…”

"A Smack of Irrelevance" in Inconsistent Mathematics?

“…The question we are interested in here is how much irrelevance is needed in logic for inconsistent mathematics, the program aiming at keeping some naïve principles (which are known to produce contradictions) with a logic weak enough to coexist with them without triviality, but strong enough to recapture a good amount of the most important part (results and proofs) of its classical counterpart while obtaining new results. 1 This is important because according to Weber ([26], [27]) and some of his collaborators (see [12], [2], [13]), a conditional that obeys an irrelevant conditional-introduction rule appears to be indispensable in mathematical proofs and some statements of the form 'φ entails ψ' are true even when the entailment makes use of background assumptions other than φ, or does not use φ at all. In a nutshell, their claim is that an irrelevant conditional is needed for inconsistent mathematics.…”

“…In fact, in the literature there are other, more liberal, necessary conditions for relevance, reflecting the differences between these conditionals. 13 A relaxed condition that can make room for (3) is ultra-weak relevance (UwR):…”

“…These conditionals are discussed again, although in a broader context, in[13].Australasian Journal of Logic (18:5) 2021, Article no 12. …”

“…One could wonder why (3) is better than (1) if both meet just one of the desiderata. The short story is that (3) scores better by relevantists' lights because in (3) not only an entailment is entailed by another entailment, unlike in (1), but also both antecedent and consequent in (3) share modal status; that a necessary entailment entails another necessary entailment is less objectionable than having an atomic, contingent proposition entailing a necessary entailment 13. For an overview of some of those conditions, see[23].Australasian Journal of Logic (18:5) 2021, Article no.…”