Adaptive Fregean Set Theory
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Adaptive Fregean Set Theory
Abstract. This paper defines provably non-trivial theories that characterize Frege’s notion of a set, taking into account that the notion is inconsistent. By choosing an adaptive underlying logic, consistent sets behave classically notwithstanding the presence of inconsistent sets. Some of the theories have a full-blown presumably consistent set theory T as a subtheory, provided T is indeed consistent. An unexpected feature is the presence of classical negation within the language. Keywords: Fregean set theories, Adaptive logics, Content guidance, Paraconsistency.
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Aim of This Paper
This paper concerns non-trivial theories that characterize Frege’s notion of a set. There are several such theories and they have remarkable properties. While the first proposed paraconsistent set theory [11] proceeded in the wake of Quine’s NF, a lot of attention was paid to paraconsistent versions of Frege’s set theory [10,17,27]. The topic is fascinating for several reasons. First, Frege’s notion of a set is generally considered natural, as is witnessed by the many mathematics curricula that offer a course on naive set theory. Next, it is important that we try to understand the way in which mathematicians moved from Frege’s inconsistent set theory to the presumably consistent set theories that succeeded it. The third reason requires a few more words. While many believe that the restrictions imposed by the ZF-axioms are sufficient to avoid the antinomies, doubts were raised, from the outset, on the issue whether those restrictions are necessary. Meanwhile paraconsistent logics came into being and gained recognition. As a result, a more general question may be raised: Is it possible to devise an inconsistent but non-trivial set theory that is based on Frege’s notion but nevertheless serves the purposes a decent set theory should serve? Some specifications are required. First, I do not aim at a set theory that Frege would have liked; as I understand Frege, he would have considered an inconsistent set theory unpalatable. I am after a set theory “based on Frege’s
Presented by Heinrich Wansing; Received September 4, 2018
Studia Logica https://doi.org/10.1007/s11225-019-09882-1
c Springer Nature B.V. 2019
D. Batens
notion”, viz. with ‘the same’ axioms as Frege’s. There is a complication because classical logic CL is unavoidably replaced by a paraconsistent logic and because Frege’s axioms, even some instances of the abstraction axiom schema Abs by themselves, are CL-trivial. Here is how to get around that. Two strings σ1 σ2 . . . σn and τ1 τ2 . . . τn are symbol-wise equivalent in CL iff (i) σi = τi if σi is not a logical symbol and (ii) σi and τi have the same meaning within CL if σi is a logical symbol.1 Desideratum 1: every axiom of the paraconsistent set theory is symbol-wise equivalent to one of Frege’s axioms and vice versa. It seems desirable to accompany the formal Desideratum 1 with some substantive demands. First, sets should be truly extensional. Desideratum 2: two sets are identical just in cas
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