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Paraconsistent Set Theory

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Paraconsistent Logic: Consistency, Contradiction and Negation

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 40))

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Abstract

This chapter offers a new approach to paraconsistent set theory by means of employing LFIs and their powerful consistency operator into sets, as well as into sentences. By assuming that not only sentences, but sets themselves can be classified as consistent or inconsistent objects, the basis for new paraconsistent set-theories that resist certain paradoxes without falling into trivialism is established.

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Notes

  1. 1.

    The notation for the axioms of this group will be explained below.

  2. 2.

    From now on \(x\not \approx y\) abbreviates the formula \(\lnot (x\approx y)\) and \(x\not \in y\) abbreviates the formula \(\lnot (x \in y)\).

  3. 3.

    We are using here the traditional terminology. If the negation \(\lnot \) is not explosive then it would be more appropriate to speak of contradictory and non-contradictory theories, respectively.

  4. 4.

    It is worth noting that in [18] it is shown that Burali-Forti’s paradox, generally regarded as the first of the set-theoretical paradoxes, was neither created by Burali-Forti nor by Cantor . It arose gradually and only acquired its contemporary form in the hands of Bertrand Russell in 1903.

  5. 5.

    L. Kronecker, a prominent German mathematician who had been one of Cantor’s teachers, even attacked Cantor personally, calling him a “scientific charlatan” a “renegade” and a “corrupter of the youth”! .

  6. 6.

    Emphasis in the original.

  7. 7.

    L. Schwarz was awarded the Fields Medal in 1950 for his work on the theory of distributions.

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Authors and Affiliations

  1. Department of Philosophy and Centre for Logic, Epistemology and the History of Science (CLE), University of Campinas (UNICAMP), Campinas, São Paulo, Brazil

    Walter Carnielli & Marcelo Esteban Coniglio

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  1. Walter Carnielli
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  2. Marcelo Esteban Coniglio
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Correspondence to Walter Carnielli .

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Carnielli, W., Coniglio, M.E. (2016). Paraconsistent Set Theory. In: Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-319-33205-5_8

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