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Rejection in Łukasiewicz’s and Słupecki’s Sense

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Part of the book series: Studies in Universal Logic ((SUL))

Abstract

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic.

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Notes

  1. 1.

    The paper is elaborated on the basis of my works [63,64,65] and Słupecki et al. [45].

  2. 2.

    Such a syntactic formalization of some propositional calculi was also, probably independently, introduced by Rudolf Carnap [6, 7]; see Citkin [10].

  3. 3.

    Słupecki’s theorem is an immediate consequence of the following theorem of the theory recursion, which we quote from Grzegorczyk’s book [14, p. 355 in the Eng. ed.]: If the union of two recursively enumerable disjoint sets T and S is computable set, then the sets T and S are also computable.

  4. 4.

    G. Bryll (as well as his book [2]) and T. Skura have been very helpful in verifying certain significant facts.

  5. 5.

    It is saved in Łukasiewicz’s, so called Polish notation.

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  1. Cardinal Stefan Wyszyński University in Warsaw, Warsaw, Poland

    Urszula Wybraniec-Skardowska

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  1. Urszula Wybraniec-Skardowska
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  1. Department of Fundamental Mathematics, UNED, Madrid, Spain

    Ángel Garrido

  2. Cardinal Stefan Wyszyński University in Warsaw, Warsaw, Poland

    Urszula Wybraniec-Skardowska

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Wybraniec-Skardowska, U. (2018). Rejection in Łukasiewicz’s and Słupecki’s Sense. In: Garrido, Á., Wybraniec-Skardowska, U. (eds) The Lvov-Warsaw School. Past and Present. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-65430-0_40

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