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Łukasiewicz and His Followers in Many-Valued Logic

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The Lvov-Warsaw School. Past and Present

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Abstract

The aim of this work is a concise introduction to the Łukasiewicz logical world: three and n-valued, n-natural or infinite, denumerable or of the power of continuum. We present Łukasiewicz inventory work and its rationale, the elaboration of original ideas and their technical complementation. Finally, we attempt to show the impact of Łukasiewicz conceptions, their development and directions of resulting applications.

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Notes

  1. 1.

    The truth-tables of binary connectives ∗ are viewed as follows: the value of α is placed in the first vertical line, the value of β in the first horizontal line and the value of α ∗ β at the intersection of the two lines.

  2. 2.

    That is, formulae taking 0 at arbitrary logical valuation.

  3. 3.

    See Łukasiewicz [21].

  4. 4.

    See Łukasiewicz [23, 1970, p. 140].

  5. 5.

    See Łukasiewicz and Tarski [24].

  6. 6.

    Łukasiewicz [23, 1970, p. 144]; no publication on the topic by Wajsberg exists.

  7. 7.

    MV algebras are presented in Sect. 4.2.

  8. 8.

    The very property applies to the logic algebras i.e. matrices without designated set of elements. Nb. the applications of Post construction are focused on algebras.

  9. 9.

    See next page.

  10. 10.

    Compare 1.3 for n = 3.

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  1. University of Łódź, Łódź, Poland

    Grzegorz Malinowski

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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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Malinowski, G. (2018). Łukasiewicz and His Followers in Many-Valued Logic. 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_24

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