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Ex Incompatibilitate Sequitur Quodlibet (The Explosiveness of Incompatibility and the Compatibility of Negation)

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Universal Logic, Ethics, and Truth

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

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Abstract

In this chapter, we explain why Ex contradictione sequitur quodlibet is a confusing expression to denote the statement \(p, \neg p \vdash q\), and we also explain why this statement is ambiguous. We start by setting out a framework about consequence relation and truth. We proceed by presenting the basic concepts of the theory of opposition and the meaning of contradiction according to this theory. We then introduce the notion of incompatibility, and, on the basis of that, we deal with explosion and introduce the notion of compatible negation. The final part of the paper is about John Corcoran.

In memory of John Corcoran

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Notes

  1. 1.

    We are using here the word “proposition,” rather than “sentence” or “formula,” not because we are dealing with a system of “propositional logic,” but because we are talking about reasoning in general, a proposition being considered, in the spirit of Frege, as a thought that can be true or false, see [48, 52] and [40].

  2. 2.

    About the history of these expressions, see [50].

  3. 3.

    The connection between lattice theory and logic has been established by Stone and Tarski. Stone showed that a complemented distributive lattice is a Boolean algebra [56], and Tarski showed that a Boolean algebra is the algebraic structure corresponding to the quotient of the classical propositional structure, the so-called Lindenbaum–Tarski’s algebra [58].

  4. 4.

    But this not the original Greek sense of the word, see [2].

  5. 5.

    About Tarski’s two ways of defining the notion of consequence, see [22] and [20].

  6. 6.

    Tarski also did not use the symbol “\(\vdash \),” neither in its original meaning when introduced by Frege nor in its abstract recent use [23].

  7. 7.

    Tarski is with Kurt Gödel the most famous logicians of the modern area. Funny enough, Tarski was born the same day of the year as Gödel died, January 14. Based on this coincidence, I launched the World Logic Day on January 14, 2019 (see [18]) and I succeeded to have it recognized by UNESCO the same year.

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Acknowledgements

I wrote a first draft of this paper in July 2022 and sent it to many colleagues. Thank you to all of them for their feedbacks and comments: Luis Felipe Bartolo Alegre, Arnon Avron, Mark van Atten, Valentin Bazhanov, Ross Brady, Julie Brumberg, Mihir Chakraborty, Oksana Cherkashina, Alexandre Costa-Leite, Luis Estrada-González, Jean-Baptiste Gourinat, Yuri Gurevich, Lloyd Humbertone, Susana Gómez, John Grant, Vedat Kamer, Jens Lemanski, David Makinson, Daniel Mundici, Raja Natarajan, Istvan Nemeti, Régis Pelliser, Stephen Read, Gianluigi Segalerba, Julio Michael Stern, Caroline Pires Ting, Denis Vernant, Jan Woleński, Jan Zygmunt.

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  1. Brazilian Academy of Philosophy, Brazilian Research Council, University of Brazil, Rio de Janeiro, Brazil

    Jean-Yves Beziau

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  1. St. John Fisher University, Philosophy, Rochester, NY, USA

    Timothy J. Madigan

  2. Departement of Philosophy, Federal University of Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, Brazil

    Jean-Yves Béziau

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Beziau, JY. (2024). Ex Incompatibilitate Sequitur Quodlibet (The Explosiveness of Incompatibility and the Compatibility of Negation). In: Madigan, T.J., Béziau, JY. (eds) Universal Logic, Ethics, and Truth. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-44461-6_3

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