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Turnstile Figures of Opposition

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The Exoteric Square of Opposition

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

Abstract

We present many figures of opposition (triangles and hexagons) for simple and double turnstiles. We start with one-sided turnstiles, corresponding to sets of tautologies, and then we go to double-sided turnstiles corresponding to consequence relations. In both cases, we consider proof-theoretic (with the simple turnstile) and model-theoretic (with the double turnstile) figures. By so doing, we discuss various central aspects of notations and conceptualizations of modern logic.

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Notes

  1. 1.

    His main book on the topic is [1], but his first works were published in the 1950s, and at this time other people had similar ideas (for details, see [2]).

  2. 2.

    Wittgenstein uses “F” and “W,” not “0” and “1.” In general, his framework is not explicitly mathematical, although he uses the notion of function, following Frege and Russell. About 0 and 1 as truth-values, the notion of truth-function, etc.; see [22] and [23].

  3. 3.

    The difference between the two levels is expressed here by doubling the horizontal line. For the turnstile, the doubling of the horizontal line is not used in this sense.

  4. 4.

    Post was using only “⊢.” As we said, “⊨” was introduced in the 1950s. Wittgenstein was using none of these symbols, and he rejected Frege’s stroke (cf. Tractatus 4.442).

  5. 5.

    In Poland during the 1930’s, the word “theory” was used in a different way: for what is nowadays called a “closed theory,” a theory such that any formula which is a consequence of the theory is in the theory.

  6. 6.

    A theory can be incomplete and decidable, a famous case is the empty theory of classical propositional logic, and an atomic formula is independent from ∅ but ∅ is decidable.

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Acknowledgments

Thanks to the participants of SQUARE’2018 and to Lloyd Humberstone for useful comments on a previous version of this paper.

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Correspondence to Jean-Yves Beziau .

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Beziau, JY. (2022). Turnstile Figures of Opposition. In: Beziau, JY., Vandoulakis, I. (eds) The Exoteric Square of Opposition. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-90823-2_10

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