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.
- 2.
- 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.
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.
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.
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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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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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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