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Composition of Deductions within the Propositions-As-Types Paradigm

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Abstract

Kosta Došen argued in his papers Inferential Semantics Došen (in Inferential semantics, Springer, Berlin 2015) and On the Paths of Categories Došen (in On the paths of categories, Springer, Cham 2016) that the propositions-as-types paradigm is less suited for general proof theory because—unlike proof theory based on category theory—it emphasizes categorical proofs over hypothetical inferences. One specific instance of this, Došen points out, is that the Curry–Howard isomorphism makes the associativity of deduction composition invisible. We will show that this is not necessarily the case.

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Notes

  1. There were, however, other crucial contributors as well, most importantly N. G. de Bruijn and Per Martin-Löf.

  2. Došen uses them interchangeably; see e.g., [7], p. 149.

  3. In compliance with the rejection of Schroeder-Heister’s first dogma of standard semantics (the priority of categorical over the hypothetical, see [30]).

  4. In CTT, \(A \supset B\) is defined via the \(\Pi \) type, i.e., the type of dependent functions, specifically as \((\Pi x : A)B\) where B does not depend on x.

  5. See [23, 25].

  6. It is difficult to surmise who was the first to suggest this reduction, however, it appears as early as the 17th century in the book Artis Logicae Compendium by Henry Aldrich (1648–1710) and the general idea was around probably even earlier.

  7. See [30].

  8. Frege’s unpublished manuscript Boole’s Logical Calculus and the Concept-Script.

  9. It is rather the other way around since the conclusion b(x) : B clearly displays its dependence on the variable x from the hypothesis.

  10. See [18, 31]. For why we say only “roughly”, see e.g., [3].

  11. More specifically, hypothetical judgments can be used to capture a primitive notion of a function which is different from the derived notion of a function captured by the \(\Pi \) type. For more, see [11, 12].

  12. Since it should be always clear from the context, we are overloading the symbol ‘\(\circ \)’ to mean any kind of composition, i.e., categorial, type-theoretical, or functional.

  13. I am indebted to Ansten Klev, who pointed this out to me and suggested a remedy utilizing the higher-order presentation of CTT presented below.

  14. See [21], p. 143.

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Acknowledgements

An earlier version of this paper was presented at a seminar organized by the Department of Logic of the Institute of Philosophy (The Czech Academy of Sciences) in Prague, February 2019. I would like to thank all the participants of this seminar for their helpful notes. A special thanks goes to Ansten Klev, whose valuable remarks helped to shape this paper.

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  1. Institute of Philosophy, The Czech Academy of Sciences, Jilská 1, 110 00, Prague 1, Czech Republic

    Ivo Pezlar

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Work on this paper was supported by Grant No. 19-12420S from the Czech Science Foundation, GA ČR.

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Pezlar, I. Composition of Deductions within the Propositions-As-Types Paradigm. Log. Univers. 14, 481–493 (2020). https://doi.org/10.1007/s11787-020-00260-3

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