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The Placeholder View of Assumptions and the Curry–Howard Correspondence (Extended Abstract)

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Logic and Argumentation (CLAR 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13040))

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Abstract

Proofs from assumptions are amongst the most fundamental reasoning techniques. Yet the precise nature of assumptions is still an open topic. One of the most prominent conceptions is the placeholder view of assumptions generally associated with natural deduction for intuitionistic propositional logic. It views assumptions essentially as holes in proofs (either to be filled with closed proofs of the corresponding propositions via substitution or withdrawn as a side effect of some rule), thus in effect making them an auxiliary notion subservient to proper propositions. The Curry-Howard correspondence is typically viewed as a formal counterpart of this conception. In this talk, based on my paper of the same name (Synthese, 198(11), 10109–10125, 2021), I will argue against this position and show that even though the Curry-Howard correspondence typically accommodates the placeholder view of assumptions, it is rather a matter of choice, not a necessity, and that another more assumption-friendly view can be adopted.

This is an extended abstract of a paper [8] with the same title published at Synthese 2021. Adapted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Synthese 198(11), 10109–10125, 2021, The placeholder view of assumptions and the Curry–Howard correspondence. Pezlar, Ivo, ©2020. Work on this paper was supported by Grant Nr. 19-12420S from the Czech Science Foundation, GA ČR.

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Notes

  1. 1.

    [13] describes this as a two-layer system. Note that, strictly speaking, the assumptions are not really withdrawn, they are rather incorporated into the propositional level in the form of an antecedent.

  2. 2.

    This non-standard behaviour is also the reason why [10] describes assumption withdrawing rules as improper rules and introduces the distinction between inference rules and deductions rules. For more, see [7, 10].

  3. 3.

    See, e.g., [1].

  4. 4.

    See, e.g., [14].

  5. 5.

    Strictly speaking, we should be writing \(A \models _\mathbb {D} B\), i.e., that \(A \models B\) can be derived with respect to a set of definitional clauses \(\mathbb {D}\) (see [12]), but for simplicity we omit these considerations.

  6. 6.

    See, e.g., Gentzen’s system NLK, discussed in [9].

  7. 7.

    See [4].

  8. 8.

    See [2, 3].

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Authors and Affiliations

  1. Institute of Philosophy, Czech Academy of Sciences, Jilska 1, 110 00, Praha, Czechia

    Ivo Pezlar

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  1. Ivo Pezlar
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Correspondence to Ivo Pezlar .

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  1. Department of Information Engineering, University of Brescia, Brescia, Italy

    Pietro Baroni

  2. Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany

    Christoph Benzmüller

  3. Department of Philosophy (Zhuhai), Sun Yat-sen University, Zhuhai, China

    Yὶ N. Wáng

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Pezlar, I. (2021). The Placeholder View of Assumptions and the Curry–Howard Correspondence (Extended Abstract). In: Baroni, P., Benzmüller, C., Wáng, Y.N. (eds) Logic and Argumentation. CLAR 2021. Lecture Notes in Computer Science(), vol 13040. Springer, Cham. https://doi.org/10.1007/978-3-030-89391-0_31

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  • DOI: https://doi.org/10.1007/978-3-030-89391-0_31

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