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A Proof-Theoretic Analysis of the Meaning of a Formula in a Combination of Intuitionistic and Classical Propositional Logic

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Logic and Engineering of Natural Language Semantics (LENLS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14213))

Abstract

This paper provides a proof-theoretic analysis of the meaning of a formula in a combination of intuitionistic and classical propositional logic, based on the analysis proposed by Restall (2009). Restall showed that his analysis is applicable to both intuitionistic and classical propositional logic separately, but this paper shows that it is also applicable to a combination of the two logics called \(\mathbf {C+J}\). In addition, two points of improvement of Restall’s analysis are mentioned, and they are overcome by employing the method provided by Takano (2018). Moreover, this paper explains how the analysis of \(\mathbf {C+J}\), which is based on Restall’s analysis and improved by Takano’s method, is related to the bilateralism-unilateralism debate. It is shown that a unilateral approach is possible for \(\mathbf {C+J}\), although Restall’s original analysis is based on bilateralism.

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Notes

  1. 1.

    This question is not a new one. Similar questions were already mentioned in [26, 35].

  2. 2.

    Some may consider ordinary intuitionistic logic itself to be a combination, since Kolmogorov-Gödel-Gentzen translation exists. A combination based on this view is studied in [31,32,33, 35]. However, such a view is criticized in [12].

  3. 3.

    It is noted that the results in \(\mathbf {C+J}\) may not be conclusive for deciding whether an intuitionistic or classical theorem should be admitted. The results in another combination also need to be considered.

  4. 4.

    Note that \(\bot \) is not considered in [39], because \(\bot \) is definable by negation and conjunction. However, the addition of \(\bot \) as a primitive symbol creates no problem.

  5. 5.

    Since the antecedent and succedent of a sequent are defined as sets, contraction and exchange rules are not necessary. It is noted that although a sequent calculus that does not contain \((w \Rightarrow )\) or \((\Rightarrow w)\) is used in [39], this difference creates no problem.

  6. 6.

    Proof-theoretic semantics, the representative of the proof-theoretic analyses of meaning, explains the meaning of a formula by using purely syntactical objects, such as arguments or proofs directly (cf. [15, 34, 43]). On the other hand, Restall’s analysis introduces the notion of a model. Thus, these two analyses are different on this point.

  7. 7.

    For example, Ripley [40, Section 3.2] argues that there is no reason to postulate it. Even Restall [38, footnote 5] himself admits that the account of assertion and denial recorded in (Cut) is a subtle one for advocates of intuitionistic logic. This paper does not discuss whether the normative constraint expressed by (Cut) is acceptable. Thus, this paper does not claim that it is unacceptable. What is shown in this paper is that it is not necessary to postulate the rule (Cut) in order to carry out a proof-theoretic analysis of the meaning of a formula.

  8. 8.

    Clearly, this point holds only in the propositional setting. Therefore, if we try to expand the analysis in this paper to the first-order setting, we need to appeal to either König’s infinite lemma or Zorn’s lemma.

  9. 9.

    Although R is defined as a preorder on W in [11], it is defined as a partial order on W in the Kripke semantics provided in [18]. It is noted that both definitions are possible for a Kripke semantics for \(\mathbf {C+J}\) (cf. [5, Section 4.5]).

  10. 10.

    The reason why this restriction on the right rule for intuitionistic implication enables us to avoid collapsing is explained in [50]. Since the right rule for intuitionistic implication is restricted compared with the original rule in \(\textbf{mLJ}\), some might wonder whether the semantic completeness of \(\mathbf {C+J}\) fails. However, the semantic completeness holds, and the detailed proof is described in [50, Section 4]. Moreover, the rule \((\Rightarrow {\rightarrow }_{\texttt{i}})\) in \(\textsf{G}(\textbf{C}+\textbf{J})\) can be regarded as the core of the ordinary right rule for implication in \(\textbf{mLJ}\), as noted in [50, p.32].

  11. 11.

    It is usually said that unilateralism fits intuitionistic logic (cf. [15]) and bilateralism fits classical logic (cf. [41]). The reason why unilateralism seems to fit intuitionistic logic but does not seem to fit classical logic lies in the fact that standard proof-theoretic semantics seems possible for the former but impossible for the latter. The reason why bilateralism seems to fit classical logic lies in the fact that by introducing the notion of denial, proof-theoretic semantics for classical logic seems possible, as Rumfitt [41] did.

  12. 12.

    It is noted that Steinberger [45] claims that Restall’s position is crucially different from the positions of Smiley [44] and Rumfitt [41].

  13. 13.

    In [38], Restall argues against the following view: if A entails B, then it ought to be the case that if you accept A, then you accept B. If we consider an interpretation of the derivability of \(\varGamma \Rightarrow \varDelta \) based on this view, we can obtain the following interpretation: it ought to be the case that if you accept all the formulas in \(\varGamma \), then you accept some formulas in \(\varDelta \). It is noted that if this interpretation is employed, the argument described here also works.

  14. 14.

    This problem also holds when another proof theory is considered. For example, if a natural deduction system is considered, an interpretation of the derivability of a formula from a set of assumptions using only the notion of assertion should contain a too strong requirement. Thus, advocates of intuitionistic logic who defend unilateralism should propose an interpretation of the derivability that does not contain a too strong requirement, although it is usually said that unilateralism fits intuitionistic logic and bilateralism fits classical logic, as noted in footnote 11.

  15. 15.

    The direction from the right to the left of Proposition 1 is the inversion of \((\lnot _{\texttt{c}} \Rightarrow )\). Inversion of rules for logical connectives is shown by induction on the construction of a derivation in [24, Theorem 3.1.1] and [51, Proposition 3.5.4]. Although rules for classical negation are not dealt with in [24, 51], we can apply this induction to show this direction. Note that although the height-preserving inversion is shown in [24, 51], the direction from the right to the left of Proposition 1 is not height-preserving.

  16. 16.

    Similar transformations are carried out in [2, 29, 46, 51], but one-sided calculi in [2, 29, 46, 51] are obtained by transmitting antecedent to succedent. Thus, the directions of transformation are different.

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Acknowledgment

This paper is based on a discussion with Katsuhiko Sano (Hokkaido University), for which I thank him. I also thank an anonymous referee for giving very helpful comments. Shunsuke Yatabe (Kyoto University) asked very interesting questions and gave very helpful comments at LENLS19, and Koji Mineshima (Keio University) informed me about [52]. Moreover, Takuro Onishi (Kyoto University) asked very interesting questions and gave very helpful comments at the 55th Annual Meeting of Philosophy of Science Society, Japan. In spite of their help, I take full responsibility for the content of this paper. This research is partially supported by Grant-in-Aid for JSPS Fellows (Grant Number JP22J20341).

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  1. Graduate School of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan

    Masanobu Toyooka

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  1. Ochanomizu University, Tokyo, Japan

    Daisuke Bekki

  2. Keio University, Tokyo, Japan

    Koji Mineshima

  3. Aoyama Gakuin University, Tokyo, Japan

    Elin McCready

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Toyooka, M. (2023). A Proof-Theoretic Analysis of the Meaning of a Formula in a Combination of Intuitionistic and Classical Propositional Logic. In: Bekki, D., Mineshima, K., McCready, E. (eds) Logic and Engineering of Natural Language Semantics. LENLS 2022. Lecture Notes in Computer Science, vol 14213. Springer, Cham. https://doi.org/10.1007/978-3-031-43977-3_7

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