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Rules of Explosion and Excluded Middle: Constructing a Unified Single-Succedent Gentzen-Style Framework for Classical, Paradefinite, Paraconsistent, and Paracomplete Logics

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Abstract

A unified and modular falsification-aware single-succedent Gentzen-style framework is introduced for classical, paradefinite, paraconsistent, and paracomplete logics. This framework is composed of two special inference rules, referred to as the rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Similar to the cut rule in Gentzen’s LK for classical logic, these rules are admissible in cut-free LK. A falsification-aware single-succedent Gentzen-style sequent calculus fsCL for classical logic is formalized based on the proposed framework. The calculus fsCL is obtained from the existing falsification-aware single-succedent Gentzen-style sequent calculus GN4 for Nelson’s paradefinite (or paraconsistent) four-valued logic N4 by adding the rules of explosion and excluded middle. A falsification-aware single-succedent Gentzen-style sequent calculus GN3 for Nelson’s paracomplete three-valued logic N3 is also obtained from GN4 by adding the rule of explosion. The cut-elimination theorems for fsCL, GN3, and some of their neighbors as well as the Glivenko theorem for fsCL are proved.

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Notes

  1. (ex-middle) and its restricted version were originally introduced by von Plato (1999) and were also presented in Negri and von Plato (2001).

  2. (cut) is well-known to be a structural rule. However, the proposal for saying that (explosion) and (ex-middle) are structural rules has an issue. These two rules involve the negation and the validity of the rules depends on the definition of the negation. Thus, in this aspect, they seem to rather logical inference rules. Besides, the corresponding properties of paraconsistency and paracompleteness are the properties that deal not only with consequence relation, but also with the negation. Thus, the issue with classification of these rules is not clear or simple.

  3. The logic N4 was originally introduced as a first-order predicate logic refereed to as N\(^-\) (Almukdad & Nelson, 1984; Nelson, 1949) which has the following axiom schemes: \(\lnot \forall x \alpha (x) \leftrightarrow \exists x \lnot \alpha (x)\) and \(\lnot \exists x \alpha (x) \leftrightarrow \forall x \lnot \alpha (x)\).

  4. The logic N3 was originally introduced as a first-order predicate logic refereed to as N Almukdad and Nelson (1984); Nelson (1949). Odintsov (2002) showed that propositional N3 can be faithfully embedded into propositional N4.

  5. Avron also used the name Pac for the implication-less fragment of this logic.

  6. Since the cut-elimination theorem for GN4 holds, it is sufficient to check the additional cases for (we-right) and (explosion) in GN3.

  7. (Peirce), (r-Peirce), (Raa), as well as (ex-middle) can be seen as the sequent calculus rule versions of the corresponding natural deduction rules. For example, (Raa) is the sequent calculus rule version of the well-known natural deduction rule of the form:

  8. In von Plato (1999); Negri and von Plato (2001), he also introduced the following natural deduction rules (Nem) and (Nem-at) that correspond to (ex-middle) and (ex-middle-at), respectively.

    where p is a propositional variable. He construct natural deduction systems (for classical logic) with these rules and proved a normalization theorem for the natural deduction system with (Nem-at).

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Acknowledgements

We would like to thank the anonymous referees for their valuable comments. Specifically, the relationship among ELP, PI\(^s\), RM\(^{\supset }_3\), Pac, and PCont was pointed out by a referee. This research was supported by JSPS KAKENHI Grant Number 23K10990.

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  1. School of Data Science, Nagoya City University, Yamanohata 1, Mizuho-cho, Mizuho-ku, Nagoya, Aichi, 467-8501, Japan

    Norihiro Kamide

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Kamide, N. Rules of Explosion and Excluded Middle: Constructing a Unified Single-Succedent Gentzen-Style Framework for Classical, Paradefinite, Paraconsistent, and Paracomplete Logics. J of Log Lang and Inf (2024). https://doi.org/10.1007/s10849-024-09416-6

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