Abstract
The paper is inspired by and explicitly presupposes the readers’ knowledge of the Kürbis normalization procedure for the Milne tree-like natural deduction system C for classical propositional logic. The novelty of C is that for each conventional connective, it has only general introduction and elimination rules, whose paradigm is the rule of proof by cases. The present paper deals with the Milne–Kürbis troublemaker—adding universal quantifier—caused by extending the normalization procedure to \(\mathbf {C^{\exists }_{\forall }} \), the first-order variant of C. For both quantifiers, \( \mathbf {C^{\exists }_{\forall }}\) has only general quantifier rules, whose paradigm is the indirect rule of existential elimination. We propose the classical first-order tree-like natural deduction system \( \mathbf {NC^{\exists }_{\forall }}\) such that it contains all the propositional and half the quantifier rules of \(\mathbf {C^{\exists }_{\forall }}\) and differs from it with respect to the other half of the quantifier rules and the notion of a deduction. We show that in the case of \(\mathbf {NC^{\exists }_{\forall }} \), whose specifics is based on the Quine treatment of flagged variables in the linear-style natural deduction system as well as on the Aguilera-Baaz treatment of characteristic and side variables for sequent-style calculi, the troublemaker doesn’t occur. In order to handle another specifics of \(\mathbf {NC^{\exists }_{\forall }}\)—an auxiliary notion of a finished deduction that makes deductions nonhereditary—we show soundness and completeness of \( \mathbf {NC^{\exists }_{\forall }} \). While emphasizing the former with the help of the Smirnov technique, we additionally indicate a fixable gap in the Quine soundness argument. At last, the philosophical significance of the main result of this paper is developed in more detail.
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Notes
As is common in the history of science, the authorship of general rules, schematic rules, or general schemata of rules is hardly determined (let alone the paradigmatic rule of disjunction elimination, or proof by cases, whose authorship is lost in the depths of centuries): Prawitz (1978, pp. 34–38) and Schroeder-Heister (1984) are cited usually (Kürbis, 2022; Negri & von Plato, 2001; von Plato, 2001; Zimmermann, 2021). There are more names on Milne’s list (Milne, 2015, p. 192). However, it undeservedly lacks (Segerberg, 1983), whose natural deduction systems seem to be the first ones that contain general rules only.
This failure might be surprising because it’s \( \exists \) rather than \(\forall \) that is a troublemaker of the seminal normalization procedure by Prawitz (1965), whose ad hoc solution was to get rid of \( \exists \) and \( \vee \) via \( \lnot ,\forall \) and \( \lnot ,\wedge \) (speaking in terms of general rules, to get rid of all general rules in his systems). von Plato’s cellar study (von Plato, 2017) of Gentzen’s unpublished manuscripts argues that Gentzen had faced this troublemaker before Prawitz, and the failure in solving it had made Gentzen abandon the normalization problem and switch to the cut-elimination problem for sequent-style calculi. Let us also note Raggio’s paper on the normalization problem which is independent and simultaneous to Prawitz (Raggio, 1965). At last, Sect. 5 discusses a final limitation-free solution of the Prawitz problem by von Plato and Siders (2012).
In this paper, “limitation” is exclusively non-formal and bears no negative connotation.
See notes to the respective chapters in Negri and von Plato (2001).
[D] traditionally means a set of occurrences of formulae of the type D.
We will use a common name parameterized rule for \( \forall I \) and \( \exists E \) to stress the fact that they have certain restrictions on parameters.
It’s a well-known minus inherent to tree-like natural deduction: its deductions start exceeding the page limit even for the textbook examples. As the authors put it in Plato and Siders (2012, p. 206): “It is convenient not to write out the degenerate derivations of minor premisses”.
We point out the readers to the correction (Kürbis, 2023).
We stress the fact that the troublemaker is not about the permutation reductions with \( \forall \).
Notice that Example 3 isn’t a \( \mathbf {C^{\exists }_{\forall }} \)-deduction. If the readers apply the needed easy changes themselves in order to obtain the counterpart \( \mathbf {C^{\exists }_{\forall }} \)-deduction, they will see that the application of \( \exists E \) isn’t a correct \( \mathbf {C^{\exists }_{\forall }} \)-deduction because a occurs in the conclusion of this application. Notice also that the application of \( \forall I \) is a correct \( \mathbf {C^{\exists }_{\forall }} \)-deduction, though: the only assumption in Example 3, where b occurs is \( [F(b)]^1 \), and it has been already discharged. This comparison between the counterpart deductions in both systems under consideration shows once more the nonhereditary specifics of the method of flagging: without falling into triviality, it allows deductions which are incorrect, according to the traditional definition of a deduction. Moreover, as Aguilera and Baaz (2019) extensively addresses, it allows shorter deductions.
This is in contrast to Quine’s alphabetical order discussed on p. 13.
Again, \( a_1 \) isn’t to mean that \( a_1 \) is a first—whatever sense usable—flagged parameter in \( \Pi \).
\( \forall q\forall rT(q,r) \) and \( \forall n\forall rT(n,r) \) are equivalent due to relettering the bound variables. The same holds for (4.2) and \( \forall n (\forall m(m \le n\supset \forall rT(m,r)) \supset \forall rT(n+1,r)) \) in I.
This clause justifies Theorem 5. Suppose the opposite, and \( a_j, j\ne i, \) is such parameter. Then the inductive hypothesis doesn’t hold for \( \Pi ^* \): both forms of it aren’t finished because the flagged \( a_j \) occurs in \( A(a_i)\supset \forall xA(x)\) and \( \exists xA(x)\supset A(a_i)\).
We needn’t pick the passive parameter c because it doesn’t affect the rank of \( \Pi \).
Quine nevertheless discriminates the standard rule of implication introduction, whose correctness could also be shown by the truth table considerations.
We, again, refer the readers to Kürbis (2022) for their definitions.
Note that the von Plato-Siders restrictions intersect with the Zimmermann limitations (see footnote 3 and Zimmermann’s quote on p. 3).
According to Negri and von Plato, “a comparison with the general elimination rules \( [\ldots ] \) displays the perfect symmetry of general introduction and elimination rules” (Negri & von Plato, 2001, p. 217). Moreover, some of these general introduction rules are \( \mathbf {C^{\exists }_{\forall } }\)-rules.
To the best of the author’s knowledge, there is no consistency proof for formal number theory which employs natural deduction.
This allusion to the famous One True logic is pure linguistic.
Following the second referee’s suggestion, we quote the Indrzejczak brief explanation of hybrid systems: “The search for a good deductive system is realized in stages. \([\ldots ]\) Closer inspection shows that natural deduction in standard form has some limitations (which should) to be overcome. In search of better deductive tools we introduce some modified versions of natural deduction and the so called hybrid systems—combinations of natural deduction with other kinds of calculi” (Indrzejczak, 2010, pp. xi–xii).
See footnotes 3 and 23.
Segerberg’s method is, arguably, is the first one chronologically (Segerberg, 1983) (see footnote 1). One could also mention Kooi and Tamminga’s method of correspondence analysis (Kooi & Tamminga, 2023) and one of its applications by Petrukhin and Shangin (2022). Geuvers et al.’s method is literally called truth table natural deduction (Geuvers et al., 2019). We emphasize the fact that in contradistinction to the other methods, it’s shown to be applicable to intuitionistic logic, whose absence of tabularity belongs to textbook knowledge. On the other hand, a specifics of Avron et al.’s method lies in dealing with nondeterministic matrices which are a generalization of the standard ones (Avron et al., 2013). To be sure, this list of methods doesn’t pretend to be exhaustive.
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Acknowledgements
The author’s many thanks go to anonymous referees for their valuable suggestions on the embryonic draft of this paper. A special thank goes to the first referee who draws the author’s attention to Kürbis (2023) which was published after the author had made this submission. The author’s cordial thanks go to Yaroslav Petrukhin and the marvelous Ksyu for comments on and the motivation for this paper, accordingly. The study was funded by RSCF, Grant No. 22–18–00450, https://rscf.ru/project/23-28-00801/.
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Shangin, V. A classical first-order normalization procedure with \(\forall \) and \(\exists \) based on the Milne–Kürbis approach. Synthese 202, 48 (2023). https://doi.org/10.1007/s11229-023-04276-2
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DOI: https://doi.org/10.1007/s11229-023-04276-2