Abstract
This paper presents a methodology to construct globally sound but possibly locally unsound analytic calculi for partial theories of Henkin quantifiers. It is demonstrated that locally sound analytic calculi do not exist for any reasonable fragment of the full theory of Henkin quantifiers. This is due to the combination of strong and weak quantifier inferences in one quantifier rule.
M. Baaz—This work is partially supported by FWF projects P 31063 and P 31955.
A. Lolic—Recipient of a DOC Fellowship of the Austrian Academy of Sciences at the Institute of Logic and Computation at TU Wien.
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Notes
- 1.
The most relevant paper is the work of Lopez-Escobar [5], describing a natural deduction system for \(Q_H\). The setting is of course intuitionistic logic. The formulation of the introduction rule for \(Q_H\) corresponds to the introduction rule right in the sequent calculus developed in this paper. The system lacks an elimination rule.
- 2.
Note that such a rule was already used by Lopez-Escobar in [5].
- 3.
In \(\mathbf{LK}\) strong quantifier inferences are \(\forall _r\) and \(\exists _l\) and weak quantifier inferences are \(\forall _l\) and \(\exists _r\).
- 4.
LK\(^+\) in [1] coincides with \(\mathbf{LK}^{++}\) with exception to the notion of regularity, which is the usual one.
- 5.
Note that usual regularity is not sufficient as the formation of quantifier blocks of inferences in Lemma 2 in cut-free \(\mathbf{LF}\)-derivations might distribute \(\exists f \exists g\) inferences by contractions to more than one branch of the proof.
References
Aguilera, J.P., Baaz, M.: Unsound inferences make proofs shorter. J. Symb. Log. 84(1), 102–122 (2019). https://doi.org/10.1017/jsl.2018.51
Baaz, M., Lolic, A.: Note on globally sound analytic calculi for quantifier macros. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds.) WoLLIC 2019. LNCS, vol. 11541, pp. 486–497. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-662-59533-6_29
Henkin, L.: Some remarks on infinitely long formulas. J. Symb. Log. 30, 167–183 (1961). https://doi.org/10.2307/2270594
Krynicki, M., Mostowski, M.: Henkin quantifiers. In: Krynicki, M., Mostowski, M., Szczerba, L.W. (eds.) Quantifiers: Logics, Models and Computation. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science), vol. 248, pp. 193–262. Springer, Dordrecht (1995). https://doi.org/10.1007/978-94-017-0522-6_7
Lopez-Escobar, E.G.K.: Formalizing a non-linear Henkin quantifier. Fundamenta Mathematicae 138(2), 93–101 (1991)
Mostowski, M., Zdanowski, K.: Degrees of logics with Henkin quantifiers in poor vocabularies. Arch. Math. Log. 43(5), 691–702 (2004). https://doi.org/10.1007/s00153-004-0220-8
Takeuti, G.: Proof Theory, 2nd edn. North-Holland, Amsterdam (1987)
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Baaz, M., Lolic, A. (2020). A Globally Sound Analytic Calculus for Henkin Quantifiers. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2020. Lecture Notes in Computer Science(), vol 11972. Springer, Cham. https://doi.org/10.1007/978-3-030-36755-8_9
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