Abstract
We introduce a two-valued and a three-valued truth-valuational substitutional semantics for the Quantified Argument Calculus (Quarc). We then prove that the 2-valid arguments are identical to the 3-valid ones with strict-to-tolerant validity. Next, we introduce a Lemmon-style Natural Deduction system and prove the completeness of Quarc on both two- and three-valued versions, adapting Lindenbaum’s Lemma to truth-valuational semantics. We proceed to investigate the relations of three-valued Quarc and the Predicate Calculus (PC). Adding a logical predicate T to Quarc, true of all singular arguments, allows us to represent PC quantification in Quarc and translate PC into Quarc, preserving validity. Introducing a weak existential quantifier into PC allows us to translate Quarc into PC, also preserving validity. However, unlike the translated systems, neither extended system can have a sound and complete proof system with Cut, supporting the claim that these are basically different calculi.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ben-Yami, Hanoch, Logic & Natural Language: On Plural Reference and Its Semantic and Logical Significance. London: Routledge (2004). https://doi.org/10.4324/9781315250267.
Ben-Yami, Hanoch, The quantified argument calculus, The Review of Symbolic Logic, 7(1):120–146 (2014). https://doi.org/10.1017/S1755020313000373.
Ben-Yami, Hanoch, The quantified argument calculus and natural logic, Dialectica 74(2):35–70 (2020). https://doi.org/10.48106/dial.v74.i2.02.
Ben-Yami, Hanoch, The Barcan formulas and necessary existence: the view from Quarc, Synthese 198(11):11029–10164 (2021). https://doi.org/10.1007/s11229-020-02771-4.
Ben-Yami, Hanoch. manuscript, Truth and Proof without Models: A Development and Justification of the Truth-Valuational Approach.
Ben-Yami, Hanoch, and Edi Pavlović, Forthcoming, Completeness of the Quantified Argument Calculus on the Truth-Valuational Approach, in Boran Berčić, Aleksandra Golubović, and Majda Trobok, (eds.), Human Rationality:Festschrift for Nenad Smokrovć, Faculty of Humanities and Social Sciences, University of Rijeka, pp. 53–67.
Chiswell, Ian, and Wilfrid Hodges. Mathematical Logic, Oxford Texts in Logic. Oxford, New York: Oxford University Press (2007).
Cobreros, Pablo, Paul Egré, David Ripley, and Robert van Rooij, Tolerant, classical, strict, Journal of Philosophical Logic 41(2):347–385 (2012). https://doi.org/10.1007/s10992-010-9165-z.
Cobreros, Pablo, Paul Egré, David Ripley, and Robert van Rooij, Reaching transparent truth, Mind 122(488):841–866 (2013). https://doi.org/10.1093/mind/fzt110.
Cobreros, Pablo, Paul Egré, David Ripley, and Robert van Rooij, Vagueness, Truth and Permissive Consequence, in Theodora Achourioti, Henri Galinon, José Martínez Fernáindez, and Kentaro Fujimoto (eds.), ]Unifying the Philosophy of Truth. Logic, Epistemology and the Unity of Science, vo. 36, Dordrecht, Springer Netherlands, pp. 409–430.
Dunn, J. Michael, and Nuel D. Belnap. 1968, The substitution interpretation of the quantifiers, Noûs 2(2):177–185. https://doi.org/10.2307/2214704.
Halldén, Sören. The Logic of Nonsense. Uppsala (1949).
Kripke, Saul A, Outline of a Theory of Truth. Reprinted in his 2011. Philosophical Troubles: Collected Papers, I. Oxford: Oxford University Press, pp. 75–98 (1975).
Kripke, Saul A, Is There a Problem About Substitutional Quantification? In Truth and Meaning, edited by Gareth Evans and John McDowell, 325–419. Oxford University Press (1976).
Lanzet, Ran, A three-valued quantified argument calculus: Domain-free model-theory, completeness, and embedding of FOL, The Review of Symbolic Logic 10(3):549–582 (2017). https://doi.org/10.1017/S1755020317000053.
Lanzet, Ran, and Hanoch Ben-Yami. 2004, Logical Inquiries into a New Formal System with Plural Reference, in Vincent Hendricks, Fabian Neuhaus, Stig Pedersen, Uwe Scheffler, and Heinrich Wansing (eds.), First-Order Logic Revisited, 173–223. Berlin: Logos Verlag.
Leblanc, Hugues, A simplified account of validity and implication for quantificational logic, The Journal of Symbolic Logic 33(2):231–235 (1968). https://doi.org/10.2307/2269868.
Leblanc, Hugues, Truth-Value Semantics. Amsterdam, New York, and Oxford: North-Holland Publishing Company (1976).
Leblanc, Hugues, Alternatives to Standard First-Order Semantics, in D. Gabbay and F. Guenthner(eds.), Handbook of Philosophical Logic: Volume I: Elements of Classical Logic, Synthese Library, Dordrecht, Springer Netherlands, pp. 189–274 (1983). https://doi.org/10.1007/978-94-009-7066-3_3.
Lewis, H. A., Substitutional Quantification and Nonstandard Quantifiers, Noûs 19(3): 447–451 (1985). https://doi.org/10.2307/2214953.
Pascucci, Matteo, An Axiomatic Approach to the Quantified Argument Calculus. Erkenntnis, January (2022). https://doi.org/10.1007/s10670-022-00519-9.
Pavlović, Edi, The Quantified Argument Calculus: An Inquiry into Its Logical Properties and Applications. PhD thesis, Central European University, Budapest (2017).
Pavlović, Edi, and Norbert Gratzl, Free Logic and the Quantified Argument Calculus, in Gabriele M. Mras, Paul Weingartner, and Bernhard Ritter(eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium, Berlin, Boston: De Gruyter (pp. 105–116)(2019a). https://doi.org/10.1515/9783110657883-007.
Pavlović, Edi, and Norbert Gratzl, Proof-Theoretic Analysis of The Quantified Argument Calculus. The Review of Symbolic Logic 12(4):607–636 (2019b). https://doi.org/10.1017/S1755020318000114.
Pavlović, Edi, and Norbert Gratzl, Abstract Forms of Quantification in the Quantified Argument Calculus. The Review of Symbolic Logic, March, 1-31 (2021). https://doi.org/10.1017/S175502032100006X.
Peano, Giuseppe, Formulaire de Mathématique. Turin: Bocca Frères, CH. Clausen (1897).
Priest, Graham, The logic of paradox, Journal of Philosophical Logic 8(1):219–241 (1979). https://doi.org/10.1007/BF00258428.
Priest, Graham, Towards Non-Being. Second Edition. Oxford, New York: Oxford University Press (2016).
Raab, Jonas, The Relationship of QUARC and Classical Logic. München: Ludwig-Maximilians-Universität München (2016).
Raab, Jonas, Aristotle, Logic, and QUARC, History and Philosophy of Logic 39(4):305–340 (2018). https://doi.org/10.1080/01445340.2018.1467198.
Strawson, P. F., On Referring, Mind 59(235):320–344 (1950).
Strawson, P. F., Introduction to Logical Theory. London: Methuen (1952).
Strawson, P. F., Identifying reference and truth-values, Theoria 30(2):96–118 (1964). https://doi.org/10.1111/j.1755-2567.1964.tb00404.x.
Westerståhl, Dag. manuscript, Foundations of First-Order Logic: Completeness, Incompleteness, Computability.
Funding
Open access funding provided by Central European University Private University
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Presented by Heinrich Wansing.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yin, H., Ben-Yami, H. The Quantified Argument Calculus with Two- and Three-valued Truth-valuational Semantics. Stud Logica 111, 281–320 (2023). https://doi.org/10.1007/s11225-022-10022-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-022-10022-5