Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T20:39:20.275Z Has data issue: false hasContentIssue false
  • English
  • Français

A THREE-VALUED QUANTIFIED ARGUMENT CALCULUS: DOMAIN-FREE MODEL-THEORY, COMPLETENESS, AND EMBEDDING OF FOL

Published online by Cambridge University Press:  08 May 2017

RAN LANZET
RAN LANZET*
Affiliation:
Tel-Aviv University and the Hebrew University of Jerusalem
*
*DEPARTMENT OF PHILOSOPHY TEL-AVIV UNIVERSITY P.O. BOX 39040 RAMAT AVIV, ISRAEL E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible.

Keywords

03B6503A0503B5003B60quantified argument calculusQuarcthree-valued logicthree-valued Quarc3-Quarcquantified argumentnatural languageanaphoradefining clausedomaindomain of quantificationuniversedomain-free model theorycompletenessreordered formSquare of Opposition
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 3 , September 2017 , pp. 549 - 582
Copyright
Copyright © Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Aoyama, H. (1994). The strong completeness of a system based on Kleene’s strong three-valued logic. Notre Dame Journal of Formal Logic, 35(3), 355–368.Google Scholar
Barwise, J. & Cooper, R. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy, 4(2), 159–219.Google Scholar
Ben-Yami, H. (2004). Logic and Natural Language: On Plural Reference and its Semantic and Logical Significance. Aldershot, Hants: Ashgate.Google Scholar
Ben-Yami, H. (2006). A critique of Frege on common nouns. Ratio, 19(2), 148–155.CrossRefGoogle Scholar
Ben-Yami, H. (2014). The quantified argument calculus. The Review of Symbolic Logic, 7(01), 120–146.Google Scholar
Francez, N. (2014). A logic inspired by natural language quantifiers as subnectors. Journal of Philosophical Logic, 43(6), 1153–1172.CrossRefGoogle Scholar
Geach, P. T. (1962). Reference and Generality: An Examination of Some Medieval and Modern Theories. Ithaca, NY: Cornell University Press.Google Scholar
Groenendijk, J. & Stokhof, M. (1991). Dynamic predicate logic. Linguistics and Philosophy, 14(1), 39–100.Google Scholar
Kearns, J. T. (1979). The strong completeness of a system for Kleene’s three-valued logic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 25(3–6), 61–68.Google Scholar
Kneale, W. & Kneale, M. (1971). The Development of Logic. London: Oxford University Press.Google Scholar
Lanzet, R. & Ben-Yami, H. (2004). Logical inquiries into a new formal system with plural reference. In Hendricks, V., Neuhaus, F., Pedersen, S. A., Scheffler, U., and Wansing, H., editors. First-Order Logic Revisited. Berlin: Logos Verlag, pp. 173–223.Google Scholar
Moss, L. S. (2010). Logics for two fragments beyond the syllogistic boundary. In Blass, A., Dershowitz, N., and Reisig, W., editors. Fields of Logic and Computation: Essays Dedicated to Yuri Gurevich on the Occasion of his 70th Birthday. Berlin: Springer, pp. 538–564.CrossRefGoogle Scholar
Pratt-Hartmann, I. & Moss, L. (2009). Logics for the relational syllogistic. Review of Symbolic Logic, 2(4), 647–683.Google Scholar
Strawson, P. F. (1950). On referring. Mind, 59(235), 320–344.Google Scholar