Abstract
The present article employs a model-theoretic semantics to interpret a fragment of the language of the Quantified Argument Calculus (\(\mathsf {Quarc}\)), a recently introduced logical system whose main aim is capturing the structure of natural language sentences in a closer way than does the language of classical logic. The main contribution is an axiomatization for the set of formulas that are valid in all standard interpretations within the employed semantics.
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Notes
In Pavlović and Gratzl (forthcoming) it is shown that both predication negation and reordered predicates can be eliminated from \(\mathsf {Quarc}\), since formulas involving these devices are logically equivalent to formulas not involving them, which are said to be in “shallow normal form”.
It is important to point out that when \(c\in SA\) and c is not a name for x in I, there is no \(I_{c\rightarrow x}\) expansion of I.
Raab employs in (Raab , 2016) two semantic operations that are closely related to the rearrangements and the expansions defined here; for analogies and differences, see his Definitions 4.3.3 and 4.3.8.
In principle, a formula \(\phi\) may occur more than once as an assumption in a derivation \({\mathcal {D}}\). Discharging \(\phi\) in \({\mathcal {D}}\) here means discharging all occurrences of \(\phi\) as an assumption in \({\mathcal {D}}\).
We would like to point out that if A9 is reformulated in such a way that \(n\ge 0\), then A1 becomes an instance of it.
Since the domain of a predicate does not change across iterated expansions (see Definition 2), this also means \(x\in I'(P)\), for any \(I'\) which is obtained from I via a finite sequence of expansions. In particular, for the aims of the present argument, \(x\in I_{n-1}(P)\).
References
Ben-Yami, H. (2004). Logic and natural language: On plural reference and its semantics and logical significance. Routledge.
Ben-Yami, H. (2009). Plural quantification logic: A critical appraisal. Review of Symbolic Logic, 2(1), 208–232.
Ben-Yami, H. (2014). The Quantified Argument Calculus. Review of Symbolic Logic, 7(1), 120–146.
Ben-Yami, H. (2021). The Barcan formulas and necessary existence: The view from Quarc. Synthese, 198, 11029–11064.
Ben-Yami, H. (forthcoming). The quantified argument calculus and natural logic. Dialectica.
Ben-Yami, H., & Pavlović, E. (forthcoming). Completeness of the quantified argument calculus on the truth-valuational approach. Manuscript.
Dekker, P. J. E. (2012). Dynamic semantics. Springer.
Freidin, R. (2007). Generative grammar: Theory and its history. London: Routledge.
Groenendijk, J., & Stokhof, M. (1991). Dynamic predicate logic. Linguistics and Philosophy, 14, 39–100.
Henkin, L. (1949). The completeness of the first-order functional calculus. The Journal of Symbolic Logic, 14(3), 159–166.
Kahrel, P. J. (1996). Aspects of negation. PhD thesis, University of Amsterdam.
Lanzet, R., & Ben-Yami, H. (2004). Logical inquiries into a new formal system with plural reference. In V. Hendricks, F. Neuhaus, U. Scheffler, S. A. Pedersen, & H. Wansing (Eds.), First-order logic revisited (pp. 173–223). Logos Verlag.
Lanzet, R. (2017). A three-valued quantified argument calculus: domain-free model theory, completeness, and embedding of FOL. Review of Symbolic Logic, 10(3), 549–582.
Leblanc, H. (1976). Truth-value semantics. North-Holland.
Pavlović, E. (2017). The Quantifed argument calculus: An inquiry into its logical properties and applications. PhD thesis, Central European University, Budapest.
Pavlović, E., & Gratzl, N. (2019). Proof-theoretic analysis of the quantified argument calculus. Review of Symbolic Logic, 12(4), 607–636.
Pavlović, E., & Gratzl, N. (forthcoming). Abstract forms of quantification in the quantified argument calculus. The Review of Symbolic Logic.
Raab, J. (2016). The relationship of QUARC and classical logic. Master’s thesis, Ludwig-Maximilians-Universität München.
Raab, J. (2018). Aristotle, logic, and QUARC. History and Philosophy of Logic, 39(4), 305–340.
van Benthem, J. (1984). Correspondence theory. In: D, Gabbay , F, Guenthner (eds.) Handbook of Philosophical Logic, (pp. 167–247). Springer.
Acknowledgements
The author would like to thank Hanoch Ben-Yami and Jonas Raab for their useful remarks on an earlier version of this work. Furthermore, he is grateful to the audience of a seminar held at Central European University in September 2021, as well as to the two anonymous reviewers for this journal, for their very helpful comments.
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Pascucci, M. An Axiomatic Approach to the Quantified Argument Calculus. Erkenn 88, 3605–3630 (2023). https://doi.org/10.1007/s10670-022-00519-9
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DOI: https://doi.org/10.1007/s10670-022-00519-9