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ABSTRACT FORMS OF QUANTIFICATION IN THE QUANTIFIED ARGUMENT CALCULUS

Part of: General logic Proof theory and constructive mathematics

Published online by Cambridge University Press:  08 March 2021

EDI PAVLOVIĆ  and
NORBERT GRATZL
EDI PAVLOVIĆ*
Affiliation:
DEPARTMENT OF PHILOSOPHY, HISTORY AND ART STUDIES UNIVERSITY OF HELSINKI P.O. BOX 24, FI-00014 HELSINKI, FINLAND
NORBERT GRATZL
Affiliation:
FAKULTÄT FÜR PHILOSOPHIE WISSENSCHAFTSTHEORIE UND RELIGIONSWISSENSCHAFT MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY (MCMP) LUDWIG-MAXIMILIANS UNIVERSITÄT MÜNCHEN GESCHWISTER SCHOLL PLATZ 1, D-80539 MÜNCHEN, GERMANY E-mail: N.Gratzl@lmu.de
*
E-mail: Edi.Pavlovic@helsinki.fi
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Abstract

The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that result.

Utilizing recent developments in structural proof theory, we develop a G3-style sequent calculus for Quarc and briefly demonstrate its structural properties. We put these properties to use immediately to construct direct proofs of the meta-theoretical properties of the system. We then incorporate an abstract (and, as we shall see, logical) predicate into the system in a way that preserves all the structural properties. This allows us to identify a system of Quarc which is deductively equivalent to PC, and also yields a constructive method of demonstrating the Craig interpolation theorem (which speaks in favor of the aforementioned predicate being logical). We further generalize this extension to develop a bivalent system of Quarc with defining clauses that still maintains all the desirable properties of a good proof system.

Keywords

quantified argument calculussequent calculus G3natural language quantificationCraig interpolationdefining clauses

MSC classification

Primary: 03F05: Cut-elimination and normal-form theorems
Secondary: 03B10: Classical first-order logic 03B65: Logic of natural languages
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 2 , June 2023 , pp. 449 - 479
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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