A three-valued quantified argument calculus: Domain-free model-theory, completeness, and embedding of fol

Review of Symbolic Logic 10 (3):549-582 (2017)
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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.

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Ran Lanzet
Hebrew University of Jerusalem

References found in this work

On referring.Peter F. Strawson - 1950 - Mind 59 (235):320-344.
Generalized quantifiers and natural language.John Barwise & Robin Cooper - 1981 - Linguistics and Philosophy 4 (2):159--219.
Dynamic predicate logic.Jeroen Groenendijk & Martin Stokhof - 1991 - Linguistics and Philosophy 14 (1):39-100.
Generalized Quantifiers and Natural Language.Jon Barwise - 1980 - Linguistics and Philosophy 4:159.
Logics for the relational syllogistic.Ian Pratt-Hartmann & Lawrence S. Moss - 2009 - Review of Symbolic Logic 2 (4):647-683.

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