This paper extends the investigations into logical properties of the quantified argument calculus (Quarc) by suggesting a series of proper subsystems which, although retaining the entire vocabulary of Quarc, restrict quantification in such a way as to make the result decidable. The proof of decidability is via a procedure that prunes the infinite branches of a derivation tree in what is a syntactic counterpart of semantic filtration. We demonstrate an application of one of these systems by showing that Aristotle’s assertoric syllogistic is embeddable within, thus also providing another method of showing its decidability.

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.

This article investigates the proof theory of the Quantified Argument Calculus (Quarc) as developed and systematically studied by Hanoch Ben-Yami [3, 4]. Ben-Yami makes use of natural deduction (Suppes-Lemmon style), we, however, have chosen a sequent calculus presentation, which allows for the proofs of a multitude of significant meta-theoretic results with minor modifications to the Gentzen’s original framework, i.e., LK. As will be made clear in course of the article LK-Quarc will enjoy cut elimination and its corollaries (including subformula property and thus consistency).

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.