Free Logic and the Quantified Argument Calculus

Abstract

The Quantified Argument Calculus (or Quarc for short) is a novel and peculiar system of quantified logic, particularly in its treatment of nonemptiness of unary predicates, as in Quarc unary predicates are never empty, and singular terms denote. Moreover, and as a consequence of this, the universally quantified formulas entail their corresponding particular ones, similar to existential import. But at the same time, Quarc eschews talk of existence entirely by having a particular quantifier instead of an existential one. To bring it back into consideration, we explicitly introduce the existence predicate, and modify the rules to make the existence assumption obvious. This, along with some modifications, leads to a version of negative free logic. A question that arises at this point, given that we are interested in free logic, is what happens when we remove the existence assumption on singular terms; here we can quite naturally choose the negative free logic framework as well. In this paper we shall therefore investigate interrelations between Quarc and free logic (especially with its negative variant), and approach these interrelations with proof-theoretic methods.

Abstract Forms of Quantification in the Quantified Argument Calculus

“…This theorem explains the existing results, namely that Quarc without instantiation is weaker than the Predicate Calculus, with its strength along the lines of negative free logic [18], and that there is an image of the Predicate Calculus within Quarc expanded with an abstract predicate [11].…”

“…In [19] the authors suggest, and it has since been shown in [18], that Quarc bears a structural similarity to free logic. This is fortunate, since recent developments in the proof theory of free logics in [20] (the same approach was used for intuitionistic logic of existence in [14]), allow us to simplify the quantifier rules, and thereby also the proofs of structural and meta-theoretic properties, as well as proof search.…”

“…The only difference is that in [19], valuation of the quantifiers is clearly defined in the absence of the semantic rule of instantiation, while in [2] this is not explicitly the case. This opens up the possibility of either approaching Quarc without the semantic rule as a free logic [18], or as three-valued [11]. In [4] both are discussed, but since here we are interested only in the bivalent approach, we have adopted the semantics from the first paper.…”

“…However, as Pavlovic and Gratzl [18] shows, it is possible to treat Quarc without instantiation as a bivalent system, in which case it displays a structural similarity to free logic and therefore meshes easily with the approach of this paper. This makes for a significantly more streamlined sequent calculus, and one which allows a simpler structural analysis (the calculus in [11] is not shown to be cut-free) and demonstration of meta-theoretical properties.…”

“…It was used to investigate assertoric syllogistic [22], natural logic [4] and necessary existence and the Barcan formulas [4]. On a more formal side a sequent calculus was developed [19], which in turn led to establishing a close connection Quarc shares with free logic [18]. Finally (but slightly earlier), a three-valued system of Quarc was developed in [11].…”