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. (shrink)

The paper has two parts: 1. A brief exposition of proof-theoretic semantics, not necessarily in connection to natural language. 2. A review, with a contrastive flavour, of some of the applications of PTS to NL with an indication of advantages of PTS as a theory of meaning for NL.

This paper investigates the meaning of restricted quantification when the embedded conditional is taken as the conditional of some first-order connexive logics. The study is carried out by checking the suitability of RQ for defining a connexive class theory, in analogy to the definition of Boolean class theory by using RQ in classical logic. Negative results are obtained for Wansing’s first-order connexive logic QC and one variant of Priest’s first-order connexive logic QP. A positive result is obtained for another variant (...) of QP. (shrink)