The intention here is that of giving a formal underpinning to the idea of ‘meaning-is-use’ which, even if based on proofs, it is rather different from proof-theoretic semantics as in the Dummett–Prawitz tradition. Instead, it is based on the idea that the meaning of logical constants are given by the explanation of immediate consequences, which in formalistic terms means the effect of elimination rules on the result of introduction rules, i.e. the so-called reduction rules. For that we suggest an extension (...) to the Curry– Howard interpretation which draws on the idea of labelled deduction, and brings back Frege’s device of variable-abstraction to operate on the labels (i.e., proof-terms) alongside formulas of predicate logic. (shrink)

The framework of natural deduction is extended by permitting rules as assumptions which may be discharged in the course of a derivation. this leads to the concept of rules of higher levels and to a general schema for introduction and elimination rules for arbitrary n-ary sentential operators. with respect to this schema, (functional) completeness "or", "if..then" and absurdity is proved.

Nontransitive responses to the validity Curry paradox face a dilemma that was recently formulated by Barrio, Rosenblatt and Tajer. It seems that, in the nontransitive logic ST enriched with a validity predicate, either you cannot prove that all derivable metarules preserve validity, or you can prove that instances of Cut that are not admissible in the logic preserve validity. I respond on behalf of the nontransitive approach. The paper argues, first, that we should reject the detachment principle for naive validity. (...) Secondly, I show how to add a validity predicate to ST while avoiding the dilemma. (shrink)

This is a comment on a translation of Franz von Kutschera's paper ‘Ein verallgemeinerter Widerlegungsbegriff für Gentzenkalküle’, which was published in German in 1969. The paper is an important predecessor of what is nowadays called ‘proof-theoretic semantics’, which describes the view that the meaning of logical connectives is determined by the rules governing their use in a proof system. Von Kutschera adopts this view in this paper, and more specifically, a bilateralist view on this subject in that his aim is (...) to give a general framework that provides generalized rule schemata for arbitrary connectives both for proving and refuting and to use this as a reference to prove completeness of certain systems of operators purely proof-theoretically. The main logical system in focus here has been shown to be equivalent to N4, Nelson's constructive logic with strong negation. In order to understand the translated paper better, a contextualizing comment is offered referring and relating it to a preceding paper by von Kutschera. (shrink)

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.

In von Kutschera 1968 a propositional semantics was outlined which takes valid inferences to be defined by derivability relations in calculi.1 It was pointed out that from this approach it is desir...

Model-theoretic proofs of functional completenes along the lines of [McCullough 1971, Journal of Symbolic Logic 36, 15–20] are given for various constructive modal propositional logics with strong negation.