We are concerned with showing how ‘labelled’ Natural Deduction presentation systems based on an extension of the so-called Curry-Howard functional interpretation can help us understand and generalise most of the deduction calculi designed to deal with the logical notion of existential quantification. We present the labelling mechanism for ‘’ using what we call ‘ɛ-terms’, which have the form of ‘a’) in a dual form to the ‘Ax.f’ terms of in the sense that the ‘witness’ is chosen at the time of (...) assertion/introduction.With the intention of demonstrating the generality of our framework, we provide a brief comparison with some other calculi of existentials, including the model-theoretic interpretations of Skolem and Herbrand, indicating the way in which some of the classical results on prenex normal forms can be extended to a wide range of logics. The essential ingredient is to look at the conceptual framework of labelled natural deduction as an attempted reconstruction of Frege's functional calculus where the devices of the Begriffsschrift work separately from, yet in harmony with, those of the Grundgesetze 1. (shrink)

SummaryLooking at proof theory as an attempt to ‘code’ the general pattern of the logical steps of a mathematical proof, the question of what kind of rules can make the meaning of a logical connective completely explicit does not seem to have been answered satisfactorily. The lambda calculus seems to have been more coherent simply because the use of ‘λ’ together with its projection 'apply' is specified by what can be called a 'reduction' rule: β‐conversion. We attempt to analyse the (...) role of proof rules, making use of a set of formal rules designed to capture both the notions of proof theory and those of the lambda‐calculus: Martin‐Löf's Intuitionistic Type Theory.RésuméSi on considère la théorie de la démonstration comme une tentative de ‘codifier’ les pas logiques d'une démonstration mathématique, on se rendra compte qu'on n'a pas répondu de façon satisfaisante à la question suivante: quelles sont les règies qui rendent complètement explicite le sens d'un connecteur logique? Le lambda‐calcul a été apparemment plus cohérent, tout simplement parce que l'utilisation du‘λ’ avec sa projection 'apply' est spécifiée par une regie de 'réduction': β‐conversion. Nous essayons d'analyser le rôle des règies de démonstration, en utilisant un système formel de règies conçu pour englober à la fois les notions de la théorie de la démonstration et celles du lambda‐calcul: la Théorie Intuitionniste des Types de Martin‐Löf.ZusammenfassungWenn man Beweistheorie als einen Versuch, ein allgemeines Muster der logischen Schritte in einem mathematischen Beweis zu kodifizieren, betrachtet, so scheint die Frage nach der Art von Regeln , die die Bedeutung von logischen Operatoren vollständig beschreiben, nicht zufriedenstellend beantwortet. Der Lambda‐Kalkül erscheint, kohärenter, einfach deshalb weil der Gebrauch von ‘λ’ zusammen mit dessen Projektion 'apply' durch Regeln bestimmt wird, die man 'Reduktions'‐Regeln nennen kann: β‐Konversion. Wir versuchen, die Rolle von Beweisregeln zu analysieren, indem wir ein Regelsystem verwenden, das sowohl die Begriffe der Beweistheorie als auch diejenigen des Lambda‐Kalküls erfasst, nämlich die Martin‐Löfsche Typentheorie. (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.

The basic ideas of game-theoretical semantics are implicit in logicians' and mathematicians' folklore but used only sporadically (e.g., game quantifiers, back-and-forth methods, partly ordered quantifiers). the general suggestions of this approach for natural languages are emphasized: the univocity of "is," the failure of compositionality, a reconstruction of aristotelian categories, limitations of generative grammars, unity of sentence and discourse semantics, a new treatment of tenses and other temporal notions, etc.

This is a purely conceptual paper. It aims at presenting and putting into perspective the idea of a proof-theoretic semantics of the logical operations. The first section briefly surveys various semantic paradigms, and Section 2 focuses on one particular paradigm, namely the proof-theoretic semantics of the logical operations.

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It (...) is argued that these two sorts of concepts must be kept strictly apart. (shrink)

According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of (...) this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book The Logical Basis of Metaphysics. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence. (shrink)