The idea of an ?inversion principle?, and the name itself, originated in the work of Paul Lorenzen in the 1950s, as a method to generate new admissible rules within a certain syntactic context. Some fifteen years later, the idea was taken up by Dag Prawitz to devise a strategy of normalization for natural deduction calculi (this being an analogue of Gentzen's cut-elimination theorem for sequent calculi). Later, Prawitz used the inversion principle again, attributing it with a semantic role. Still working (...) in natural deduction calculi, he formulated a general type of schematic introduction rules to be matched ? thanks to the idea supporting the inversion principle ? by a corresponding general schematic Elimination rule. This was an attempt to provide a solution to the problem suggested by the often quoted note of Gentzen. According to Gentzen ?it should be possible to display the elimination rules as unique functions of the corresponding introduction rules on the basis of certain requirements?. Many people have since worked on this topic, which can be appropriately seen as the birthplace of what are now referred to as ?general elimination rules?, recently studied thoroughly by Sara Negri and Jan von Plato. In this study, we retrace the main threads of this chapter of proof-theoretical investigation, using Lorenzen's original framework as a general guide. (shrink)

This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.

This new work provides an approachable introduction to the complex system that Making It Explicit mapped out.

Our topic is the notion of logical consequence: the link between premises and conclusions, the glue that holds together deductively valid argument. How can we understand this relation between premises and conclusions? It seems that any account begs questions. Painting with very broad brushtrokes, we can sketch the landscape of disagreement like this: “Realists” prefer an analysis of logical consequence in terms of the preservation of truth [29]. “Anti-realists” take this to be unhelpful and o:er alternative analyses. Some, like Dummett, (...) look to preservation of warrant to assert [9, 36]. Others, like Brandom [5], take inference as primitive, and analyse other notions in terms of it. There is plenty of disagreement on the “realist” side of the fence too. It is one thing to argue that logical consequence involves preservation of truth. It is another to explain how far truth must be preserved. Is the preservation essentially modal (in all circumstances [25]) or analytic (vouchsafed by.. (shrink)

Michael Dummett is a leading contemporary philosopher whose work on the logic and metaphysics of language has had a lasting influence on how these subjects are conceived and discussed. This volume contains some of the most provocative and widely discussed essays published in the last fifteen years, together with a number of unpublished or inaccessible writings. Essays included are: "What is a Theory of Meaning?," "What do I Know When I Know a Language?," "What Does the Appeal to Use Do (...) for the Theory of Meaning?," "Language and Truth," "Truth and Meaning," "Language and Communication," "The Source of the Concept of Truth," "Mood, Force, and Convention," "Frege and Husserl on Reference," "Realism," "Existence," "Does Quantification Involve Identity?," "Could there be Unicorns?," "Causal Loops," "Common Sense and Physics," "Testimony and Memory," "What is Mathematics About?," "Wittgenstein on Necessity: Some Reflections," and "Realism and Anti-Realism." Serving well as a companion to Dummett's other collections, the essays in this volume are not forbiddingly technical or specialized, and have relevance to many areas of analytic philosophy. (shrink)

Such a conception, says Dummett, will form "a base camp for an assault on the metaphysical peaks: I have no greater ambition in this book than to set up a base ...

Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such systems from logic (...) to mathematics, and on the connection between the two main forms of structural proof theory - natural deduction and sequent calculus. The authors emphasize the computational content of logical results. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated. (shrink)

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have (...) at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933. (shrink)

In his penetrating and thought-provoking article “What Is Logic?” Ian Hacking flags an issue that he leaves undiscussed.

This highly acclaimed book is a major contribution to the philosophy of language as well as a systematic interpretation of Frege, indisputably the father of ...

In the proof-theoretic semantics approach to meaning, harmony , requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony , requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony : imposes the existence of certain transformations of derivations, known as reduction and expansion . We propose (...) a construction of the E-rules (in GE-form) from given I-rules, and prove that the constructed rules satisfy also local intrinsic harmony. The construction is based on a classification of I-rules, and constitute an implementation to Gentzen’s (and Pawitz’) remark, that E-rules can be “read off” I-rules. (shrink)

What binds the constituents of a state of affairs together and provides unity to the fact they constitute? I argue that the fact that they are related is basic and fundamental. This is the thesis of Factualism: the world is a world of facts. I draw three corollaries: first, that the Identity of truth is mistaken, in conflating what represents (the proposition) with what is represented (the fact). Secondly, a popular interpretation of Wittgenstein's Tractatus, due to Steinus, whereby false propositions (...) are taken to picture non-existent state of affairs, cannot be right. For Wittgenstein, propositions had two poles, and a proposition and its negation picture the same fact. Finally, the metaphysics of modal realism must be wrong, for there are no non-actual states of affairs to constitute any world other than the actual world. (Published Online October 13 2005). (shrink)

The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.Through the condition that in a cut-free derivation of the sequent Γ⇒C, no inactive weakening (...) or contraction formulas remain in Γ, a correspondence with the formal derivability relation of natural deduction is obtained: All formulas of Γ become open assumptions in natural deduction, through an inductively defined translation. Weakenings are interpreted as vacuous discharges, and contractions as multiple discharges. In the other direction, non-normal derivations translate into derivations with cuts having the cut formula principal either in both premisses or in the right premiss only. (shrink)

Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use. I argue that Dummett gives a mistaken elaboration (...) of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen's. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic. (shrink)

in die operative Logik und Mathematik Zweite Auflage Springer-Verlag Berlin Heidelberg New York 1969 Paul Lorenzen o. Prof. der Philosophie an der Universitat Erlangen Geschaftsfilhrende Herausgeber: Prof. Dr. B. Eckmann Eidgenossische Technische Hochschule Zurich Prof. Dr. B. L. van cler Waerclen Mathematisches Institut der Universitat ZUrich ISBN 978-3-642-86519-0 ISBN 978-3-642-86518-3 (eBook) DOl 10.1007/978-3-642-86518-3 Aile Rechte vorbehalten. Kein Teil dieses Buches darf ohne schriftliche Genehmigung des Springer-Verlages ubersetzt oder in irgendeiner Form vervielfaltigt werden © by Springer-Verlag Berlin· Heidelberg 1955 und 1969 (...) Softcover reprint of the hardcover 2nd edition 1969 Library of Congress Catalog Card Number 73-76724 Titel-Nr. 5061 Vorwort zur zweiten Auflage. Fiir die Neuauflage ist der Text nur unwesentlich geandert worden. Es ist - neben der Korrektur einiger Ungenauigkeiten - vor allem die Terminologie und Symbolik an meine spateren Arbeiten angeglichen. Obwohl ich - verstandlicherweise - jetzt die Ansatze der spateren Arbeiten fiir "sachgemaBer" halte, z. B. eine Logik der Dialoge statt einer Logik der Kalkiile, die Verwendung indefiniter Quantoren statt einer expliziten Konstruktion von Sprachschichten, enthalt diese Neu­ auflage den Inhalt der 1. Auflage unverandert. Der Leser kann also einen Vergleich mit meinen spateren Arbeiten (vgl. Literaturverzeichnis) seIber durchfiihren. Mein Dank gilt wiederum dem Verlag fiir seine entgegenkommende Mitarbeit bei der Vorbereitung dieser Neuauflage. Erlangen, den 1. November I968. PAUL LORENZEN. Vorwort zur ersten Auflage. (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.