A more general general proof theory

The goal of this paper is to introduce the reader to the distinction between intensional and extensional as a distinction between different approaches to meaning. We will argue that despite the common belief, intensional aspects of mathematical notions can be, and in fact have been successfully described in mathematics. One that is for us particularly interesting is the notion of deduction as depicted in general proof theory. Our considerations result in defending a) the importance of a rule-based semantical approach and b) the position according to which non-reductive and somewhat circular explanations play an essential role in describing intensionality in mathematics.

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Some Facts from the Theory of Proofs and Some Fictions from General Proof Theory

Essays on Mathematical and Philosophical Logic ◽

10.1007/978-94-009-9825-4_1 ◽

1979 ◽

pp. 3-23 ◽

Cited By ~ 1

Author(s):

G. Kreisel

Keyword(s):

Proof Theory ◽

General Proof

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The Fundamental Problem of General Proof Theory

Studia Logica ◽

10.1007/s11225-018-9785-9 ◽

2018 ◽

Vol 107 (1) ◽

pp. 11-29 ◽

Cited By ~ 3

Author(s):

Dag Prawitz

Keyword(s):

Proof Theory ◽

Fundamental Problem ◽

General Proof

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Reprint of: A more general general proof theory

Journal of Applied Logic ◽

10.1016/j.jal.2017.12.003 ◽

2017 ◽

Vol 25 ◽

pp. S25-S47 ◽

Cited By ~ 1

Author(s):

Heinrich Wansing

Keyword(s):

Proof Theory ◽

General Proof

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An Approach to General Proof Theory and a Conjecture of a Kind of Completeness of Intuitionistic Logic Revisited

Trends in Logic - Advances in Natural Deduction ◽

10.1007/978-94-007-7548-0_12 ◽

2014 ◽

pp. 269-279 ◽

Cited By ~ 8

Author(s):

Dag Prawitz

Keyword(s):

Intuitionistic Logic ◽

Proof Theory ◽

General Proof

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Resolution and the Origins of Structural Reasoning: Early Proof-Theoretic Ideas of Hertz and Gentzen

Bulletin of Symbolic Logic ◽

10.2178/bsl/1182353872 ◽

2002 ◽

Vol 8 (2) ◽

pp. 246-265 ◽

Cited By ~ 20

Author(s):

Peter Schroeder-Heister

Keyword(s):

Normal Form ◽

Logic Programming ◽

Proof Theory ◽

Structural Rules ◽

General Proof ◽

Early Proof ◽

Sld Resolution ◽

Structural Reasoning ◽

Structural Proof Theory

AbstractIn the 1920s, Paul Hertz (1881–1940) developed certain calculi based on structural rules only and established normal form results for proofs. It is shown that he anticipated important techniques and results of general proof theory as well as of resolution theory, if the latter is regarded as a part of structural proof theory. Furthermore, it is shown that Gentzen, in his first paper of 1933, which heavily draws on Hertz, proves a normal form result which corresponds to the completeness of propositional SLD-resolution in logic programming.

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GÖDEL’S NOTRE DAME COURSE

Bulletin of Symbolic Logic ◽

10.1017/bsl.2016.36 ◽

2016 ◽

Vol 22 (4) ◽

pp. 469-481 ◽

Cited By ~ 3

Author(s):

MILOŠ ADŽIĆ ◽

KOSTA DOŠEN

Keyword(s):

Proof Theory ◽

Natural Deduction ◽

Deduction System ◽

General Proof ◽

Natural Deduction System ◽

University Of Notre Dame ◽

Proper Perspective ◽

The University

AbstractThis is a companion to a paper by the authors entitled “Gödel’s natural deduction,” which presented and made comments about the natural deduction system in Gödel’s unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to Gödel and general proof theory, one can find a brief summary of Gödel’s notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.

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