Proof Theory

Proof-theoretical analysis: weak systems of functions and classes

The term proof theory, in German Beweistheorie, was introduced by Hilbert [1] in 1922 as the name of a subject which he had already described in a lecture [2] in 1917 as ‘an important, new field of research’ where ‘we must make the concept of specific mathematical proof itself object of investigation, just as also the astronomer pays attention to his place of observation, the physicist must care about the theory of his instrument, and the philosopher criticizes reason itself’.

Philosophical aspects of proof theory

Despite the fact that for many years intuitionistic logic has served its function primarily in relation to foundational questions in mathematics, there has been a significant revival of interest over the last couple of decades stimulated by the application of intuitionistic formalisms in computer science (1982) . It is beyond the scope of this chapter to comment on these applications in detail which, broadly speaking, either exploit formalisations of the intuitionistic meaning of general mathematical abstractions as programming logics [Martin-Lof, 1984; Martin-Lof, 1996; Constable et al., 1986] , or exploit the similarity of systems of formal intuitionistic proofs under cut-elimination to systems of typed lambda terms under various forms of reduction (e.g. [Howard, 1980; Girard, 1989; Coquand, 1990]) .1 Both types of application rely on the rich proof theory possessed by intuitionistic formalisms in comparison with their classical counterparts.2

Tableaux for Intuitionistic Logics

/pdf/light-logic-and-polynomial-time-computation-17yahv56yx.pdf

Light Logic and Polynomial Time Computation

We present a method for defining logical frameworks as a collection of features which are defined and behave independently of one another. Each feature is a set of grammar clauses and rules of deduction such that the result of adding the feature to a framework is a conservative extension of the framework itself. We show how several existing logical frameworks can be so built, and how several much weaker frameworks defined in this manner are adequate for expressing a wide variety of object logics.

/pdf/a-modular-hierarchy-of-logical-frameworks-31fsuvoy74.pdf

Proof Theory

Citations

Proof-theoretical analysis: weak systems of functions and classes

Philosophical aspects of proof theory

Tableaux for Intuitionistic Logics

Light Logic and Polynomial Time Computation

A Modular Hierarchy of Logical Frameworks

Related Papers (5)

Untersuchungen über das logische Schließen. II

Introduction to Metamathematics

The collected papers of Gerhard Gentzen

Basic proof theory

Linear logic