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Proof Theory

01 Jan 1975-
About: The article was published on 1975-01-01 and is currently open access. It has received 770 citations till now. The article focuses on the topics: Proof theory & Gentzen's consistency proof.
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Journal ArticleDOI
TL;DR: In this article, the authors propose a method to solve the problem of homonymity in homonymization, i.e., homonym-of-subjects-with-objectivity.
Abstract: ion

1,268 citations

Proceedings Article
01 Jan 1987
TL;DR: The Edinburgh Logical Framework (LF) as discussed by the authors provides a means to define (or present) logics, based on a general treatment of syntax, rules, and proofs by means of a typed l-calculus with dependent types.
Abstract: The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed l-calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Lo¨f's system of arities. The treatment of rules and proofs focuses on his notion of a judgment. Logics are represented in LF via a new principle, the judgments as types principle, whereby each judgment is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgments and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools, such as proof editors and proof checkers, can be constructed.

1,144 citations

Journal ArticleDOI
02 Jan 1993
TL;DR: The Edinburgh Logical Framework provides a means to define (or present) logics through a general treatment of syntax, rules, and proofs by means of a typed λ-calculus with dependent types, whereby each judgment is identified with the type of its proofs.
Abstract: The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed l-calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Lo¨f's system of arities. The treatment of rules and proofs focuses on his notion of a judgment. Logics are represented in LF via a new principle, the judgments as types principle, whereby each judgment is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgments and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools, such as proof editors and proof checkers, can be constructed.

929 citations

Book
01 Sep 1985
TL;DR: This advanced text for undergraduate and graduate students introduces mathematical logic with an emphasis on proof theory and procedures for algorithmic construction of formal proofs and basics of automatic theorem proving.

490 citations

Journal ArticleDOI
TL;DR: This paper finds the logic LP of propositions and proofs and shows that Godel's provability calculus is nothing but the forgetful projection of LP, which achievesGodel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s.
Abstract: In 1933 G¨ odel introduced a calculus of provability (also known as modal logic S4 )a nd left open the question of its exact intended semantics. In this paper we give as olution to this problem. We find the logic LP of propositions and proofs and show that G¨ odel's provability calculus is nothing but the forgetful projection of LP .T his also achieves G¨ odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov styleprovability semantics forIntwhich resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and ! -calculus.

448 citations