https://hal.inria.fr/inria-00076024/document

The calculus of constructions

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.

A Framework for Defining Logics

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A framework for defining logics

/pdf/logic-for-computer-science-foundations-of-automatic-theorem-5an9u1ga8c.pdf

Logic for computer science: foundations of automatic theorem proving

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.

https://logic.uconn.edu/wp-content/uploads/sites/508/2014/01/Milnikel2.pdf

Gaisi Takeuti

Papers

Proof Theory