Type theories (1995) [2 citations — 0 self]
|
Abstract:
Abstract. Deduction modulo is a way to express a theory using computation rules instead of axioms. We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive occurrences, while others can only be used at negative ones. We show that all theories in propositional calculus can be expressed in this framework and that cuts can always be eliminated with such theories. Mathematical proofs are almost never built in pure logic, but besides the deduction rules and the logical axioms that express the meaning of the connectors and quantiers, they use something else- a theory- that expresses the meaning of the other symbols of the language. Examples of theories are equational theories, arithmetic, type theory, set theory,... The usual denition of a theory, as a set of axioms, is sucient when one is interested in the provability relation, but, as well-known, it is not when one is interested in the structure of proofs and in the theorem proving process. For
Citations
| 3 | A normalization theorem for set theory – Bailin - 1988 |
| 2 | Non-normalisation de la théorie de Zermelo – Crabbe - 1974 |
| 2 | Strati and cut-elimination – Crabbe - 1991 |
| 1 | Axioms vs. rewrite rules: from completeness to cut elimination – Dowek |
