In logic, the contraction rule states that any valid deduction that uses a premise more than once remains valid using that premise only once. Along with the weakening rule and the exchange rule, it is one of the most commonly adopted structural rules.
The contraction rule is not used in all logical frameworks. For instance in linear logic it is discarded.
Exactly how this looks depends on the logic used.
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This rule corresponds to having diagonals for the monoidal structure corresponding to the logical binary operator at hand.
The notion of contraction as a structural inference rule originates (under the German name Zusammenziehung) with:
Gerhard Gentzen, §1.2.1 in: Untersuchungen über das logische Schließen I Mathematische Zeitschrift 39 1 (1935) [doi:10.1007/BF01201353]
Gerhard Gentzen, §1.21 in: Investigations into Logical Deduction, in M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, Studies in Logic and the Foundations of Mathematics 55, Springer (1969) 68-131 [ISBN:978-0-444-53419-4, pdf]
On the categorical semantics:
Discussion in dependent type theory:
Bart Jacobs, p. 122, 585 in: Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141, Elsevier (1998) [ISBN:978-0-444-50170-7, pdf]
(emphasis on the categorical model of dependent types)
Univalent Foundations Project, §A of Homotopy Type Theory – Univalent Foundations of Mathematics (2013) [web, pdf]
(in the context of homotopy type theory)
Egbert Rijke, Dependent type theory [pdf], Lecture 1 in: Introduction to Homotopy Type Theory, lecture notes, CMU (2018) [pdf, pdf, webpage]
(in the context of homotopy type theory)
Last revised on August 25, 2023 at 15:11:42. See the history of this page for a list of all contributions to it.