nLab contraction rule

The contraction rule

Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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The contraction rule

Idea
Statements
Semantics
Related concepts
References

Idea

In formal logic and type theory, 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: In substructural logics like linear logic and affine logic it is discarded.

Statements

Exactly how this looks depends on the logic used.

…

Semantics

In categorical semantics, the contraction rule corresponds to having diagonals for the monoidal structure corresponding to the logical binary operator at hand.

Related concepts

References

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:

Bart Jacobs, Semantics of weakening and contraction, Annals of Pure and Applied Logic 69 1 (1994) 73-106 [doi:10.1016/0168-0072(94)90020-5]

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 September 6, 2024 at 12:26:41. See the history of this page for a list of all contributions to it.