nLab structural rule

Structural rules


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

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

syntax object language

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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Structural rules


(from Gentzen 1935)


In formal logic and type theory, by the structural inference rules [Gentzen 1935 §1.2.1, 1969, §1.21] of a deductive system one means those inference rules which make no reference to logical operations but only to the unstructured premises of a deduction.

In standard intuitionistic logic (intuitionistic type theory) the structural rules notably include the weakening rule and the contraction rule in the antecedent, which exhibit context extensions ΓΓ,P\Gamma \,\mapsto\, \Gamma, P as admitting natural diagonal and projection maps, respectively, hence as admitting interpretation as cartesian products Γ×P\Gamma \times P (cf. Jacobs 1994):

Discarding these rules leads to linear logic (linear type theory).

Generally, logical systems discarding some structural rules are called substructural logics.

In dependent type theory

In (intuitionistic) dependent type theory the structural rules include (e.g. Jacobs (1998), p. 122, UFP13, A.2.2, Rijke (2018), Sec. 1.4):

shown in the following table together with their categorical semantics of dependent types:


Sub-structural logic

If some of the structural rules are not imposed in a formal logic one speaks of substructural logic.

For instance, if the weakening rule and contraction rule are omitted, one speaks of linear logic/linear type theory (see there), since then the logical conjunction is no longer consrained to behave like a Cartesian product but may behave like a non-cartesian tensor product.


The notion of the structural inference rules originates (under the German name Struktur-Schlußfiguren) with


Their categorical semantics is made explicit in:

Discussion in dependent type theory:

Last revised on September 4, 2023 at 09:56:46. See the history of this page for a list of all contributions to it.