nLab weakening rule

The weakening rule


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


The weakening rule


In formal logic and type theory, by “weakening rules” one means inference rules for context extension, which state that any premises may be added to the hypotheses (antecedents) of a valid judgement.

Along with the variable rule, the contraction rule and the exchange rule, the weakening rule is one of the most commonly adopted structural rules. But the weakening rule is not used in all logical frameworks, for instance in linear logic it is discarded.

Concretely, in intuitionistic dependent type theory the weakening rule is the inference rule

WΓ,Δj:JΓA:TypeΓ,A,Δj:J W \frac{ \Gamma ,\, \Delta \; \vdash \; j \colon J \;\;\;\;\;\;\;\; \Gamma \; \vdash \; A \colon Type }{ \Gamma ,\; A ,\, \Delta \; \vdash \; j \colon J }

As usual in dependent type theory, the meaning of this rule is a little less trivial than it may superficially seem, due to the generic type dependency involved: In extending the contents in the antecedent we are implicitly making the succedent depend on this extended context, albeit trivially.

Accordingly, in the categorical semantics of dependent types the weakening rule corresponds to pullback of bundles/display maps to the fiber product which interprets the extended context:


Weakening rules correspond to having projections for the monoidal structure that corresponds to the logical binary operator at hand.


The notion of weakening as a structural inference rule originates (under the German name Verdünnung) with:

On the categorical semantics

Discussion in intuitionistic dependent type theory:

Last revised on August 25, 2023 at 15:08:55. See the history of this page for a list of all contributions to it.