nLab logical relation



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





In proof theory and the theory of programming languages, the term logical relations refers to a certain style of argument used in proving results such as strong normalization or observational equivalence?. Although there is no precise definition, typically a logical relation (or logical predicate) corresponds to a family of (sometimes unary) relations defined by induction on types, and such that the definition of the relation at a particular type mirrors the logical structure of that type in an appropriate sense. This methodology goes by many names historically, such as Tait’s method of computability and reducibility candidates.


As originally described by (Tait 1967) (and presented in (Girard 1990)), logical predicates can be used to prove strong normalization for the simply-typed lambda calculus. The idea is to define a family of predicates RED TRED_T over terms, indexed by types TT, such that:

  • (T=XT = X an atomic type): tRED Xt \in RED_X iff tt is strongly normalizing

  • (T=UVT = U \to V a function type): tRED UVt \in RED_{U \to V} iff u\forall u, uRED Uu \in RED_U implies t(u)RED Vt(u) \in RED_V.

By induction on TT, one establishes the following three conditions:

  1. if tRED Tt \in RED_T, then tt is strongly normalizing

  2. if tRED Tt \in RED_T and ttt \to t', then tRED Tt' \in RED_T

  3. if tt is neutral (i.e., either a variable or application) and for all ttt \to t' we have tRED Tt' \in RED_T, then tRED Tt \in RED_T

Finally, one proves the “fundamental lemma”

  • if t:Tt:T then tRED Tt \in RED_T

which by condition (1) implies that every simply-typed term is strongly normalizing.


  • W. W. Tait. Intensional Interpretations of Functionals of Finite Type I, JSL 32:2, June 1967. (JSTOR)

  • Gordon Plotkin. Lambda-definability and logical relations, unpublished manuscript, Edinburgh 1973. (pdf)

  • John C. Reynolds, Types, Abstraction and Parametric Polymorphism Information Processing 83(1) (1983), pp. 513-523.

  • Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and Types, 1990. (web)

For a recent perspective:

  • Claudio Hermida, Uday Reddy, E. Robinson, section 2 of Logical Relations and Parametricity - A Reynolds Programme for Category Theory and Programming Languages, Electronic Notes in Theoretical Computer Science (2013) (pdf)

See also

  • Patricia Johann, Neil Ghani, Logical Relations for Program Verification, research proposal (pdf)

  • Thierry Coquand, from slide 63 on in Equality and dependent type theory (pdf)

  • Florian Rabe, Kristina Sojakova, Logical Relations for a Logical Framework (pdf)

  • Carsten Schürmann, Jeffrey Sarnat, Structural logical relations (pdf)

  • Lau Skorstengaard, An Introduction to Logical Relations: Proving Program Properties Using Logical Relations, (pdf)

  • Karl Crary, Logical Relations and a Case Study in Equivalence Checking, in Benjamin Pierce (ed.), Advanced Topics in Types and Programming Languages

  • Amal Ahmed, Lectures on logical relations at OPLSS 2015, (lecture recordings)

Further discussion:

Last revised on May 19, 2022 at 20:32:12. See the history of this page for a list of all contributions to it.