propositional extensionality



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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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


Equality and Equivalence


Disambiguation: This should not be confused with propositional extensionality in extensional type theory.



In formal logic propositional extensionality holds when any two propositions PP and QQ are identified, P=QP = Q, precisely if they imply each other, (PQ)(P \leftrightarrow Q) (hence if they are logically equivalent (PQ)(P \simeq Q)), i.e.

(P=Q)(PQ). (P=Q) \simeq (P\leftrightarrow Q) \,.


In type theory and Relation to univalence

In type theory the expression (P=Q)(P=Q) is the identity type Id Prop(P,Q)Id_{Prop}(P,Q) of the universe of propositions (or of the whole universe of types, under propositions as types), with PP and QQ substituted. One might more precisely write (P=Q)('P'='Q') here, with the quotation marks indicating that this is the name of the proposition, namely a term of the universe type, rather than the proposition/type itself.

On the other hand, the expression (PQ)(P \leftrightarrow Q) in homotopy type theory is the type of equivalences (PQ)(P \simeq Q) between the two propositions, hence the subtype? of the function type (PQ)(P \to Q) on those terms that, in particular, have a homotopy inverse.

Hence propositional extensionality in type theory is the statement that

(P=Q)(PQ). ('P' = 'Q') \simeq (P \simeq Q) \,.

(e.g. Sozeau-Tabareau, section 3.2)

In homotopy type theory, the assertion of this equivalence is a special case of the univalence axiom which asserts this equivalence for all types PP,QQ, not necessarily propositions, with the identity type of the full universe of types on the left.

(e.g. Sozeau-Tabareau, section 3.9)


Specializing to the case where one of the propositions is ‘true’, George Boole can be taken (see Voevodsky 14, slide 8) to be talking about propositional extensionality when he writes (Boole 1853, p. 53):

If instead of the proposition, “The sun shines,” we say, “It is true that the sun shines,” we then speak not directly of things, but of a proposition concerning things, viz., of the proposition “The sun shines.” And, therefore, the proposition in which we thus speak is a secondary one. Every primary proposition may thus give rise to a secondary proposition, viz., to that secondary proposition which asserts its truth, or declares its falsehood.

Later this became Alfred Tarski‘s material adequacy condition, also known as Convention T:

any viable theory of truth must entail, for every sentence PP of a language, a sentence of the form:

P'P' is true if, and only if, PP.

(see Wikipedia – Semantic theory of truth – Tarski’s theory)

This may be regarded as the above equivalence of propositional extensionality for the case that QQ \coloneqq true:

(P=true)(Ptrue). ('P' = 'true') \simeq (P \simeq true) \,.


  • George Boole, An Investigation of the Laws of Thought, (1853) (retyped pdf)

  • Matthieu Sozeau and Nicolas Tabareau, Univalence For Free (pdf)

  • Vladimir Voevodsky, Foundations of Mathematics: their past, present and future, Part II, (slides)

Last revised on January 2, 2020 at 05:37:48. See the history of this page for a list of all contributions to it.