nLab propositional extensionality

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

Idea

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

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

Properties

In type theory and Relation to univalence

In type theory the expression $(P=Q)$ is the identity type $Id_{Prop}(P,Q)$ of the universe of propositions (or of the whole universe of types, under propositions as types), with $P$ and $Q$ substituted. One might more precisely write $('P'='Q')$ 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 $(P \leftrightarrow Q)$ in homotopy type theory is the type of equivalences $(P \simeq Q)$ between the two propositions, hence the subtype of the function type $(P \to Q)$ on those terms that, in particular, have a homotopy inverse.

Hence propositional extensionality in type theory is the statement that

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

(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 $P$,$Q$, not necessarily propositions, with the identity type of the full universe of types on the left.

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

History

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 $P$ of a language, a sentence of the form:

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

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

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

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

See also

• extensionality

• set extensionality?

References

• 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)