nLab type of propositions

Idea

In type theory a type of propositions — typically denoted $Prop$ or $\Omega$ — corresponds, under categorical semantics, roughly to a subobject classifier.

(To be precise, depending on the type theoretic rules and axioms this may not be quite true: one needs propositional resizing, propositional extensionality, and — in some type theories where “proposition” is not defined as an h-proposition, such as the calculus of constructions — the principle of unique choice.)

Its generalization from propositions to general types is a type universe.

The subtypes of a type $A$ may typically be identified with the function types into the type of propositions

$$Sub(A) \;\; = \;\; Prop_A \;\; \equiv \;\; A \to Prop \;\; \equiv \;\; A \to \Omega \,.$$

In dependent type theory such a $P \,\colon\, A \to Prop$ is equivalently an $A$-dependent proposition, to be understood as assingning to any term $a \colon A$ the assertion that “$a$ is contained in the given subtype”.

Definition

There are many different ways of defining the isProp modality in dependent type theory. Some of them include

$$\mathrm{isProp}(A) \coloneqq \prod_{x:A} \prod_{y:A} x =_A y$$

$$\mathrm{isProp}(A) \coloneqq \prod_{x:A} \prod_{y:A} \mathrm{isContr}(x =_A y)$$

$$\mathrm{isProp}(A) \coloneqq A \to \mathrm{isContr}(A)$$

We shall be agnostic about the different definitions of $\mathrm{isProp}(A)$, and just directly use $\mathrm{isProp}(A)$.

The type of all propositions

The type of all propositions $\mathrm{Prop}$ in a dependent type theory could be presented either as a Russell universe or a Tarski universe. The difference between the two is that in the former, every mere proposition in the type theory is literally an element of the type of all propositions, while in the latter, elements of $\mathrm{Prop}$ are only indices of a (-1)-truncated type family $\mathrm{El}$; every mere proposition in the type theory is only essentially $\mathrm{Prop}$-small for weak Tarski universes or judgmentally equal to an $\mathrm{El}(P)$ for $P:\mathrm{Prop}$ for strict Tarski universes.

As a strict Tarski universe

As a strict Tarski universe, the type of all propositions is given by the following natural deduction inference rules:

Formation rules for the type of all propositions:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Prop} \; \mathrm{type}}$$

Introduction rules for the type of all propositions:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toProp}_A:\mathrm{isProp}(A) \to \mathrm{Prop}}$$

Elimination rules for the type of all propositions:

$$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash \mathrm{El}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash \mathrm{proptrunc}(A):\mathrm{isProp}(\mathrm{El}(A))}$$

Computation rules for the type of all propositions:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \mathrm{El}(\mathrm{toProp}_A(p)) \equiv A \; \mathrm{type}}$$

• Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \mathrm{proptrunc}(\mathrm{toProp}_A(p)) \equiv p:\mathrm{isProp}(A)}$$

• Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \beta_\mathrm{Prop}^{\mathrm{proptrunc},A}(p):\mathrm{proptrunc}(\mathrm{toProp}_A(p)) =_{\mathrm{isProp}(A)} p}$$

Uniqueness rules for the type of all propositions:

• Judgmental computation rules:

  $$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash A \equiv \mathrm{toProp}_{\mathrm{El}(A)}(\mathrm{proptrunc}(A)):\mathrm{Prop}}$$

• Typal computation rules:

  $$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash \eta_{\mathrm{Prop}}(A):A =_{\mathrm{Prop}} \mathrm{toProp}_{\mathrm{El}(A)}(\mathrm{proptrunc}(A))}$$

Extensionality principle of the type of all propositions:

$$\frac{\Gamma \vdash A:\mathrm{Prop} \quad \Gamma \vdash B:\mathrm{Prop}} {\Gamma \vdash \mathrm{ext}_\mathrm{Prop}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

As a weak Tarski universe

As a weak Tarski universe, the type of all propositions is given by the following natural deduction inference rules:

Formation rules for the type of all propositions:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Prop} \; \mathrm{type}}$$

Introduction rules for the type of all propositions:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toProp}_A:\mathrm{isProp}(A) \to \mathrm{Prop}}$$

Elimination rules for the type of all propositions:

$$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash \mathrm{El}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash \mathrm{proptrunc}(A):\mathrm{isProp}(\mathrm{El}(A))}$$

Computation rules for the type of all propositions:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \beta_\mathrm{Prop}^{\mathrm{El}, A}(p):\mathrm{El}(\mathrm{toProp}_A(p)) \simeq A \; \mathrm{type}}$$

• Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \mathrm{congform}_\mathrm{isProp}(\beta_\mathrm{Prop}^{\mathrm{El}, A}(p))(\mathrm{proptrunc}(\mathrm{toProp}_A(p))) \equiv p:\mathrm{isProp}(A)}$$

• Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \beta_\mathrm{Prop}^{\mathrm{proptrunc},A}(p):\mathrm{congform}_\mathrm{isProp}(\beta_\mathrm{Prop}^{\mathrm{El}, A}(p))(\mathrm{proptrunc}(\mathrm{toProp}_A(p))) =_{\mathrm{isProp}(A)} p}$$

where the equivalence

$$\mathrm{congform}_\mathrm{isProp}(\beta_\mathrm{Prop}^{\mathrm{El}, A}(p)):\mathrm{isProp}(\mathrm{El}(\mathrm{toProp}_A(p))) \simeq \mathrm{isProp}(A)$$

can always be constructed in a type theory with dependent product types, dependent sum types, and identity types, and function extensionality, as given types $A$ and $B$ and an equivalence $e:A \simeq B$, it is always possible to form the equivalence

$$\mathrm{congform}_\mathrm{isProp}(e):\mathrm{isProp}(A) \simeq \mathrm{isProp}(B)$$

For example, for $\mathrm{isProp}$ defined as the dependent product type $\prod_{x:A} \prod_{y:A} x =_A y$, we have that the application of the equivalence $e:A \simeq B$ to identifications is itself an equivalence:

$$\mathrm{ap}_e(x, y):(x =_A y) \simeq (e(x) =_B e(y))$$

Then by the typal congruence rules of dependent function types, there is an equivalence

$$\mathrm{congform}_\mathrm{isProp}(e) \equiv \mathrm{congform}_{\prod}(e, \lambda x:A.\mathrm{congform}_{\prod}(e, \lambda y:A.\mathrm{ap}_e(x, y))):\left(\prod_{x:A} \prod_{y:A} x =_A y\right) \simeq \left(\prod_{x:B} \prod_{y:B} x =_B y\right)$$

And for any other definition of $\mathrm{isProp}$, we have an equivalence $\delta_{\mathrm{isProp}(A)}:\mathrm{isProp}(A) \simeq \prod_{x:A} \prod_{y:A} x =_A y$ for type $A$, and so there is an equivalence

$$\mathrm{congform}_\mathrm{isProp}(e) \equiv \delta_{\mathrm{isProp}(B)}^{-1} \circ \mathrm{congform}_{\prod}(e, \lambda x:A.\mathrm{congform}_{\prod}(e, \lambda y:A.\mathrm{ap}_e(x, y))) \circ \delta_{\mathrm{isProp}(A)}:\mathrm{isProp}(A) \simeq \mathrm{isProp}(B)$$

Uniqueness rules for the type of all propositions:

• Judgmental computation rules:

  $$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash A \equiv \mathrm{toProp}_{\mathrm{El}(A)}(\mathrm{proptrunc}(A)):\mathrm{Prop}}$$

• Typal computation rules:

  $$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash \eta_{\mathrm{Prop}}(A):A =_{\mathrm{Prop}} \mathrm{toProp}_{\mathrm{El}(A)}(\mathrm{proptrunc}(A))}$$

Extensionality principle of the type of all propositions:

$$\frac{\Gamma \vdash A:\mathrm{Prop} \quad \Gamma \vdash B:\mathrm{Prop}} {\Gamma \vdash \mathrm{ext}_\mathrm{Prop}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

As a Russell universe

As a Russell universe, the type of all propositions is given by the following natural deduction inference rules:

Formation rules for the type of all propositions:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Prop} \; \mathrm{type}}$$

Introduction rules for the type of all propositions:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toProp}_A:\mathrm{isProp}(A) \to \mathrm{Prop}}$$

Elimination rules for the type of all propositions:

$$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash A \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash \mathrm{proptrunc}(A):\mathrm{isProp}(A)}$$

Computation rules for the type of all propositions:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \mathrm{toProp}_A(p) \equiv A \; \mathrm{type}}$$

• Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \mathrm{proptrunc}(\mathrm{toProp}_A(p)) \equiv p:\mathrm{isProp}(A)}$$

• Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \beta_\mathrm{Prop}^{\mathrm{proptrunc},A}(p):\mathrm{proptrunc}(\mathrm{toProp}_A(p)) =_{\mathrm{isProp}(A)} p}$$

Uniqueness rules for the type of all propositions:

• Judgmental computation rules:

  $$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash A \equiv \mathrm{toProp}_{A}(\mathrm{proptrunc}(A)):\mathrm{Prop}}$$

• Typal computation rules:

  $$\frac{\Gamma \vdash A:\mathrm{Prop}}{\Gamma \vdash \eta_{\mathrm{Prop}}(A):A =_{\mathrm{Prop}} \mathrm{toProp}_{A}(\mathrm{proptrunc}(A))}$$

Extensionality principle for the type of all propositions:

$$\frac{\Gamma \vdash A:\mathrm{Prop} \quad \Gamma \vdash B:\mathrm{Prop}} {\Gamma \vdash \mathrm{ext}_\mathrm{Prop}(A, B):(A =_\mathrm{Prop} B) \simeq (A \simeq B)}$$

Other types of propositions

Other types of propositions include

• the boolean domain, which is the type of decidable propositions.

• any dominance, which is the type of open propositions, or the type of semi-decidable propositions.

• For a Russell universe $U$, the type $\sum_{A:U} \mathrm{isProp}(A)$ is the type of $U$-small propositions. Similarly, for a Tarski universe $(U, T)$, the type $\sum_{A:U} \mathrm{isProp}(T(A))$ is the type of $U$-small propositions

• sProp, which is the type of strict propositions

• any contractible type, which is (equivalent to) the type of pointed propositions or the type of contractible types.

In general, these types of propositions can also be distinguished between their Russell and Tarski variants, where elements of the type of propositions are actual types, or indices for the type family $\mathrm{El}$.

Most generally, a type of propositions is a type $\Omega$ with a type family $T$ such that

• for every elements $P:\Omega$, the type $T(P)$ is an h-proposition:
  $$\prod_{P:\Omega} \prod_{a:T(P)} \prod_{b:T(P)} a =_{T(P)} b$$
• the transport function
  $$\mathrm{trans}^T(P, Q):P =_\Omega Q \to (T(P) \simeq T(Q))$$

  is an equivalence of types for all elements $P:\Omega$ and $Q:\Omega$

  $$\prod_{P:\Omega} \prod_{Q:\Omega} \mathrm{isEquiv}(\mathrm{trans}^T(P, Q))$$

These axioms imply that $(\Omega, T)$ satisfy propositional extensionality and thus that $\Omega$ is an h-set.

Predicate logic

In this section, we assume that function extensionality or weak function extensionality is true for all dependent function types.

The empty type and falsehood

For Russell types of all propositions, the type $\prod_{P:\mathrm{Prop}} P$ is the empty type $\emptyset$, which by weak function extensionality is a proposition, so is already in the type of all propositions $\left(\prod_{P:\mathrm{Prop}} P\right):\mathrm{Prop}$. Similarly, for Tarski types of all propositions, the type $\prod_{P:\mathrm{Prop}} \mathrm{El}(P)$ is the empty type, which by weak function extensionality is a proposition, and so one has the element $\left(\prod_{P:\mathrm{Prop}} \mathrm{El}(P)\right)_\mathrm{Prop}:\mathrm{Prop}$.

In either case, the definition implies that if the empty type has an element, then every proposition has an element and is thus contractible. This implies that the empty type represents falsehood $\bot$ in the type of all propositions.

The type of pointed propositions and truth

The type of pointed propositions $\mathrm{Prop}_*$ is given by the type $\sum_{P:\mathrm{Prop}} P$ for Russell types of propositions and $\sum_{P:\mathrm{Prop}} \mathrm{El}(P)$ for Tarski types of propositions, and because pointed propositions are contractible types, pointed propositions are equivalent to each other, and thus by the extensionality principle of $\mathrm{Prop}$, they are equal to each other; thus, the type of pointed propositions is a proposition.

Because $\mathrm{Prop}_*$ is itself a proposition, $\mathrm{Prop}_*$ is an element of itself, and thus a pointed proposition and a contractible type. This implies that the type of pointed propositions represents truth $\top$ in the type of all propositions.

Propositional truncations

Given the type of all propositions $\mathrm{Prop}$ and given a type $A$, the propositional truncation of a type $A$, which turns $A$ into an element of $\mathrm{Prop}$, is given for Russell types of all propositions by the type

$$[A] \equiv \prod_{P:\mathrm{Prop}} (A \to P) \to P$$

and for Tarski types of all propositions by the type

$$[A] \equiv \prod_{P:\mathrm{Prop}} (A \to \mathrm{El}(P)) \to \mathrm{El}(P)$$

By weak function extensionality, both of these types are propositions.

Universal quantification

Given any type $A$ and family of propositions $P(x)$ indexed by $x:A$, by weak function extensionality, the dependent function type $\prod_{x:A} P(x)$ is also a proposition. This means that the propositional truncation of $\prod_{x:A} P(x)$ is equivalent to $\prod_{x:A} P(x)$, and the dependent function type is used as the universal quantifier.

More generally, given any type $A$ and family of propositions $B(x)$ indexed by $x:A$, the universal quantifier is given by propositional truncation of the dependent product type:

$$\forall x:A.B(x) \equiv \left[\prod_{x:A} B(x)\right]$$

Implication and negation

Given any proposition $Q$ and $P$, by weak function extensionality, the function type $Q \to P$ is also a proposition. This means that the propositional truncation of $A \to P$ is equivalent to $Q \to P$, and the function type is used as implication. If $P$ is the empty type or falsehood defined above, then the function type represents the negation of $Q$, $\neg Q \equiv Q \to \emptyset$.

More generally, given any type $A$ and $B$, implication is given by propositional truncation of the function type:

$$A \implies B \equiv \left[A \to B\right]$$

Biconditionals

Given any proposition $P$ and $Q$, by weak function extensionality, the equivalence type $P \simeq Q$ is also a proposition. This means that the propositional truncation of $P \simeq Q$ is equivalent to $P \simeq Q$, and the equivalence type is used as the biconditional.

More generally, given any type $A$ and $B$, the biconditional is given by propositional truncation of the equivalence type:

$$A \iff B \equiv \left[A \simeq B\right]$$

Existential quantification

Given any type $A$ and family of types $B(x)$ indexed by $x:A$, the existential quantifier is given by the type

$$\exists x:A.B(x) \equiv \prod_{P:\mathrm{Prop}} \left(\prod_{x:A} B(x) \to P\right) \to P$$

for Russell types of propositions and

$$\exists x:A.B(x) \equiv \prod_{P:\mathrm{Prop}} \left(\prod_{x:A} B(x) \to \mathrm{El}(P)\right) \to \mathrm{El}(P)$$

for Tarski types of propositions.

If the type theory also has dependent sum types, the existential quantifier is equivalently given by the propositional truncation of the dependent sum type

$$\exists x:A.B(x) \equiv \left[\sum_{x:A} B(x)\right]$$

since by currying there is an equivalence

$$\left(\left(\sum_{x:A} B(x)\right) \to P\right) \simeq \left(\prod_{x:A} B(x) \to P\right)$$

Conjunction

Given types $A$ and $B$, the conjunction of $A$ and $B$ is given by the type

$$A \wedge B \equiv \prod_{P:\mathrm{Prop}} \left(A \to (B \to P)\right) \to P$$

for Russell types of propositions and

$$A \wedge B \equiv \prod_{P:\mathrm{Prop}} \left(A \to (B \to \mathrm{El}(P))\right) \to \mathrm{El}(P)$$

for Tarski types of propositions.

If the type theory also has product types, the existential quantifier is equivalently given by the propositional truncation of the product type

$$A \wedge B \equiv \left[A \times B\right]$$

since by currying there is an equivalence

$$\left((A \times B) \to P\right) \simeq \left(A \to (B \to P)\right)$$

Disjunctions

Given types $A$ and $B$, the disjunction of $A$ and $B$ is given by the type

$$A \vee B \equiv \prod_{P:\mathrm{Prop}} (A \to P) \to (B \to P) \to P$$

for Russell types of propositions and

$$A \vee B \equiv \prod_{P:\mathrm{Prop}} (A \to \mathrm{El}(P)) \to (B \to \mathrm{El}(P)) \to \mathrm{El}(P)$$

for Tarski types of propositions.

If the type theory also has product types, the existential quantifier is equivalently given by the type

$$A \vee B \equiv \prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P$$

for Russell types of propositions and

$$A \vee B \equiv \prod_{P:\mathrm{Prop}} ((A \to \mathrm{El}(P)) \times (B \to \mathrm{El}(P))) \to \mathrm{El}(P)$$

for Tarski types of propositions,

and if the type theory also has coproduct types, the existential quantifier is equivalently given by the propositional truncation of the coproduct type

$$A \vee B \equiv \left[A + B\right]$$

since by currying there is an equivalence

$$\left((A \to P) \times (B \to P)\right) \simeq \left((A + B) \to P\right)$$

Decidable propositions and booleans

A decidable proposition is a proposition $P$ with a witness $p:P \vee \neg P$ that the disjunction of $P$ and its negation holds.

The type of booleans is the type of all decidable propositions in $\mathrm{Prop}$,

$$\mathrm{Bool} \equiv \sum_{P:\mathrm{Prop}} P \vee \neg P$$

for Russell types of propositions, and

$$\mathrm{Bool} \equiv \sum_{P:\mathrm{Prop}} \mathrm{El}(P) \vee \neg \mathrm{El}(P)$$

for Tarski types of propositions.

The law of excluded middle states that every proposition is decidable $\mathrm{lem}:\prod_{P:\mathrm{Prop}} P \vee \neg P$, or equivalently that there is an equivalence $\mathrm{lem}':\mathrm{Bool} \simeq \mathrm{Prop}$ between the type of propositions and the type of booleans.

Impredicative polymorphism

Given a universe of all propositions $\mathrm{Prop}$, impredicative polymorphism of $\mathrm{Prop}$ is equivalent to weak function extensionality.

More generally, a universe of propositions $\Omega$ has impredicative polymorphism if and only if $\Omega$ is closed under dependent product types of predicates valued in $\Omega$.

Propositional resizing

Theorem

Any universe of propositions satisfying propositional resizing is a contractible type.

Proof

Given a universe of propositions $\Omega$, the type of all propositions in $\Omega$ is $\Omega$ itself. Then propositional resizing says that $\Omega$ has an element $\Omega':\Omega$ such that its type reflection is $\Omega$ itself. This implies that $\Omega$ itself a mere proposition by definition of universe of proposiions, which implies that $\Omega$ is a contractible type by the element $\Omega':\Omega$ and the fact that for all other elements $P:\Omega$, $P = \Omega'$.

Theorem

Any universe of propositions $\Omega$ closed under the empty type does not satisfy propositional resizing.

Proof

Suppose that $\Omega$ is closed under the empty type $\emptyset$ represented in $\Omega$ by the element $\bot:\Omega$, and satisfies propositional resizing. Then one has that $\bot = \Omega'$ since $\Omega$ is a mere proposition, and by transport one has $\emptyset \simeq \Omega$, but since $\Omega$ is contractible by propositional resizing, the empty type is also contractible, which is false.

Theorem

The universe of all propositions $\mathrm{Prop}$ does not satisfy propositional resizing.

Directed univalence

Suppose that the dependent type theory has hom-types $\mathrm{hom}_A(a, b)$. Then directed univalence for a type of propositions $(\Omega, \mathrm{El})$ with $P:\Omega \vdash \mathrm{El}(P)$ a covariant type family states that for all propositions $P$ and $Q$, there is an equivalence of types between $\mathrm{hom}_\Omega(P, Q)$ and $\mathrm{El}(P) \to \mathrm{El}(Q)$. Directed univalence for the type of propositions implies propositional extensionality.

Categorical semantics

The categorical semantics of the type of all propositions is the subobject classifier.

Related concepts

• boolean domain

• subtype, subobject

• subobject classifier

• dominance

• propositional extensionality

• type universe

• type of classes

• type of strict propositions

• propositional logic as a dependent type theory

• type of entire relations

• local propositional resizing?

• impredicative polymorphism

• impredicative dependent type theory

• sigma-frame of propositions

References

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013) [web, pdf]

Detailed discussion of the type of propositions in Coq is in

• Adam Chlipala, Certified programming with dependent types – Library Universes