nLab type of propositions

Idea

In type theory a type of propositions $Prop$ or $\Omega$ corresponds roughly under categorical semantics 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 the type universe.

Definition

The type of all propositions

The type of all propositions is given by the following rules:

Formation rules:

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

Type reflection:

$$\frac{\Gamma \vdash A:\Omega}{\Gamma \vdash El(A) \; \mathrm{type}}$$

Introduction rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \mathrm{proptrunc}_A:\mathrm{isProp}(A)}{\Gamma \vdash A_\Omega:\Omega}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \mathrm{proptrunc}_A:\mathrm{isProp}(A)}{\Gamma \vdash \delta_A:El(A_\Omega) \simeq A}$$

Propositional truncation for each type reflection

$$\frac{\Gamma \vdash A:\Omega}{\Gamma \vdash \mathrm{proptrunc}(A):\mathrm{isProp}(El(A))}$$

Univalence:

$$\frac{\Gamma \vdash A:\Omega \quad \Gamma \vdash B:\Omega} {\Gamma \vdash \mathrm{univalence}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

The type of all decidable propositions

The type of all decidable propositions is given by the following rules:

Formation rules:

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

Type reflection:

$$\frac{\Gamma \vdash A:\Omega}{\Gamma \vdash El(A) \; \mathrm{type}}$$

Introduction rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \mathrm{proptrunc}_A:\mathrm{isProp}(A) \quad \Gamma \vdash \mathrm{lem}_A:A \vee \neg A}{\Gamma \vdash A_\Omega:\Omega}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \mathrm{proptrunc}_A:\mathrm{isProp}(A) \quad \Gamma \vdash \mathrm{lem}_A:A \vee \neg A}{\Gamma \vdash \delta_A:El(A_\Omega) \simeq A}$$

Propositional truncation for each type reflection

$$\frac{\Gamma \vdash A:\Omega}{\Gamma \vdash \mathrm{proptrunc}(A):\mathrm{isProp}(El(A))}$$

Excluded middle for each type reflection

$$\frac{\Gamma \vdash A:\Omega}{\Gamma \vdash \mathrm{lem}(A):El(A) \vee \neg El(A)}$$

Univalence:

$$\frac{\Gamma \vdash A:\Omega \quad \Gamma \vdash B:\Omega} {\Gamma \vdash \mathrm{univalence}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

The type of booleans

The type of booleans or booleans type is given by the following rules:

Formation rules:

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

Type reflection:

$$\frac{\Gamma \vdash A:\mathrm{bool}}{\Gamma \vdash El(A) \; \mathrm{type}}$$

Introduction rules:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{false}:\mathrm{bool}} \quad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \delta_\mathbb{0}:El(\mathrm{false}) \simeq \mathbb{0}}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{true}:\mathrm{bool}} \quad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \delta_\mathbb{1}:El(\mathrm{true}) \simeq \mathbb{1}}$$

Univalence:

$$\frac{\Gamma \vdash A:\mathrm{bool} \quad \Gamma \vdash B:\mathrm{bool}} {\Gamma \vdash \mathrm{univalence}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

Unlike a generic two-valued type defined as the sum type of two copies of the unit type or directly defined in terms of natural deduction rules, having a type of booleans is inconsistent with every type being an h-proposition. If $\mathrm{bool}$ were an h-proposition, that means that the identity type $\mathrm{false} = \mathrm{true}$ is pointed, and by transport across $\mathrm{El}$, there is an equivalence between the empty type and the unit type, which is a contradiction.

Other types of propositions

We work in an intensional type theory with propositional truncations $\left[T\right]$ for types $T$. A type of propositions is a type $\Omega$ with a type family $T$ such that

• for every proposition $P:\Omega$, 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 propositions $P:\Omega$ and $Q:\Omega$

  $$\prod_{P:\Omega} \prod_{Q:\Omega} \mathrm{isEquiv}(\mathrm{trans}^T(P, Q))$$
• $(\Omega, T)$ is weakly closed under truth: there is an element $\top_\Omega:\Omega$ and an equivalence $\mathrm{canonical}_\top: T(\top_\Omega) \simeq \mathbb{1}$
• $(\Omega, T)$ is weakly closed under conjunction: there is a function $(-)\wedge(-):\Omega \times \Omega \to \Omega$ and an equivalence $\mathrm{canonical}_\wedge(P, Q): T(P \wedge Q) \simeq \left[T(P) \times T(Q)\right]$
• $(\Omega, T)$ is weakly closed under implication: there is a function $(-)\implies(-):\Omega \times \Omega \to \Omega$ and an equivalence $\mathrm{canonical}_\implies(P, Q): T(P \implies Q) \simeq \left[T(P) \to T(Q)\right]$
• $(\Omega, T)$ is weakly closed under falsehood: there is an element $\bot_\Omega:\Omega$ and an equivalence $\mathrm{canonical}_\bot: T(\bot_\Omega) \simeq \mathbb{0}$
• $(\Omega, T)$ is weakly closed under disjunction: there is a function $(-)\vee(-):\Omega \times \Omega \to \Omega$ and an equivalence $\mathrm{canonical}_\vee(P, Q): T(P \vee Q) \simeq \left[T(P) + T(Q)\right]$

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

Examples

The type of all internal propositions

$$\sum_{A:U} \mathrm{isProp}(T(A))$$

in a Tarski universe $(U, T)$ is a type of propositions. If the Tarski universe has all propositions, then

$$\sum_{A:U} \mathrm{isProp}(T(A))$$

is the type of all propositions.

Related concepts

• dominance

• propositional extensionality

• subobject classifier

• type universe

• type of classes

• propositional logic as a dependent type theory

References

Detailed discussion of the type of propositions in Coq is in

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