mere proposition (changes) in nLab

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Mere Propositions

Idea

In homotopy type theory, the notion of mere proposition is an internalization of the notion of proposition / (-1)-truncated object. Mere propositions are also called h-propositions, types of h-level 11, (−1)(-1)-truncated types, or even just propositions if no ambiguity will result. Semantically, mere propositions correspond to (-1)-groupoids or more generally (-1)-truncated objects.

Mere propositions are one way to embed logic into homotopy type theory, via the propositions as some types / bracket types approach. For many purposes, this is the “correct” way to embed logic (as opposed to the propositions as types approach — for instance, the PAT version of excluded middle contradicts the univalence axiom, whereas the mere-propositional version does not).

Definition

Consider intensional type theory with dependent sums, dependent products, and identity types.

There are many equivalent definitions of a type being an h-proposition. Perhaps the simplest is as follows.

In propositions as types language, this can be pronounced as “every two elements of AA are equal”.

The first fits into the general inductive definition of n-groupoid: an ∞\infty-groupoid is an nn-groupoid if all its hom-groupoids are (n−1)(n-1)-groupoids.

The second says that being a proposition is equivalent to being “contractible if inhabited”, and when isContr is expanded out to ∑ x:A∏ y:Ax= Ay\sum_{x:A} \prod_{y:A} x =_A y, the resulting type A→∑ x:A∏ y:Ax= AyA \to \sum_{x:A} \prod_{y:A} x =_A y is just the type of the graphs of dependent functions of ∏ x:A∏ y:Ax= Ay\prod_{x:A} \prod_{y:A} x =_A y.

The third states that being a proposition is the same as being a bool\mathrm{bool}-local type: the function

const 𝟚,A≡λx:A.λb:v.x:A→(bool→A)\mathrm{const}_{\mathbb{2}, A} \equiv \lambda x:A.\lambda b:v.x:A \to (\mathrm{bool} \to A)

which takes elements x:Ax:A to constant functions from the boolean domain with value xx is an equivalence of types.

The fourth states that being a proposition is equivalent to the condition that for every function from the boolean domain, f(0)=f(1)f(0) = f(1), and is interderivable from the usual definition of isProp via the recursion principle of the boolean domain.

The fifth states that being a proposition is equivalent to the condition that for every function from the boolean domain, f(0)=f(1)f(0) = f(1) is contractible, and is interderivable from the first via the recursion principle of the boolean domain.

The sixth states that being a proposition is equivalent to the condition that every function from the boolean domain is a weakly constant function, and can be proven to be equivalent to the fourth via double induction on the boolean domain and the properties of identity types.

The seventh states that being a proposition is equivalent to the condition that every function from the boolean domain to AA factors through the interval type via the function rec 𝟚 𝕀(0 𝕀,1 𝕀):𝟚→𝕀\mathrm{rec}_\mathbb{2}^\mathbb{I}(0_\mathbb{I}, 1_\mathbb{I}):\mathbb{2} \to \mathbb{I} defined by the recursion principle of the boolean domain.

Rules for isProp

The usual definition of isProp is given by the following rules:

Γ⊢AtypeΓ⊢isProp(A)typeΓ⊢AtypeΓ⊢isProp(A)≡∏ x:A∏ y:Ax= Aytype\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \equiv \prod_{x:A} \prod_{y:A} x =_A y \; \mathrm{type}}

If the dependent type theory doesn’t have dependent function types, but still has identity types one could still define isProp by adding the formation, introduction, elimination, computation, and uniqueness rules for isProp:

Formation rules for isProp types:

Γ⊢AtypeΓ,x:A,y:A⊢x= AytypeΓ⊢isProp(A)type\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash x =_A y \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \; \mathrm{type}}

Introduction rules for isProp types:

Γ,x:A,y:A⊢p(x,y):x= AyΓ⊢λ(x).λ(y).p(x,y):isProp(A)\frac{\Gamma, x:A, y:A \vdash p(x, y):x =_A y}{\Gamma \vdash \lambda(x).\lambda(y).p(x, y):\mathrm{isProp}(A)}

Elimination rules for isProp types:

Γ⊢p:isProp(A)Γ⊢a:AΓ⊢b:AΓ⊢p(a,b):a= Ab\frac{\Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma \vdash p(a, b):a =_A b}

Computation rules for isProp types:

Γ,x:A,y:A⊢p(x,y):x= AyΓ⊢a:AΓ⊢b:AΓ⊢β isProp:(λ(x).λ(y).p(x,y))(a,b)= a= Abp(a,b)\frac{\Gamma, x:A, y:A \vdash p(x, y):x =_A y \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma \vdash \beta_\mathrm{isProp}:(\lambda(x).\lambda(y).p(x, y))(a, b) =_{a =_A b} p(a, b)}

Uniqueness rules for isProp types:

Γ⊢p:isProp(A)Γ⊢η isProp:p= isProp(A)λ(x).λ(y).p(x,y)\frac{\Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \eta_\mathrm{isProp}:p =_{\mathrm{isProp}(A)} \lambda(x).\lambda(y).p(x, y)}

Properties

• We can prove that for any AA, the type isProp(A)isProp(A) is an h-proposition, i.e. isProp(isProp(A))isProp(isProp(A)). Thus, any two inhabitants of isProp(A)isProp(A) (witnesses that AA is an h-proposition) are “equal”.

• Every function between h-propositions AA and BB is an embedding of types.

• If AA and BB are h-props and there exist maps A→BA\to B and B→AB\to A, then AA and BB are equivalent. This follows from the fact that if AA and BB are types and there exists embeddings A→BA\to B and B→AB\to A, then AA and BB are equivalent.

Relation to the principles of omniscience

Suppose that AA is a mere proposition. Then AA is equivalent to the existential quantifier ∃x:A.(λt.1)(x)=1\exists x:A.(\lambda t.1)(x) = 1, where λt.1\lambda t.1 is the constant function from AA to the booleans which takes elements of AA to the boolean 1:bool1:\mathrm{bool}. Then, one can show that various principles of omniscience are equivalent to weak versions of excluded middle, via the special case of the constant function λt.1\lambda t.1:

• That the limited principle of omniscience holds for all mere propositions is equivalent to excluded middle:

(∏ A:Prop∏ f:A→2∃x:A.f(x)=1∨¬(∃x:A.f(x)=1))≃(∏ A:PropA∨¬A)\left(\prod_{A:\mathrm{Prop}} \prod_{f:A \to 2} \exists x:A.f(x) = 1 \vee \neg (\exists x:A.f(x) = 1)\right) \simeq \left(\prod_{A:\mathrm{Prop}} A \vee \neg A\right)

• That the weak limited principle of omniscience holds for all mere propositions is equivalent to weak excluded middle:

(∏ A:Prop∏ f:A→2¬(∃x:A.f(x)=1)∨¬¬(∃x:A.f(x)=1))≃(∏ A:Prop¬A∨¬¬A)\left(\prod_{A:\mathrm{Prop}} \prod_{f:A \to 2} \neg (\exists x:A.f(x) = 1) \vee \neg \neg (\exists x:A.f(x) = 1)\right) \simeq \left(\prod_{A:\mathrm{Prop}} \neg A \vee \neg \neg A\right)

• That the lesser limited principle of omniscience holds for all mere propositions is equivalent to de Morgan's law:

(∏ A:Prop∏ B:Prop∏ f:A→2∏ g:B→2¬((∃x:A.f(x)=1)∧(∃x:B.g(x)=1))→(¬(∃x:A.f(x)=1)∨¬(∃x:B.g(x)=1)))≃(∏ A:Prop∏ B:Prop¬(A∧B)→(¬A)∨(¬B))\left(\prod_{A:\mathrm{Prop}} \prod_{B:\mathrm{Prop}} \prod_{f:A \to 2} \prod_{g:B \to 2} \neg ((\exists x:A.f(x) = 1) \wedge (\exists x:B.g(x) = 1)) \to (\neg (\exists x:A.f(x) = 1) \vee \neg (\exists x:B.g(x) = 1))\right) \simeq \left(\prod_{A:\mathrm{Prop}} \prod_{B:\mathrm{Prop}} \neg (A \wedge B) \to (\neg A) \vee (\neg B)\right)

• That Markov's principle holds for all mere propositions is equivalent to the double negation law:

(∏ A:Prop∏ f:A→2¬¬(∃x:A.f(x)=1)→∃x:A.f(x)=1)≃(∏ A:Prop¬¬A→A)\left(\prod_{A:\mathrm{Prop}} \prod_{f:A \to 2} \neg \neg (\exists x:A.f(x) = 1) \to \exists x:A.f(x) = 1\right) \simeq \left(\prod_{A:\mathrm{Prop}} \neg \neg A \to A\right)

Categorical semantics

We discuss the categorical semantics of h-propositions.

Let 𝒞\mathcal{C} be a locally cartesian closed category with sufficient structure to interpret all the above type theory. This means that CC has a weak factorization system with stable path objects, and that trivial cofibrations are preserved by pullback along fibrations between fibrant objects. (We ignore questions of coherence, which are not important for this discussion.)

Then for a fibrant object AA, the object isProp(A)isProp(A) defined above (with the first definition) is obtained by taking dependent product of the path-object A I→A×AA^I \to A\times A along the projection A×A→1A\times A\to 1. By adjunction, a global element of isProp(A)isProp(A) is therefore equivalent to a section of the path-object, which exactly means a (right) homotopy between the two projections A×A⇉AA\times A\;\rightrightarrows\; A. The existence of such a homotopy, in turn, is equivalent to saying that any two maps X⇉AX\;\rightrightarrows\; A are homotopic.

More generally, we may apply this locally. Suppose that A→BA\to B is a fibration, which we can regard as a dependent type

x:B⊢A(x):Type.x\colon B \vdash A(x)\colon Type.

Then we have a dependent type

x:B⊢isProp(A(x)):Typex\colon B \vdash isProp(A(x))\colon Type

represented by a fibration isProp(A)→BisProp(A)\to B. By applying the above argument in the slice category 𝒞/B\mathcal{C}/B, we see that isProp(A)→BisProp(A)\to B has a section exactly when any two maps in 𝒞/B\mathcal{C}/B into A→BA\to B are homotopic over BB. We can also construct the type

∏ x:BisProp(A(x))\prod_{x\colon B} isProp(A(x))

in global context, which has a global element precisely when isProp(A)→BisProp(A)\to B has a section.

• subsingleton, subterminal object

• propositional truncation

• isEquiv

• isContr

• isProp

• center of contraction

• propositional logic as a dependent type theory

• coinduction

• strict proposition

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level n+2n+2	nn-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-nn-groupoid
h-level ∞\infty	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-∞\infty-groupoid

References

• section 3.3 of Homotopy Type Theory -- Univalent Foundations of Mathematics

• HoTT repository