Context
Universes
Type theory
Contents
Idea
In type theory a type of propositions — typically denoted or — 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 may typically be identified with the function types into the type of propositions
In dependent type theory such a is equivalently an -dependent proposition, to be understood as assingning to any term the assertion that “ is contained in the given subtype”.
Definition
The type of all propositions
The type of all propositions 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 are only indices of a (-1)-truncated type family ; every mere proposition in the type theory is only essentially -small for weak Tarski universes or judgmentally equal to an for for strict Tarski universes.
As a Russell universe
As a Russell universe, the type of all propositions is given by the following rules:
Formation rules:
Type reflection:
Propositional truncation for each type reflection
Introduction rule:
Univalence:
As a Tarski universe
As a Tarski universe, the type of all propositions is given by the following rules:
Formation rules:
Type reflection:
Propositional truncation for each type reflection
Introduction rule:
Essential smallness of propositions (for weak types of all propositions) or judgmental equality (for strict types of all propositions):
Univalence:
The empty proposition and falsehood
To ensure that the type of all propositions isn’t a contractible type, one could include the following axiom, which states that there exists a proposition satisfying ex falso quodlibet:
The type of all decidable propositions
Something similar holds for the type of all decidable propositions in a dependent type theory, which could be presented either as a Russell universe or a Tarski universe. The difference between the two is that in the former, every decidable proposition in the type theory is literally an element of the type of all decidable propositions, while in the latter, elements of are only indices of a (-1)-truncated type family ; every decidable proposition in the type theory is only essentially -small for weak Tarski universes or judgmentally equal to an for for strict Tarski universes.
As a Russell universe
As a Russell universe, the type of all decidable propositions is given by the following rules:
Formation rules:
Type reflection:
Propositional truncation for each type reflection
Excluded middle for each type reflection
Introduction rules:
Univalence:
As a Tarski universe
As a Tarski universe, the type of all decidable propositions is given by the following rules:
Formation rules:
Type reflection:
Propositional truncation for each type reflection
Excluded middle for each type reflection
Introduction rules:
Essential smallness of propositions (for weak types of all decidable propositions) or judgmental equality (for strict types of all decidable propositions):
Univalence:
The type of booleans
The type of booleans or booleans type is given by the following rules:
Formation rules:
Type reflection:
Introduction rules:
Univalence:
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 were an h-proposition, that means that the identity type is pointed, and by transport across , 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 for types . A type of propositions is a type with a type family such that
These axioms imply that satisfy propositional extensionality and that is an h-set and a Heyting algebra.
Examples
The type of all internal propositions
in a Tarski universe is a type of propositions. If the Tarski universe has all propositions, then
is the type of all propositions.
References
Detailed discussion of the type of propositions in Coq is in