Contents
Context
Universes
Type theory
Contents
Idea
In type theory a type of propositions or 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:
Type reflection:
Introduction rules:
Propositional truncation for each type reflection
Univalence:
The type of all decidable propositions
The type of all decidable propositions is given by the following rules:
Formation rules:
Type reflection:
Introduction rules:
Propositional truncation for each type reflection
Excluded middle for each type reflection
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