Sentence

A sentence is the result of a hypothesis considered irrespective of the particular hypothesis. Analogously, a presentation is the result of an operator considered irrespective of the particular operator. We will compare these two notions on the way to testing just what we mean by result; provisionally, let us take it as what "follows" a hypothesis or operator.

Sentences are not propositions, following the use-mention distinction. Propositions, i.e. appeals, describe the content or meaning of the sentence.

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So every particular sentence is, in fact, the type of some collection or class. Given some additional assumptions, we can construct an equivalence relation on these types (for example, when we pass from preorders to partial orders by taking equivalence classes). This strategy of associating sentences with the types of their proofs is developed in homotopy type theory.

In the provisional formalism, a sentence is written as $$R^n$$.

Sentences vs. presentations
As a general rule, we take hypotheses (what says something) as continuous with operators (what does something). This is not the case when we compare sentences and presentations. Strictly speaking, we will always take a sentence as the abstraction of a presentation.

Homotopy type theory
See the article on univalence.

In intuitionistic type theory, every proposition is associated with the type of its proofs, and propositional equality is equated with homotopy equivalence.

As a class
What is the type of a sentence, i.e. what is the type of a type?