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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeDec 1st 2016

Is there any point to having both type of propositions and Prop? The analogous page-name Type is a redirect to type of types.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 1st 2016

Not sure why this happened. I have merged the two entries now.

1. added a section on the definition of types of propositions

Anonymous

2. added rules for the type of all propositions

Anonymous

• CommentRowNumber5.
• CommentAuthorGuest
• CommentTimeOct 17th 2022

Should be possible to define the type of propositions via the universal property of the subobject classifier, something like

$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \Omega \; \mathrm{type}}$ $\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{true}:\Omega}$ $\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash i:A \hookrightarrow B}{\Gamma \vdash \chi_B:B \to \Omega}$

et cetera

• CommentRowNumber6.
• CommentAuthorChristian Sattler
• CommentTimeOct 17th 2022
• (edited Oct 17th 2022)

That’s effectively being done already when you say every proposition is classified by $\Omega$. To derive your proposed rule, consider the proposition in context $\Gamma.(b : B)$ of preimages of $b$ under $i$.

Side note: we don’t currently have models of the type of all propositions (as currently defined, with a meta-equality $\mathsf{El}(A_\Omega) \equiv A$) in HoTT.

3. added section on the type of all decidable propositions, or the type of booleans

Anonymous