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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


    diff, v9, current

  2. added rules for the type of all propositions


    diff, v13, current

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeOct 17th 2022

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

    ΓctxΓΩtype\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \Omega \; \mathrm{type}} ΓctxΓtrue:Ω\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{true}:\Omega} ΓAtypeΓBtypeΓi:ABΓχ B:BΩ\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 Ω\Omega. To derive your proposed rule, consider the proposition in context Γ.(b:B)\Gamma.(b : B) of preimages of bb under ii.

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

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


    diff, v17, current