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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeDec 1st 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIs there any point to having both [[type of propositions]] and [[Prop]]? The analogous page-name [[Type]] is a redirect to [[type of types]].

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 1st 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexNot sure why this happened. I have merged the two entries now.

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

3. • CommentRowNumber3.
   • CommentAuthornLab edit announcer
   • CommentTimeOct 6th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadded a section on the definition of types of propositions Anonymous <a href="https://ncatlab.org/nlab/revision/diff/type+of+propositions/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/type+of+propositions/9">v9</a>, <a href="https://ncatlab.org/nlab/show/type+of+propositions">current</a>

   added a section on the definition of types of propositions

   Anonymous

   diff, v9, current

4. • CommentRowNumber4.
   • CommentAuthornLab edit announcer
   • CommentTimeOct 17th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadded rules for the type of all propositions Anonymous <a href="https://ncatlab.org/nlab/revision/diff/type+of+propositions/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/type+of+propositions/13">v13</a>, <a href="https://ncatlab.org/nlab/show/type+of+propositions">current</a>

   added rules for the type of all propositions

   Anonymous

   diff, v13, current

5. • CommentRowNumber5.
   • CommentAuthorGuest
   • CommentTimeOct 17th 2022
   • PermaLink
   Author: Guest
   Format: MarkdownItexShould 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

   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

6. • CommentRowNumber6.
   • CommentAuthorChristian Sattler
   • CommentTimeOct 17th 2022
   • (edited Oct 17th 2022)
   • PermaLink
   Author: Christian Sattler
   Format: MarkdownItexThat'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.

   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.

7. • CommentRowNumber7.
   • CommentAuthornLab edit announcer
   • CommentTimeDec 1st 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadded section on the type of all decidable propositions, or the type of booleans Anonymous <a href="https://ncatlab.org/nlab/revision/diff/type+of+propositions/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/type+of+propositions/17">v17</a>, <a href="https://ncatlab.org/nlab/show/type+of+propositions">current</a>

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

   Anonymous

   diff, v17, current