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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeOct 18th 2009
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   Author: TobyBartels
   Format: MarkdownI wrote [[two-valued topos]] to help me tighten up Mike's latest edit to [[cocomplete well-pointed topos]].

   I wrote two-valued topos to help me tighten up Mike's latest edit to cocomplete well-pointed topos.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeOct 18th 2009
   • (edited Oct 18th 2009)
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   Author: TobyBartels
   Format: MarkdownShould we clarify that a consistent logic is proof-theoretically two-valued if the only closed terms for propositions are true and false? Then even intuitionistic propositional logic (but not intuitionistic higher-order logic) is two-valued. And then a model is two-valued if the only propositions in it are true and false, so intuitionistic propositional logic has no complete two-valued models. (Of course, it has an incomplete two-valued model, the two-valued model of classical logic, and classical logic has many models that are not two-valued.) Then we can extend the internal/external distinction between boolean and two-valued that is clear for toposes. Is there anything standard in the literature about this?

   Should we clarify that a consistent logic is proof-theoretically two-valued if the only closed terms for propositions are true and false? Then even intuitionistic propositional logic (but not intuitionistic higher-order logic) is two-valued. And then a model is two-valued if the only propositions in it are true and false, so intuitionistic propositional logic has no complete two-valued models. (Of course, it has an incomplete two-valued model, the two-valued model of classical logic, and classical logic has many models that are not two-valued.) Then we can extend the internal/external distinction between boolean and two-valued that is clear for toposes. Is there anything standard in the literature about this?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeOct 19th 2009
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   Author: Mike Shulman
   Format: Markdown>Is there anything standard in the literature about this? I don't know; I think you know more about proof theory and non-topos-theoretic intuitionistic logic than I do. But it seems like a reasonable thing to mention.

   Is there anything standard in the literature about this?

   I don't know; I think you know more about proof theory and non-topos-theoretic intuitionistic logic than I do. But it seems like a reasonable thing to mention.