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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 20th 2019
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   Author: David_Corfield
   Format: MarkdownItexAndrej Bauer was going to speak about 'General type theories' at the [HoTTEST seminar](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html) but has delayed. One meta-theorem to be proved is > 1. Presupposition theorem: if a judgement is derivable then so are its presuppositions. Does anyone know how 'presupposition' is defined here?

   Andrej Bauer was going to speak about ’General type theories’ at the HoTTEST seminar but has delayed. One meta-theorem to be proved is

   1. Presupposition theorem: if a judgement is derivable then so are its presuppositions.

   Does anyone know how ’presupposition’ is defined here?

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 20th 2019
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   Author: David_Corfield
   Format: MarkdownItexSo I see there was considerable discussion at [Initiality Project: plans](https://nforum.ncatlab.org/discussion/9183/initiality-project-type-theory/) with some variation of point of view. Is there anything to draw from this which has a bearing on the discussion of presuppositions in informal settings, as we [discussed once](https://golem.ph.utexas.edu/category/2013/02/have_you_left_off_beating_your.html). I like the idea of making sense of a proposition by constructing a context in which it may be formed.

   So I see there was considerable discussion at Initiality Project: plans with some variation of point of view. Is there anything to draw from this which has a bearing on the discussion of presuppositions in informal settings, as we discussed once. I like the idea of making sense of a proposition by constructing a context in which it may be formed.

3. • CommentRowNumber3.
   • CommentAuthoratmacen
   • CommentTimeJun 20th 2019
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   Author: atmacen
   Format: MarkdownItexAt the initiality project, I preferred the terms "validity in" and "validity out" for two ways of reducing presuppositions to ordinary logic. These both come in syntactic and semantic versions. Having a "presupposition theorem" indicates that Bauer's framework is using (syntactic) validity out. From your nCafe post, Russell's reading: > There is something which is presently king of France, anything else which is presently king of France is the same as that thing, and it is bald. is an instance of semantic validity out. For "The present king of France is bald." to be valid, the presupposition "There is something which is presently king of France, [and] anything else which is presently king of France is the same as that thing." (or just "There's a unique present king of France.") has to be true. So Russell turned the statement with the presupposition to a statement that instead implies the presupposition, and doesn't itself have any presupposition. The semantic validity in reduction would be "If there's a unique present king of France, it's bald.". (I guess "validity in/out" are pretty bad terms for general presuppositions. At initiality project, the presuppositions were things like context validity and type validity.) Jon Sterling had a blog, which disappeared, but it had a post about presuppositions. He cited something about "invincible quantifiers" or something, which supposedly has to do with the semantic validity in and out reductions. A lot of his blog is on the Wayback machine, but apparently not that post. Maybe he'll show up and remind me. I bring it up because it sounded like a categorical explanation of the reductions.

   At the initiality project, I preferred the terms “validity in” and “validity out” for two ways of reducing presuppositions to ordinary logic. These both come in syntactic and semantic versions. Having a “presupposition theorem” indicates that Bauer’s framework is using (syntactic) validity out.

   From your nCafe post, Russell’s reading:

   There is something which is presently king of France, anything else which is presently king of France is the same as that thing, and it is bald.

   is an instance of semantic validity out.

   For “The present king of France is bald.” to be valid, the presupposition “There is something which is presently king of France, [and] anything else which is presently king of France is the same as that thing.” (or just “There’s a unique present king of France.”) has to be true. So Russell turned the statement with the presupposition to a statement that instead implies the presupposition, and doesn’t itself have any presupposition.

   The semantic validity in reduction would be “If there’s a unique present king of France, it’s bald.”. (I guess “validity in/out” are pretty bad terms for general presuppositions. At initiality project, the presuppositions were things like context validity and type validity.)

   Jon Sterling had a blog, which disappeared, but it had a post about presuppositions. He cited something about “invincible quantifiers” or something, which supposedly has to do with the semantic validity in and out reductions. A lot of his blog is on the Wayback machine, but apparently not that post. Maybe he’ll show up and remind me. I bring it up because it sounded like a categorical explanation of the reductions.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 21st 2019
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   Author: David_Corfield
   Format: MarkdownItexThanks for this. It sounds rather like: Someone makes a claim $P$. We look for a context in which it makes sense, say $\Gamma \vdash P true$. Then the Russellian 'validity out' approach turns this into a large dependent sum made from $\Gamma$ and $P$, while the 'validity in' approach forms a dependent product out of the dependent sum that is $\Gamma$ and $P$.

   Thanks for this. It sounds rather like:

   Someone makes a claim $P$. We look for a context in which it makes sense, say $\Gamma \vdash P true$. Then the Russellian ’validity out’ approach turns this into a large dependent sum made from $\Gamma$ and $P$, while the ’validity in’ approach forms a dependent product out of the dependent sum that is $\Gamma$ and $P$.

5. • CommentRowNumber5.
   • CommentAuthoratmacen
   • CommentTimeJun 21st 2019
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   Author: atmacen
   Format: MarkdownItexYes, that sounds about right. But the technical point of the reductions is to avoid some extra typing-like machinery in the first place. BTW when dealing with type theory itself, the presuppositions have to do with judgments, not types. So to analyze the use of judgment presuppositions in terms of a typing context, this would be a context in a metalanguage or [[logical framework]], using a judgments-as-types representation. So for example the judgment that some object-level context is valid, is a type in the metalanguage, which can be put in a metalanguage context. (This would be *not* using HOAS. If you use HOAS, you don't deal with object language contexts.)

   Yes, that sounds about right. But the technical point of the reductions is to avoid some extra typing-like machinery in the first place.

   BTW when dealing with type theory itself, the presuppositions have to do with judgments, not types. So to analyze the use of judgment presuppositions in terms of a typing context, this would be a context in a metalanguage or logical framework, using a judgments-as-types representation. So for example the judgment that some object-level context is valid, is a type in the metalanguage, which can be put in a metalanguage context. (This would be not using HOAS. If you use HOAS, you don’t deal with object language contexts.)