Indeed, I echo Thorsten's comment — to put it another way, even being able to tell whether these rules are derivable or only admissible is like knowing what an angel's favorite TV show is (in other words, a form of knowledge that cannot be applied toward anything by human beings). At least for structural type theory, there is nothing worth saying that cannot be phrased in a way that does not depend on whether structural rules are admissible or derivable. It may be that admissiblity of structural rules starts to play a role in substructural type theory, however, but this is not my area of expertise.
It is revealing that nobody has proposed a notion of **model** of type theory in which the admissible structural rules do not hold; this would be the necessary form taken by any evidence for the thesis that it is important for structural rules to not be derivable. Absent such a notion of model and evidence that it is at all compelling/useful, we would have to conclude that worrying about admissibility vs. derivability of structural rules in the official presentation of type theory is fundementally misguided.
On 16 Nov 2022, at 4:52, 'Thorsten Altenkirch' via Homotopy Type Theory wrote:
That depends on what presentation of Type Theory you are using. Your remarks apply to the extrinsic approach from the last millennium. More recent presentation of Type Theory built in substitution and weakening and use an intrinsic approach which avoids talking about preterms you don’t really care about.
https://dl.acm.org/doi/10.1145/2837614.2837638
Cheers,
Thorsten
From: homotopytypetheory@googlegroups.com <homotopytypetheory@googlegroups.com> on behalf of andrej.bauer@andrej.com <andrej.bauer@andrej.com>
Date: Tuesday, 15 November 2022 at 22:39
To: Homotopy Type Theory <homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] Question about the formal rules of cohesive homotopy type theory> Does this also include the structural rules of type theory such as the substitution and weakening rules?
I would just like to point out that substutition and weakening typically are not part of the rules. They are shown to be admissible. In this spirit, the question should have been: what is the precise version of substitution and weakening (which is a special case of substitution) that is admissible in cohesive type theory?
With kind regards,
Andrej
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