Re: [HoTT] Question about the formal rules of cohesive homotopy type theory

[Michael Shulman <shulman@sandiego.edu>]

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As far as the mathematical study of type theories and their models goes,
that may be true.  But I believe that when talking about the way type
theories are used in practice, either on paper or in a proof assistant,
there is still a difference.

Suppose I am teaching a calculus class, and I define f(x) = x^2 + 1 and I
want to evaluate f(3).  I don't write

f(3) = (x^2+1)[3/x] = (x^2)[3/x] + 1[3/x] = 3^2 + 1 = 9 + 1 = 10.

Instead, I jump right to f(3) = 3^2+1, because substitution is an operation
that happens immediately in my head, not a computational step analogous to
3^2 = 9.  Similarly, the user of a proof assistant never types or sees
substitution as part of the syntax; it is an operation *on* syntax that
happens behind the scenes.

Yes, this is a property of a particular *presentation* of a free structure
rather than a property of the structure itself, but that doesn't
intrinsically make it unimportant.  Groups and group presentations are both
important.  Maybe you want to say that "a type theory" refers to the free
structure rather than its presentation, but choosing to use words in that
way doesn't by itself make "presentations of type theories" (or whatever
you call them) less important.  Maybe you want to define an "admissible
rule" to be a property that holds in the syntax but fails in some actual
models; but that doesn't make "rules that hold in all models and can be
made to hold in a particular presentation of the free model without being
given explicitly as generating operations/equalities" (or whatever you call
them) less important.

I do agree that Andrej's formulation sounds backwards.  In practice (in my
experience) one doesn't write the rules down first and then try to figure
out what kind of substitution is admissible.  Instead one decides what the
substitution rule should be, and *then* (hopefully) works out a way of
presenting the syntax to make that substitution rule admissible.  This is
particularly tricky for modal type theories, where the categorically
"obvious" rules do not admit substitution, and leads to the introduction of
split contexts as used in the BFP paper.  I have trouble imagining how I
could have written that paper if I had been forced to write explicit
substitutions everywhere.  Thorsten and Jon, do you maintain that all the
work that's gone into figuring out ways to present modal type theories with
"admissible substitution" is meaningless?

On Thu, Nov 17, 2022 at 5:37 AM Jon Sterling <jon@jonmsterling.com> wrote:

> 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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