OK. So it sounds like definitional equality is another way of thinking about equality of sense, and is generally not the same as strict equality. That's a relief.

But the use of judgmental equality for capturing a system of paths that's fully coherent is actually about strict equality, in general; not necessarily judgmental or definitional equality.

So to bring things back to HoTT, what are people's opinions about the best use of these three equalities?

My opinion is that strict equality should be implemented as judgmental equality, which should be richer than definitional equality, by using a 2-level system with reflective equality. This is the case in HTS and computational higher dimensional type theory. We would still probably want to consider different theories of strict equality, but there would be no obligation to implement them as equality algorithms. Definitional equality would be a quite separate issue, pertaining to proof automation.

On Tuesday, February 12, 2019 at 12:54:24 PM UTC-5, Michael Shulman wrote:

For sure definitional equality doesn't have to do with models.  Like
all the kinds of equality we are discussing, it is a syntactic notion.
Actually I would say it is a *philosophical* notion, and as such is
imprecisely specified; syntactic notions like judgmental equality can
do a better or worse job of capturing it in different theories (and in
some cases may not even be intended to capture it at all).

> what's the difference between "denoting by definition" and regular denoting?

x+(y+z) and (x+y)+z denote the same natural number for any natural
numbers x,y,z, because we can prove that they are equal.  But they
don't denote the same natural number *by definition*, because this
proof is not just unfolding the meanings of definitions; it involves
at least a little thought and a pair of inductions.

For a more radical example, "1+1=2" and "there do not exist positive
integers x,y,z,n with n>2 and x^n+y^n=z^n" denote the same
proposition, namely "true".  But that's certainly not the case by
definition!  Same reference; different senses.