I would object to the characterization of definitional identity as identity of sense. Sense is often glossed as mode of givenness; or one might say that the sense of an expression determines a route to its reference. Now take terms such as 7+5 and 9+3. These are definitionally identical, but they present the number twelve in different ways; qua senses each determines a different route to ssssssssssss(0). Hence they are not the same senses. Identity of sense is, to my mind, a stricter relation than definitional identity.

Very often, also in this thread, the notions of judgemental identity and definitional identity seem to be treated as one and the same thing. But by judgemental identity we should mean a certain form of identity assertion that is different from the assertion that an identity proposition/type is true/inhabited; and by definitional identity we should mean one among several relations on terms that may properly be called a relation of identity.

Here is an argument for the need of a form of identity assertion other than p : Id(A,a,b). The proposition Id(A,a,b) is formed by applying the function Id to the type of individuals (set in Martin-Löf's terminology) A, and the elements a,b : A. The notion of a function, however, presupposes a notion of identity: f is a function only if, when applied to identical arguments a, b, we get identical results f(a), f(b). On pain of circularity, the notion of identity appealed to here cannot be propositional identity.

A notion of identity is also implicit in the notion of domain presupposed by the notion of a function: f is a function from the domain A to the domain B. To have the right to speak of A as a domain we need to know what it is for elements of A to be identical, we need to know a criterion of identity for A.

In type theory the alternative form of identity assertion, used for instance in the explanation of what a function is, is the identity judgement a = b : A. There is nothing in the argument just given that forces the relation picked out by this form of identity assertion to be definitional identity; but there may be other arguments why such an interpretation is preferable.

On Wed, Feb 13, 2019 at 7:37 AM Matt Oliveri <atmacen@gmail.com> wrote:

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.

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