This question is partially related to How does "is" work? and What is the difference between the "is" of predication and the "is" of identity?, but more specifically it concerns statements about physical quantities and measurements.
[Question heavily re-edited because unclear]
Consider the statement "the instantaneous velocity of body B is 1 m/s". This is usually expressed in a mathematical form like "v = 1 m/s" or similar.
The "=" in that form seems to suggest that this statement expresses a relation. But is this really the case? or does that statement express a predicate?
The question is also related to the fact that we can consider the statements
"the instantaneous velocity of body B is 1 m/s"
and
"the mean velocity of body A is 1 m/s"
and deduce
"the instantaneous velocity of body B is the same as the mean velocity of body A
All three statements – which are perfectly fine – can be expressed in deceivingly similar mathematical forms: "v = 1 m/s", "u = 1 m/s", and "v = u".
The last expression is particularly nasty, because "=" or "is the same as" really means "has the same value as", and not "is the same (thing) as". The latter would obviously be false, because mean and instantaneous velocity are different notions, and moreover they refer to different bodies.
By the same token, "the instantaneous velocity of body B is 1 m/s" seems to really mean "the instantaneous velocity of body B has the value 1 m/s".
Clearly(?) the statement "v = u" expresses a relation ("has the same value as"), and the relation is symmetric, reflexive, and transitive; an equivalence relation.
But does "v = 1 m/s" also express a relation? It seems to me that it doesn't. The chain of reasoning {"v = 1 m/s", "u = 1 m/s", "v = u"} seems to use transitivity and symmetry, and suggests that "=" is the same along the chain, and it's therefore a relation.
And yet I think that "=" denotes different things in the chain. It's mixing apples and oranges. And yet, graphically at least, it seems to work.
My questions are:
Does the statement "v = 1 m/s" (or "the instantaneous velocity of body B has the value 1 m/s") express a predicate? Or a relation? Or something else? Or is the intepretation arbitrary? (I would be inclined to see it as a predicate, but in a second-order logic with an infinity of such predicates, over which we can use quantifiers.)
How does an inference chain such as {"v = 1 m/s", "u = 1 m/s", "v = u"} work, if the "=" has different meanings in it? (eg if it expresses a predicate in the first two statements and a relation in the last.)
Can you suggest any literature on this issue, in relation to measurement statements in science?
I can add that a similar question arises in probability theory, which typically has statements about "random variables" of the form "X = x". This would mean "the quantity X has value x". But such an interpretation has lead to a great debate about causality and correlation. The problem is that more precisely "X = x" can mean two things: "X has been observed to have the value x" and "X has been set to the value x". These two very different statements have hugely different consequences, for example if they appear in the conditional of the probability of another statement.
[Note: I'm a physicist and this question is very important for my field, so I'd like to keep some science- or physics-related tag for this question.]