Does English have frequently used ordinary words that distinguish between equality and equivalence?
For example:
It was the same man on the photo.
Equality. The two persons are identical.
She ordered the same dish as her cousin.
Equivalence. The two dishes are the same kind of food, but they are two objects.
Best Answer
Interesting question.
There really aren't any common words that express the difference between equality and equivalence. Wearing the same hat, eating the same food, driving the same car — all of these things point to equivalence rather than equality or identity.
What the hat, food and car represent here are instances of classes, but not the same instances. To express that one instance of car is identical to another instance — for example, that you and I were driving the same Ford Fusion, California License Plate No. FOOBAR1 (sorry if that is a real plate number), on the same day, I in the morning and you in the afternoon — we would have to go out of our way to express that by actually citing the plate number or explaining that I loaned you my car or you loaned me yours.
Even to say we were driving the identical car would not cause the listener, at first, to suspect we meant the exact same car with the same plate number (and serial number). Identical here would be understood only to mean we were driving the same make, model, year, and color vehicle. Even saying "the exact same" car would still be understood to mean a car exactly like the other car, not the car itself.
Look at NOAD's list of synonyms for identical:
Not one of those synonyms expresses anything like the Law of Identity (A = A) in mathematics or the strict equality operator in some programming languages (=== instead of ==), even though the root of the word identical is, in fact, the same as for identity: Latin idem meaning "the same".
Even when we speak of things that point to identity, such as fingerprints or DNA, saying that a sample of DNA is identical to the DNA found at a crime scene does not mean the strands are the same strands, but that they come from the same person.