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.
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:
identical adjective 1 wearing identical badges: indistinguishable, (exactly) the same, uniform, twin, duplicate, interchangeable, synonymous, undifferentiated, equivalent, homogeneous, of a piece, cut from the same cloth; alike, like, matching, like (two) peas in a pod; similar.
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.
I don't think English has any "frequently-used ordinary words" to make this distinction.
How often does it come up in ordinary speech that this distinction is important? My guess is rarely if ever. In certain types of technical discussion this distinction is important, and that's why there are technical terms to distinguish equivalence and equality.
However, distinctions that are not relevant in ordinary conversations don't generally have frequently-used ordinary words to distinguish them. On the occasions that this distinction does come up and is not obvious from context (such as the situations invented in some of the answers and comments here), the distinction is made either using technical terms or using description.
I can't think of any words that would be a direct replacement for “same” in your examples, but it's possible to express the distinction.
Alice and Bob share a car.
A single car between them. In Python syntax, Alice.car is Bob.car.
Alice and Bob have identical cars.
Two cars (note the plural form) that have the same year, make, model, and color. Alice.car == Bob.car.
How about the very same?
I'm wearing the same dress as Mary.
And
I'm wearing the very same dress I wore a year ago.
The very same will be the identical object.
very same dress as Mary. It would be a tight fit. How would our two necks fit trough the collar? And the waist would be way too tight to accommodate both of us. No, really, the very same dress is the identical object. I cannot wear the very same dress as another person, unless we snuggle up and spoon into it and it's made of stretch lycra.
– teylyn
Jun 08 '11 at 11:50
I am assuming following meanings for equality and equivalence
Neither of these definitions that I chose are complete and you can find other, different meaning for these words, especially in technical sense (logic, philosophy, mathematics, science). However, I choose these as I think these are close to what you meant and my answer is based on them.
Regarding these differences, let me mention for example, in mathematics equality is a relation that defines equivalence class (any relation that is reflexive, symmetric and transitive i.e. partitions a set so that every element of the set is a member of one and only one cell of the partition, is equivalence relation).
In logic the definition of equivalence is very different (see material equivalence and logical equivalence), and I think it does not apply here, though some usage of the word can imply logical equivalence.
Regarding the equality in the sense of philosophical identity or exact sameness, usually in English that is easily expressed. Your first example can be shortened further
It was the same man.
and it still expresses this1.
The definite article and adjective "same" applied to non-abstract words express the exact sameness2. If you substitute "man" with "car" or even seemingly indistinguishable "ant" you will express the exact sameness.
You can try to extends it to abstract concepts
It was the same proof.
However, abstract concepts are abstracted and the exact sameness holds for the concept, not for the instance. Instances can indeed be different and, at the same time, equivalent. But, the concept of the proof remains one (and only, hence exactly same to itself).
Now, for the equivalence - let me here rewrite your second example in the same structure
It was the same dish that her cousin had ordered.
what I propose is that here with the term "a dish that her cousin had ordered" the exact sameness remains. "A dish" here is not the food on the plate, but "food prepared and cooked in a particular way", 2nd meaning of the noun. In this case, context and frequency determine that we are not talking about the 1a meaning: "the food contained in a dish".
A more ambiguous example would be
It was the same car that her cousin had ordered.
In this case car can indeed refer to an instance of a car or a type of the car (model, color, etc) and in this case it is really not possible to distinguish what is meant.
In such cases what FumbleFingers and teylyn mention does help
It was the very same car that her cousin had ordered.
Also here there are other helping phrases: the exact same car, the very same car, literally the same car, actually the same car. All these try to emphasize that the meaning of "the same" is more true than without them. However, they also work in constructions where you want to emphasize that it is the same model (especially if such context had been implied, for example by talking about very rare, discontinued model).
Now, I believe you can be more specific by dropping "the same" and say:
It was the very car that her cousin had ordered.
where without the relation "the same" you can only interpret it as an actual material instance of an object (and avoid hyperbolic interpretations 3). Terms actual and exact can be interchanged here with the same meaning (also "It was literally the car...").
1 However, do note that the context remains, it is only unspecified now. We still compare one idea with another: the man is the same as the man we saw previously, or the man in photograph, or the man in your dream, etc.
2 Definitive article comes from demonstratives meaning that.
3 For example "literally" is so often used figuratively (as hyperbole) that the dictionary entires list the second meaning which is opposite to the original meaning! See here.
I can't think of one word to generically distinguish equivalence and equality, but here are a few suggestions to help clarify:
Example for : Equality. The two persons are identical.
If you mean the two persons are actually the one person:
The man in the photo was the same person.
If you meant that the man in the photo looked alike to someone:
It was a matching man on the photo.
Example For: " Equivalence. The two dishes are the same kind of food, but they are two objects."
If you meant that there were two dishes, but the same flavour:
She ordered the same type of dish as her cousin.
If you want to mean there was only one dish:
She ordered a dish with her cousin.
Well, there is simile and metaphor. You couldn't really call them common words, but they are common forms of speech and the difference is taught in every High School English class I know of. Generally, this boils down to using "is like a" vs. using "is a" in a sentence.
Yes, there are frequently used ordinary words that can be used to distinguish between equality and equivalence.
Unfortunately, a lot of words are used badly / improperly in common English usage.
To work from your example:
As you have already highlighted, "she ordered the same dish as her cousin" is not quite right. The choice is the same, so "she chose the same dish as her cousin" would be a better statement. Similarity is the concept to be conveyed, but it is difficult to use "similar" in exactly the same way - you would need to say something like "her cousin ordered a prawn cocktail, she ordered something similar" to avoid clumsy English usage.
There's an emerging trend of using "virtual / virtually" to signify closeness and equivalence, in circumstances where "practically" is what is actually meant.
Too often, people say that two items are the same when the intended meaning is that they are similar, practically the same, or effectively the same.
LOL the words are there, people often do not appreciate the precision available to them.
