What is the difference between these, the "is" of predication and the "is" of identity?
For example, when I say, "my pet is a cat", am I using "is" as an identity or as a predicate?
In "Paris is the capital of France", "is" is used to mean identity.
In "my pet is a cat", "is" is used to mean predication : my pet belongs to the class of cats.
See Ludwig Wittgenstein's Tractatus :
3.323 In everyday language it very frequently happens that the same word has different modes of signification—and so belongs to different symbols—or that two words that have different modes of signification are employed in propositions in what is superficially the same way.
Thus the word ‘is’ figures as the copula, as a sign for identity, and as an expression for existence; ‘exist’ figures as an intransitive verb like ‘go’, and ‘identical’ as an adjective; we speak of something, but also of something’s happening.
The idea that there is a fundamental difference is known as the Frege-Russell "is" ambiguity thesis, Corazzon's webpage is a very good source on it. In addition to the is-es of predication and identity they distinguished the is of existence, and the is of subsumption. However, from Aristotle to Frege philosophers did not draw such distinctions. The need for them is transparently linked to the needs of formalizing logic into predicate calculus with quantifiers, where predication and identity are expressed by different means, P(x) vs x=y. There are even more options in set theory, where one can talk of inclusion of classes defined by predicates, and use "is" for that. There is no general fact of the matter as to what is/are "really" stand for, it depends on context and how one wishes to interpret the claim when formalizing. Here is from Hintikka's Meinong in a Long Perspective:
"This history involves more changes and contrasts than one might perhaps expect. For one thing, we twentieth-century philosophers are wont to approach the notion of being by means of the Frege-Russell ambiguity thesis. As we all know this thesis concerns the notion of being, as codified in verbs for being in languages like the English is, German ist or the ancient Greek estin. What it asserts is that these verbs are multiply ambiguous. We have to distinguish (according to this thesis) from each other the ises of existence, identity, predication and subsumption. Indeed we are in fact supposed to have learned to distinguish them from each other in practice, for we have all been taught to use first-order logic as our canonical notation in logic and logical analysis. It is in order for me to emphasize the word ambiguity here. Every half-way sensitive analyst (or perhaps I should say, every sensible analyst) will grant that verbs for being like the English is are used in different ways on different occasions. What the Frege-Russell thesis does, is to blame these differences in use on the ambiguity of a single word, instead of explaining the difference in use away in some other way, for instance by reference to the context of use."
Aristotle considered the distinction and rejected it. He followed Greek grammar in treating "is" (ἐστιν) always as a copula, i.e. always as means of predication. The other "meanings" were treated in terms of force, existential, etc., which may or may not be resent in specific uses of ἐστιν. Medieval scholastics followed Aristotle. In his syllogistic there is no distinction between predication, existence and identity because it does not have means (or need) to express such a distinction. "My cat is a pet" is in a typical form of syllogistic premise, but it makes no difference whether this attaches predicate to a subject (Aristotle's preference) or subsumes the pet under the concept of a cat (we have no identity here). The same with "all cats are animals" or "bachelors are unmarried man" (here we do have identity).
In fact, the extension/intension (class/predicate) ambiguity was common in 19th century logic and mathematics and persisted until extensionalization of set theory by Hausdorff. But the unraveling started earlier with Kant, who famously declared "'Being' is obviously not a real predicate; that is, it is not a concept of something which could be added to the concept of a thing". Frege and Russell developed this into the use of the existential quantifier, which expresses existence without a predicate. The move is now controversial, see What are the counterexamples to Kant's argument that existence is not a predicate?
'My pet is a cat' involves the 'is' of predication; you predicate - say of - your pet that it is a cat. Your pet has the attribute or property of being a cat. If you say 'The Morning Star is the Evening Star' (Frege's example) then this is the 'is' of identity : the Morning Star and the Evening Star are one and the same object.
Put the point like this. The property of being a cat is not identical with the property of being your pet since your pet might equally well be something different - an alligator instead. In contrast, the Morning Star can never be different from the Evening Star : they are identical, one and the same thing with (as it happens) different names.
The difference between the "is" of identity and the "is" of predication is the difference between being an object and being the property of an object.
As Graham Priest puts it (page 84)
We must distinguish between an object and its properties. When we say that you, with a different hair-style, are different, we are saying that you have different properties. It does not follow that you are literally a different person, in the way that I am a different person from you.
The reason this can be confusing is the English word "is" can be used to refer to both objects and properties of objects.
When I refer to the same object saying my cat is Kiki, where Kiki is the name of my cat, this is the "is" of identity.
When I am talking about a property of the object and claim that my pet is a cat, this is the "is" of predication. The property of being a cat is a property of the object, my pet.
Priest, G. (2010). Logic: A brief insight.
The distinction lies between "it is a thing", and "it is the thing".
Prediation indicates the entity belong to a collection of entities with shared properties; it is one of maybe many. "Donald is a Duck."
Identity indicates that an entity known by one name, and an entity known by another name are in fact the same unique entity. "Bruce Wayne is the Batman!"