The concept of identity (sameness) is the most fundamental (foundational, basic, primitive) concept.
Since confusion about fundamental concepts infects understanding that uses derivative concepts, it’s important to get clear about identity.
Is it even a concept, a principle of classification? [See the post ‘Define Understanding.’]
To be clear about terminology, let’s use ‘object’ to refer to anything we are able to pick or single out as the focus of attention, anything we are able to perceive, imagine, conceive, or remember. Let’s use ‘pure object’ to refer to an object considered in isolation from all other objects.
Do we classify objects based on the idea of (numerical) sameness?
Yes, in fact, we do. We do it whenever we think that, for example, we are speaking with the same person today who we talked with yesterday or we are seeing the same planet that we saw yesterday. While such judgments are commonplace, they are puzzling.
How is it possible to think about two objects a and b (such as the Morning Star and the Evening star) as if they were different when in fact, and we may even understand the fact, they are the same? Hume, Hegel, Frege, Russell, and Wittgenstein all wondered about this.
Suppose you claim that the Morning Star is the Evening Star. How many objects are you thinking about? If your judgment is true, it’s about one thing, yet, if it’s false, it’s about two things. How could what your judgment is about depend at all on its truth value?
The concept of sameness is the idea of two-in-one. If true, an identity judgment is about one thing, but it nevertheless appears to be about two things.
Your claim that the Morning Star is the Evening Star obviously appears to be about two objects. Someone who did not understand the astronomical fact that the planet Venus may be seen both in the morning and in the evening would naturally take your claim to be about two objects.
However, how can identicals appear distinct?
I agree with Panayot Butchvarov’s solution in Being Qua Being.
The Morning Star and the Evening Star are different pure objects. That’s a fact about our apprehension of the world that gets reflected in our language.
It’s also an ordinary astronomical fact that those two objects are literally the same thing, namely, the planet Venus. “They” are, in reality, indiscernible. Instead of using the word ‘thing’ in such contexts, let’s use with ‘entity.’
The critical function of the concept of identity is simplification.
From a logical point of view, the starting point of all our experiences is the domain of pure objects. Each pure object is itself and different from every other pure object. That domain is unintelligible since intelligibility requires recurrence (of, minimally, qualities).
That domain is not the world. The world is the entire set of inter-connected entities. An entity is the subject matter of a true identity judgment. So our conceptual apprehension of the world requires us to use the concept of sameness.
What are the criteria for using it correctly? Without criteria, there would be unlimited conceptual freedom. In fact, our use of the concept of identity is governed by paradigms, but there is nothing logically necessary about those paradigms.
In other words, it is logically possible for any pure object to be identical with any other pure object. In fact, it is logically possible that all identity judgments are true, which is a fascinating idea!