To lay it on the table, I will approach this from a pragmatist's perspective (see Dewey, James, Quine, etc.).
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1) If x is identical to y it does not seem, intuitively, that there would be a possible situation in which we had x but not y or had y but not x.
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But, in terms of possible worlds, this strict correspondence may only be maintained for "simple" concepts or objects, ie those that may not be broken down into others. 2 lines of argument follow:
1. Let's take the example of water. We can imagine a possible world where a wet, largely inert, and opulent polar solvent serves as the basis of life on an alternate Earth, but it is NOT h2o. We may alternately imagine a world where h20 is a sandy solid at room temperature.
This is a restatement of Atlas's argument.
Water is not a simple object. Are the brain and mind?
2. I don't think that concepts are inert "things" that exist outside of the subject-environment interaction. Social context, goals, novel interactions and events, etc. anchor and re-anchor what concepts mean vis-a-vis both the environment and other concepts. Concepts are their use. Thus, there can be no simple concepts of the sort you need.
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For the sake of example and because I don't know your real name lets just pretend it is 'Charlie'. Can you come up with a situation in which you, ebola, were not also Charlie (pretending this is your name)?>>
But what if we interrogate the assumption that I am a unitary entity? Perhaps the ebola that posts is a different self from the Andy that teaches social theory which is a different self from the Charlie that I'm imagining for your example

Sure, they all are capable of doing similar searches of my memory, but they're not identical, nor are the proclivities of my selves. I am speaking quite literally.
Ironically, all these minds share a single body with the same brain.
What if the flux is even more ubiquitous? What if I am born anew to each moment, but born into a set of memories that creates the illusion of continuity of self?
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Because you are identical to yourself, and you are called both 'Charlie' and 'ebola', there is no possible situation in which we have Charlie but not ebola or ebola but not Charlie. You're one and the same person! Therefore, if ebola is identical to Charlie there is no possible situation in which they are not identical. To say that there is no possible situation in which something is not identical to itself is the logical equivalent of saying that the identity is necessary.>>
Not "necessarily".

Your example works well for specific people (in terms of proper names) who conform to your assumptions of personhood. What about specific objects? What about conceptual categories?
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If you apply this understanding to the mind/body problem you should be able to understand why the mind is not identical to the brain.>>
Oh, I'm with you on this. I just don't think that this fact stems from a condition where all identities are necessary. It's kind of like nuking NY to eliminate a single trash collector.

I do think that some identities are empirical, but in the pragmatist sense, not the way a logical positivist (like yourself?) would use the term.
>>2+2 is not 4>>
This is an interesting one. While I think that the axioms that allow for the derivation of this statement (and all arithmetic) are "necessary" if we are to get a coherent system of logic off the ground, I don't think that they are necessary in any transcendental way. If we shed our logical assumptions, we open the door to experience (usually framed as mystical) that cannot be captured by logical description.
Thanks for making me think!
ebola
np: mindless self-indulgence (as is seen in this post)