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Suppose some condition A is identical to some condition B; to be concise, let's write A=B. It seems obvious, then, that A is necessary and sufficient for B; or more compactly, A B.
On the other hand, that implication's converse (i.e. that A B implies A=B) seems like it isn't right, because we can easily come up with counter-examples. Take my mother, for example; she is always saying, "eating spinach everyday is a necessary and sufficient condition for becoming strong." In other words, she claims that you will become strong if, and only if, you eat spinach everyday. Surely it does not follow that becoming strong is identical to eating spinach...right?
Now I am tempted to consider the question in the context of sets. Suppose you want to prove that two sets S and T are equal. Then it is sufficient to prove that membership in one follows from membership in the other, and vice versa. I.e. x is an element of S x is an element of T. So it appears that the "=" relation follows from " " relation.
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A nice question. Yes, if the predicate "F" and the predicate "G" are co-extensive (i.e., are true of exactly the same things), it would be wrong to conclude that the property corresponding to "F" is the same as the property corresponding to "G". (Nick gives some good examples of this in his response.) You seem to think that the set example conflicts with this observation, but it doesn't. If we establish that x is an element of S if and only if x is an element of T, we can indeed infer that S equals T. But that's different from inferring that the property of being an element of S is the same as the property of being an element of T. And it's that inference that would conflict with our observation. Perhaps you think that this last claim can nevertheless be inferred because you think that if S is identical to T, then the property of being an element of S is identical to the property of being an element of T. But that isn't right. Washington, D.C. is identical to the capital of the...

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