Logic

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. Unfortunately, I do not understand where I am going wrong. If it is easy to see why I am confused, could you help clarify this for me?

Comments

It may be that two distinct properties (in your example, the property of being a spinach-eater and the property of being strong, if your mother were right--which, I fear, she is not!) have the same extensions--that is, may apply to all and only the same things in the world. In this case, the set of all spinach-eaters would be identical to the set of all strong things, since as you say, the identity of sets is determined wholly by membership. But that does not mean that the property of being a spinach-eater is the same as the property of being strong. One reason for thinking that it is not is the two properties would appear to have different causal or explanatory relations--one becomes strong by eating spinach (eating spinach is what explains becoming strong) , but one does not become a spinach-eater by (first) becoming strong (becoming strong is not what explains becoming a spinach-eater).

Another famous example from philosophy: assuming there is an omniscient and omnibenevolent God, then it will be true that the set of all good things and the set of all God-approved things will be identical, because such a God would approve of all and only good things. But that does not prove (despite some theists' claims to the contrary) that the property of being good just is (that is, is identical to) the property of being God-approved. One reason for thinking that it is not is that the two properties appear to have different causal or explanatory relations: God approves of things because they are good, but they are not good because God approves of them.

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 United States, but the property of living in Washington is not the same as the property of living in the capital of the United States.

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