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If some condition $P$ is necessary and sufficient for $Q$, why is it the case that $P$ if and only if $Q$?

I can understand if $P$ is sufficient for $Q$, then $P \implies Q$, but am not sure why $P$ being necessary for $Q$ implies $Q \implies P$?

It seems that if $P$ is necessary, it doesn't necessarily imply anything about $Q$ causing $P$. Is there a nice example to understand why?

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  • $\begingroup$ What does "cause" have to do with "$P$ if and only if $Q$"? Do you think that "$P$ if and only if $Q$" means that $P$ and $Q$ cause each other? $\endgroup$ – David K Jan 1 '17 at 0:05
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To expand on djechlin's answer: you are mixing up "implies" and "causes". By "$A$ implies $B$", all we mean is that if $A$ is true then $B$ is true as well. On the other hand, by "$A$ causes $B$" (nonstandard terminology!), we usually mean "if $A$ is true, then $B$ is true by some mechanism which relies on the fact that $A$ is true".

It's a fact that "$1 = 1$ implies that my name is Patrick", because these statements are both true; it's not true that $1=1$ causes my name to be Patrick. This is ultimately a matter of making sure we all use the same definition of "implies", and that is the definition we have settled on.

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"Cause" is a loaded word. "If p > 2 is prime, then it is necessarily odd." Being odd certainly doesn't cause a number > 2 to be prime. But it is a necessary condition for this to happen.

In fact I think "sufficient" is much more closely related to "causes," with the understanding that there are possibly other possible causes. "If $f$ is a nonzero polynomial, to show that $f$ has at most 10 zeroes, it is sufficient to show $\deg f \leq 10$." Of course, $x^{100} - 1$ has 2 zeroes, so there are many reasons a polynomial can have at most $10$ zeroes, but if you happen to be able to show that $\deg f \leq 100$, that of course is sufficient.

But I would still be cautious in using the word "causes." It seems to imply that one of the two conditions happens before the other. If "A is necessary and sufficient for B," which one is the cause? It could be either. It's probably whichever you are able to prove first when you actually apply the theorem. This is why we tend to prefer the words "necessary" and "sufficient," precisely because they express something important without imposing a causal story.

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Do a proof by contradiction.

Suppose $Q\implies P$ was false, then by contrapostive,

$\lnot P\implies \lnot Q$. We want to show this statement is false.

By hypothesis, $P$ is necessary and sufficient for $Q$. This means that $P\implies Q$. By contrapositive again, this means that $\lnot Q\implies \lnot P$. But since $\lnot Q\implies \lnot P$ is true, then clearly $\lnot P\implies \lnot Q$ is false, this means that $Q\implies P$ cannot be false, and so indeed $Q\implies P$. We have an "if and only if" statement.

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I think this is a semantic problem.

‘$P$ is necessary and sufficient for $Q$’ says two things:

  • $P$ is sufficient ( or is a sufficient condition) for $Q$ does mean, as you explained that $P \implies Q$, or that we have $P$ ‘only if’ we have $Q$.
  • $P$ is necessary for (or is a necessary consequence of) $Q$ means that $Q \implies P$ , or that we have $P$ if we have $Q$.
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