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This my be a simple question and I am missing something fundamental. When doing if and only if proofs, doesn't necessity imply sufficiency, so why does the 'if' have to be proven if the 'only if' is true?

AppleBottoms4
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  • No, in P iff Q, you need to show that P implies Q and Q implies P both. – Sean Roberson Mar 08 '21 at 17:45
  • "$A$ if $B$" means "$B\implies A$" (i.e. $A$ is necessary for $B$). "$A$ only if $B$" means "$A\implies B$" (i.e. $A$ is sufficient for $B$) . Therefore, "$A$ if and only if $B$" means "$A\iff B$", i.e. $A$ is necessary and sufficient for $B$. – Surb Mar 08 '21 at 17:46
  • Necessity does not imply sufficiency. Example: equilateral triangle: two sides equal is necessary, but not sufficient. – herb steinberg Mar 08 '21 at 17:47
  • Besides the examples given, the following "visual" example might help. Let $A$ and $B$ be sets. Then "$x \in A$ if $x \in B$" means $B \subseteq A,$ and "$x \in A$ only if $x \in B$" means $A \subseteq B.$ – Dave L. Renfro Mar 08 '21 at 17:52

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Necessity certainly does not imply sufficiency. As an example:

The planet Earth existing is necessary for a unicorn to exist and live a happy life, as they cannot survive in the vacuum of space.

The planet Earth existing is not sufficient for a unicorn to exist and live a happy life, as we can see, unicorns do not exist.

Duncan Ramage
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