If $\forall x\in X:P(x)$ is true for all $X\neq\emptyset$, is it true when $X=\emptyset$?

For the statement to be true, there at least must be an element of $X$ such that $P(x)$ holds, but there is no element in the empty set, so is it true, false or undefined?

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    $\begingroup$ $\forall x \in \varnothing (P(x))$ is true no matter what $P(x)$ is. $\endgroup$ – amrsa Jan 12 '18 at 20:33
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    $\begingroup$ $\forall x\in X: P(x)$ is short for $\forall x (x\in X\implies P(x))$... $\endgroup$ – Robert Wolfe Jan 12 '18 at 21:12
  • $\begingroup$ @Robert Thanks. $\endgroup$ – Garmekain Jan 12 '18 at 21:23

No, in fact $\forall x\in X :P(x)$ is always true for $X=\emptyset$, regardless of whether it's true for $X\ne\emptyset$. Your reasoning is backwards; in fact for it to be false would mean that $P(x)$ was false for some $x\in\emptyset$.


No, the statement $\forall x \in X : P(x)$ does not require that there be an $x \in X$ (at least, under the standard interpretation). Generally, $\forall$ is taken to mean the same as $\neg\exists\neg$ - in other words, $\forall x \in X : P(x)$ should be the same as $\neg\exists x \in X : \neg P(x)$. In order for this new sentence to be false, we would need to have an $x \in X$ with $\neg P(x)$; for $X = \emptyset$, there is no such $x$, so $\neg\exists x \in X : \neg P(x)$ can't be false, so $\forall x \in X : P(x)$ must be true.

Another way of thinking about it is this: if I say "all unicorns are blue", the only way to prove me wrong is to supply a unicorn which is not blue - I'm not claiming that there is a unicorn at all, just that any that do exist are blue. Since there are no unicorns, you can't provide a counterexample; so my statement was true.


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