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Given that $\forall xQ(x)$ is true, is $\exists xQ(x)$ also true?

Thanks in advance.

Mauro ALLEGRANZA
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gachaSalt
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    It depends what the domain of discourse is. If the set to which the $x$ may belong is empty, then $\forall x \ Q(x)$ can be true while $\exists x \ Q(x)$ is not true. – Charles Hudgins Feb 08 '20 at 20:34
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    It also depends on the exact formulation of first-order logic: some require a non-empty domain of discourse, while other formulations allow an empty domain of discourse. (And then, if the first-order logic language you're considering includes a nullary function symbol, or in other words a constant symbol, that always means that the domain of discourse is necessarily nonempty in any model.) – Daniel Schepler Feb 08 '20 at 20:39
  • No, it is not necessarily true. – amWhy Apr 26 '20 at 23:31

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Yes—as long as the set that the universal quantifier $\forall$ is ranging over is nonempty. For exmaple, $\forall x\in[0,1]\ Q(x)$ does imply $\exists x\in[0,1]\ Q(x)$, but $\forall x\in\emptyset\ Q(x)$ does not imply $\exists x\in\emptyset\ Q(x)$.

Greg Martin
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  • Hence it is not necessarily true. Thanks for verifying that, in general, the proposition is not true. – amWhy Apr 26 '20 at 23:33