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Translating these statements from predicate logic to English: are these the saying the same thing essentially or is there a difference?

$\forall x~(Fx \vee \neg Fx)$

$\forall x~Fx ~\lor~ \forall x~\lnot Fx$

Graham Kemp
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mayalarson
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    Where are you having trouble? Please show your attempt at translation and how you compare those claims. People prefer to help those they see trying to help themselves. – Graham Kemp Oct 08 '19 at 02:02

2 Answers2

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There is a big difference between these statements.

$\forall x (Fx \lor \neg Fx)$ is saying for all $x$ either $F x$ is true or $F x$ is false.

$\forall x F x \lor \forall x \neg F x$ can be rewritten as $(\forall x F x) \lor (\forall x \neg F x)$ and can be read as "Either $F x$ is true for ever $x$, or $F x$ is false for every $x$."

Will
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Different. For example, if $Fx$ is “x is even” then the first statement is true for $\mathbb{N}$, but not the second, which would require either all natural numbers to be even or all not to be.

Stephen K.
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