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From my course text:

If the universe of discourse is composed of the natural numbers then we can write:

∀X (odd(X) ⇒ (∃Y (X = 2 x Y)))

If the universe of discourse is everything then we can write:

∀X ∃Y(odd(X) ⇒ (X = 2 x Y))

I know that the difference is the position of the "∃Y". But I do not understand why this occurs. Can someone explain to me why "∃Y" needs to be shifted to the front?

Thanks!

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It doesn't need to be shifted as far as I can see. The formulas you gave are logically equivalent, by which I mean, the first is satisfied by a model M, if and only if the second is also satisfied by the model M.

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    @HenningMakholm Mendelson, Enderton, van Dalen ... All allow you to *drop* bracketing (when no harm is done), some allow you to change the flavour of brackets, none allow you to *add* bracketing. The first displayed formula wouldn't be a wff on their stories, or on others I'm familiar with. Hence my remark about the dialect used being non-standard. Of course, you might think that we ought to be more relaxed about allowing additional brackets when that makes for readability. And I *agree* (and do it e.g. in my Gödel book). But readability considerations don't seem to apply here.2012-09-02