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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?


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up vote 3 down vote accepted

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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Matt is of course right. And I can't help wondering what course text that might be. The syntax is pretty non-standard too. – Peter Smith Sep 2 '12 at 10:28
There doesn't seem to be anything non-standard about the syntax as far as I can see. – Henning Makholm Sep 2 '12 at 12:00
@HenningMakholm The clause after the arrow? Why the brackets around? Which standard logic text asks for this? – Peter Smith Sep 2 '12 at 17:29
@PeterSmith: Which standard logic text forbids superfluous parentheses? And why? – Henning Makholm Sep 2 '12 at 20:17
@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. – Peter Smith Sep 2 '12 at 23:04

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