Is it permissible and normal to express the prop. ∀x(Px ≡ Qx) simply as Px ≡ Qx? That is, to treat the univ. quantifier as implicit if its scope is all the rest of the prop.?
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As a complement to the accepted answer below: the first proposition can be translated "all Ps are Qs and conversely" and the second "if it's a P, it's a Q and conversely" where the reference of "it" is unspecified. – Quentin Ruyant Jan 26 '16 at 20:43
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While the answer depends a bit on context,
generally no, you cannot express ∀x(Px ≡ Qx) as simply Px ≡ Qx . The reason has to do with bound and unbound variables. (http://www.cs.odu.edu/~cs381/cs381content/logic/pred_logic/quantification/quantification.html)
Thus, seeing x without the ∀ means something different than (∀x) precisely because x in the former would represent a specific x whereas in the latter use it does not refer to a specific thing but rather is a variable standing over a range .
virmaior
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