It's typically correct to say something like "all cats are beautiful" as ∀x(CAT(x)->BEAUTIFUL(x)). It is incorrect to say ∀x(CAT(x) & BEAUTIFUL(x)) because you're effectively saying "everything in the world is a beautiful cat". However, I have seen quantifiers used with set notation in some compsci classes (e.g. ∀x ∈ A , CAT(x)->BEAUTIFUL(x) where A is the set of animals.) If you're using this kind of notation, then would it be equally correct to translate "all cats are beautiful" as simply ∀x ∈ C , BEAUTIFUL(x) (where C is the set of cats)?
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1I'd say so, yes. – Raymond Timmermans Oct 14 '17 at 9:52
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Mathematically, saying "for all X, [sometthing]" is incorrect because the universal quantifier requires a bounding set. In other words, the correct form is "for all X member of S, [something]". So you're correct, but mostly because your first two formulations are not mathematically valid. – barrycarter Oct 16 '17 at 15:32
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Restricted/relativized quantification is very common in 'everyday' mathematical practice. You can read more at:
https://math.stackexchange.com/a/2101961/359302
https://en.wikipedia.org/wiki/Quantifier_(logic)#Equivalent_expressions
https://www.encyclopediaofmath.org/index.php/Restricted_quantifier
