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. – user935 Oct 16 '17 at 15:32
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@barrycarter What's wrong with unrestricted quantification? In the logic classes I studied, we were aware that restricted quantification was often better-behaved, but generally speaking "for all x in C, P(x)" was regarded as a shorthand for "for all x, x in C implies P(x)" – Ben Millwood Aug 17 '19 at 8:40
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@BenMillwood You run into things like Russell's Paradox if you allow things "for all x, if x is a set then...". The problem with "for all x" is that the nature of x can be arbitrarily strange, which leads to paradoxes. – user935 Aug 17 '19 at 17:48
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@barrycarter the paradox only arises when you permit the formation of the set of all x satisfying some arbitrary condition. It's not that unrestricted quantifiers are forbidden, it's just that you can't form sets from them, only subsets of "existing" sets. – Ben Millwood Aug 19 '19 at 15:05
The most common way I've seen to do this is to just define the set of cats in the following way:
{X:Cx & Bx}
with C and B standing for "is a cat" and "is beautiful" respectively. This defines the set of all X that are cats and are beautiful.
For the sake of interest: if we consider all the Xs such that Cx and the Xs such that Bx as sets called "C" and "B", then the above definition is equivalent to the intersection of C and B: "C ∩ B".
Perhaps I'm misunderstanding the question though because if understand you correctly, this is the obvious way to rephrase "∀x(Cx-->Bx)", but no one else has given this answer. Assuming that the set of cats is the intersection of C and B does the same job of saying that for every object, if it's a cat then its beautiful.
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Would it be the "union" or "intersection" of C and B in the paragraph before the last where it is called "intersection"? – Frank Hubeny Aug 16 '19 at 16:20
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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
