# Alternate form of “all x are y”

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)?

• I'd say so, yes. – Raymond Timmermans Oct 14 '17 at 9:52
• 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
• @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 at 8:40
• @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. – barrycarter Aug 17 at 17:48
• @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 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.

• 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 at 16:20
• I think intersection. let me fix that – Daniel Prendergast Aug 16 at 16:50

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