# What does 'any' mean in predicate calculus

I need to translate an English sentence into a well-formed predicate calculus formula.

The sentence starts off as:

Any tiger who chases every creature also chases itself.

Does 'any' translate to 'for all' or 'there exists' in predicate calculus?

"Any" is sometimes an ambiguous word in English.

"You can cash this check at any bank" means no matter which bank it is, you can do it, so every banks is one at which you can cash the check. But that's not the same as "You can cash this check at every bank."

"Any member of the club can be chairman" means that every member has that ability, but it does not mean "Every member of the club can be chairman", which would imply all at the same time.

"Is there any number that satisfies this equation?" means "Is there some number that satisfies this equation?". But "Any number satisfies this equation" means in effect that every number satisfies this equation.

"Prove that any number is purple" is ambiguous: It could mean "Pick any number --- it doesn't matter which one --- and prove that it is purple", or it could mean "Prove that if you pick any number at all --- it doesn't matter which one --- then it is purple", and that would mean that every number is purple.

In some contexts the word "any" should be avoided because of the ambiguity.

• I can't avoid it because the sentence is given to me. Im assuming in this case it means "for all"? – user2510809 Oct 29 '14 at 2:31
• In this case it means every tiger. – Michael Hardy Oct 29 '14 at 2:33
• It is even worse. We agree that the sentence "Any number satisfies the equation $0x = 0$" means "Every number satisfies ...". But we also have: "an element $x$ of a commutative ring is a zero divisor if any nonzero $y$ in the ring satisfies $xy = 0$." So the broader context in which the phrase "any number satisfies ..." affects the meaning of the word "any" in the English phrase. – Carl Mummert Oct 29 '14 at 11:24

"For any even integer, x, you can write it as two times some integer k."

$\forall x:~x$ is even$, \exists k\in\mathbb{Z}~$s.t. $x=2k$

"There aren't any real numbers that satisfy the equation $x^2 = -1$."

$\nexists x\in\mathbb{R}: x^2=-1$

Any tiger who chases every creature also chases itself.

Does "any" translate to "for all" or "there exists" in predicate calculus.

Since there is no suggestion that we are talking about a particular tiger, but about tigers in general, I would translate it as:

$\forall a:[Tiger(a) \implies \forall b:[Creature(b) \implies Chases(a,b)]]$

$\implies \forall a:[Tiger(a) \implies Chases(a,a)]$

You could derive (or prove) this statement if you first assumed as a kind of axiom that:

$\forall a:[Tiger(a)\implies Creature(a)]$