# Reference request for the logical analysis of the word "any"

I have recently noticed a curious phenomenon with the word "any", in that it sometimes does not behave like the universal quantifier "every". Consider this pair of sentences: "John can outrun every man on the sports team" and "John can outrun any man on the sports team". In this particular case, the word "any" functions as a universal quantifier, and so the sentences mean the same thing. However, now consider this pair of sentences: "John cannot outrun every man on the sports team" and "John cannot outrun any man on the sports team". Those two sentences do not mean the same thing. So, my real question is, are there papers that do a logical analysis of the word "any"? I would be very interested in such texts.

• The issue is that "any" works as an unspecified variable. Thus, we can read the statement as implicitly universally quantified. The issue is when there is a negation: is it inside or outside the quantifier? This is the reason why we need quantifiers :-) Commented Dec 29, 2022 at 18:18
• Using set theory jargon “every” could be material while “any” could be said to be more structural. In the sense of the usual quantified logics there’s no such analysis therein for natural language terms… Commented Dec 29, 2022 at 21:28
• Levy wrote a 190pp thesis on it:"English any has been noted to be associated with two uses, a so called negative polarity (NPI) use and a so called Free choice (FC) use... On its NPI use any is most naturally interpreted as having existential force and is typically found in so called downward entailing contexts (Ladusaw 1979). On FC use any is most naturally interpreted as a universal quantifier and is typically found in contexts which can be interpreted as intensional." Commented Dec 30, 2022 at 12:47

If you are asking a question about how 'every' and 'any' work in the English language, and the differences between them, then this article has some useful information.

They tend to differ when negated, or when used in a conditional, or when embedded in another expression, or when used in a question. E.g. contrast:

``````I do not own any book written by Schopenhauer.
I do not own every book written by Schopenhauer.

If any person here is from New Zealand, that would be surprising.
If every person here is from New Zealand, that would be surprising.

This piano is too heavy for any man here to lift.
This piano is too heavy for every man here to lift.

Is any book in this library by a Greek author?
Is every book in this library by a Greek author?
``````

Another difference is that 'any' can be used with mass nouns, but 'every' cannot. "Is there any water?" makes sense, but "Is there every water?" does not.

For added confusion, 'each' is different again, and is not always identical with 'every'.

If you asking about how these terms differ when rendering them into formal logic, then there are some subtleties to observe. 'Any' usually suggests an arbitrary item, and can be ambiguous. 'Every' indicates an entire class of things. 'Each' may differ from 'every'. For example, there is a difference between asserting that F(n) holds for each natural number n, and asserting that (∀x)F(x) holds, which would naturally be read as that F(x) holds for every x. The difference is to be found in the ω-rule.

• What’s the exact difference btw each and every in your final section referencing omega rule? Even in a nonstandard model of PA one can still say each member n… Commented Dec 29, 2022 at 21:05
• The ability to prove F(n) for any/each value of n falls short of being able to prove (∀x)F(x). The ω-rule covers the gap. It seems natural to express the former as 'for each n' or 'for any n' and the latter as 'for all x' or 'for every x'. Commented Dec 30, 2022 at 3:30
• Not seems natural for me at linguistic level (any seems more strong for me linguistically, and one can certainly say every n) and not related to ω-rule which is at logic level... Commented Dec 30, 2022 at 5:35