# Unusual change of meaning of word "any" in negative sentences form "for all" to "there exists". Predicate logic

Question. Why does the word "any" in negative sentences changes its meaning from "for all" to "there exists"?

Origin of the question. I have a question about translating English to predicate logic. I don't understand well enough how the word "any" should be translated in predicate logic in general. I know that this word has principal meaning "for all", but I have found one example where it, in my opinion, means "there is at least one".

None of Occam's followers likes any realists. This example is from Suppss's "introduction to logic"

(x)(Ox->(y)(Ry->Lxy))

In my opinion this is equivalent to: "There is not at least one follower of Occam who likes at least one realist".

My attempt to solve this issue. So let us consider simpler example. Say, the sentence: No dogs likes any cat. (x)(Dx->(y)(Cy->Lxy)) In my opinion it is equivalent to: "There is no single dog who likes at least one cat" and not equivalent to: "There is no single dog who likes all the cats".

First one. There is a set of all the dogs. There is a set of things that like at least one of the cats. So intersection of these sets is empty.

Second one. There is a set of all the dogs. There is a set of all the things that like all of the cats (even mad and ill ones). So intersection of these sets is empty.

From second one we can infer that there might be a dog who likes some cats, but not all of them (It might like only fluffy ones). From first one it follows that there can't be any cat that is liked by any dog.

So I concluded that "any" changes it's meaning from "for all" to "there exists" in some sentences, possibly in all negative sentences.

My research. There is a similar question: Reference request for the logical analysis of the word "any" In my opinion it doesn't fully answer the question, they say that it is some artifact of English language. So there is grammar book, that tells that in these cases we use "any" as existential and in other cases as universal quantifier.

I think that it deeper phenomenon than just some grammatical artifact of English. I know that there is similar issue with translation of "any" in other different languages. So in these languages there is the same issue.

P.S.: I have read 6 books in logic, I know what a square of opposition is. I am a chemist who has logic as a hobby.

• It is not unusual, "any" generally behaves universally in affirmative contexts and existentially in negative contexts, and not just in English. Defining what "affirmative" and "negative" (downward entailing) mean in general is subtle, see Levy's dissertation, pp. 13-20 for details, but it does depend only on the logical form of the sentence. Commented Feb 25 at 8:33
• The difficulties lie in the context since natural language is not context free while FOL is, thus 'any' can be simply regarded as 'all', like 'each' and 'every', just beware of context during whole sentence translation. For examples discussed below, "If you can speak any language, you must be a genius" the entity of interest is the person 'you', thus it's really a nested conditional. While "If you can use any programming language, you can write a program to solve this problem", the entity of interest is programming language, thus it's really a simpler single conditional. 'you' is irrelevant... Commented Apr 3 at 5:41
• Have you understood my answer? Commented Jun 8 at 15:07

Conifold's comment is incorrect. See this post for a more accurate representation of the meaning of "any" in English. That post also gives a sufficient counter-example:

(1) She does not have any disease.
(2) She does not have some disease.
(3) For any disease D, she does not have D.

(1) does not mean (2), but rather (3). Applying the same explanation to your example:

(4) No dog likes any cat.
≡ ∀A∈Cats ¬∃B∈Dogs ( B likes A ).

Your idea (i.e. that "any" changes it's meaning from "for all" to "there exists" in some sentences, possibly in all negative sentences.) is therefore wrong when implemented in the natural way (i.e. changing (1) to (2)). Here are yet more counter-example of different kinds:

(5) If any person does not understand English, he/she would answer your question wrongly.
≡ ∀R∈People ( ¬( R understands English ) ⇒ R would answer your question wrongly ).

(6) If this job can be done by anyone, we would not need to hire any expert.
≡ ∀Y∈Experts ( ( ∀X∈People ( this job can be done by X ) ) ⇒ we would not need to hire Y ) ).

(7) If this job had been done by anyone, we would not have any problem now.
≡ ∀X∈People ∀Y∈Problems ( this job had been done by X ⇒ we would not have Y now ).

As explained at the linked post, in (6) the "anyone" is stuck under the modal "can", whereas the "any expert" is not under any modal boundary so it goes all the way to the global level. And (7) looks superficially similar to (6) but lacks the modal "can" and so the "anyone" does go all the way to the global level!

• If I understand you correctly, you are saying that 'any' behaves differently from 'some' and 'all' when it comes to crossing a modal or conditional context. That is quite an interesting approach. I'm not sure how that would explain the difference between: "If you can speak any language, you must be a genius", and, "If you can use any programming language, you can write a program to solve this problem". The first indicates "any and every" while the second indicates "an arbitrary example". The difference appears to be resolved by the context. Commented Apr 1 at 22:39
• @Bumble: Your first example fits my explanation. Your second example is actually ambiguous in an interesting way; my explanation is not clearly invalid, but some people might indeed interpret your way instead based on some expectations. I would not be hasty to claim that it is just a matter of context, however, because this reminds me of the phenomenon that there is a spectrum of logicality of the various English dialects and registers. In some dialects you say "Not everything that glitters is gold." but in less logical dialects you would say "Everything that glitters is not gold.". [cont] Commented Apr 2 at 7:31
• [cont] I would personally consider the second illogical for the obvious reason, but yet an existing usage of English that we need to be aware of. There are logically precise versions of your second example: (a) "If you can use any programming language at all"; (b) "If there is any programming language that you can use"; (c) "If you can use a programming language". Just like the glitter-gold example, which is an example of a lay person's invalid switch of quantifiers, I would consider it invalid or at least imprecise to write your can-use-any-prog-language example. Commented Apr 2 at 7:41