Ambiguity of the word 'any'?

Three more possibilities can be added:

Should an employee of Company X be allowed to assume absolute authority in any project with Company X's name associated?

Should an employee of Company X be allowed to assume absolute authority in a project with Company X's name associated?

Should any employee of Company X be allowed to assume absolute authority in a project with Company X's name associated?

For interpretation #2, the second option above is more suitable.

For interpretation #1, there are two sub-interpretations 1) if someone is an employ then (if the person wishes) allow .. (same as 'everyone is allowed') -- if yes, then let anyone take it up! 2) By asking the question, one is trying to find out if there is one particular (any) employee who can be allowed.. if yes, then pick that one!

The question

Should any employee of Company X be allowed to assume absolute authority in any project with Company X's name associated?

does indeed permits at least two very different readings. To simplify the analysis, let's consider a similar question that uses any just once:

Should any employee of Company X have the authority to turn off the lights when he or she is the last person to leave the building?

There are two possible ways to read this question. The first is to read it as asking whether every employee should have the authority to turn off the lights if he or she happens to be the last person to leave the building. In short, any = every. The implied alternative answers in this case would be

Yes, every employee of Company X should have the authority to turn off the lights when he or she is the last person to leave the building.

and

No, only certain designated employees of Company X should have the authority to turn off the lights when they are the last person to leave the building.

The second ways to read this question is as asking whether a particular designated employee should have the authority to turn off the lights. In short, any = even one. The implied alternative answers in this case would be

Yes, a particular designated employee of Company X should have the authority to turn off the lights when he or she is the last person to leave the building.

and

No, no employee of Company X should have the authority to turn off the lights when he or she is the last person to leave the building.

In the poster's original example, readers face the same interpretative choice between any as "every" and any as "even one." To avoid the ambiguity, you would have to reframe the question in a way that clarified which sense of any you intended. For example, you might say

Should every employee of Company X be allowed to assume absolute authority over a project that has Company X's name associated with it?

if you had any = every in mind, or you might say

Should no employee of Company X be allowed to assume absolute authority over a project that has Company X's name associated with it?

if you had any = even one in mind.

Your wording very strongly suggests;

for any employee, e, and any project, p, e may assume absolute responsibility for p.

If you mean anything else you should reword it.

The following are three propositions but reworded slightly different. The first proposition is the OP's one

1. Should any (whoever) employee of Company X be allowed to assume absolute authority in any (whatsoever) project with Company X's name associated?

In other words, does it matter who has the authority in a project, as long as the person is an employee of Company X, and the project is connected to the company's name?

— Somebody voting "yes" agrees there should be no constriction other than those previously mentioned.

2. Should an employee (specific) of Company X be allowed to assume absolute authority in any (whatsoever) project with Company X's name associated?

Is it permissible for an employee of Company X to assume authority in any single project, as long as that project is connected with the company's name on it?

— A person voting "yes" agrees that an employee should be given the opportunity to take charge in a project that has the company's name connected to it.

3. Should an employee (specific) of Company X be allowed to assume absolute authority in a project (specific) with Company X's name associated?

Could an employee (under special circumstances) be given the opportunity to take charge in a project, if that project is connected with Company X's name?

— A person voting "yes" agrees there may be instances when this authority may be granted to an individual employee.

The following answer is particularized to a question from user Ken N which has been closed but referred to the question on the present page. Its text is repeated here for convenience.

QUESTION

An important legal result hinges on the meaning of the word "any" in this sentence:

"If the effect of the domiciliary requirement under subparagraph (A) is to render the debtor ineligible for any exemption, the debtor may elect to exempt property that is specified under subsection (d)."

My opposition argues that because the debtor is still eligible for some exemptions, he cannot elect to exempt property that is specified under subsection (d). I argue that the effect of the domiciliary requirement rendered my client ineligible for some exemptions, therefore I can elect to exempt property that is specified under subsection (d). It seems like it should be a simple issue, but it has now confused judges all the way up to the 9th Circuit Court of Appeals

In other words, one party argues that this sentence means: As long as the debtor is eligible for at least one exemption, the [result occurs].

The other party argues that this sentence means: If the debtor is ineligible for at least one exemption, the [result occurs].

Which is correct, and why?

ANSWER

I    According to the principles to be read in CoGEL, "any" is non-ambiguous and means "it doesn't matter matter what".

"Any", as a determiner, has two meanings.

1. The first meaning of "any" as a central determiner is found in so-called non-assertive territory, that is, essentially, the grammatical contexts of negative sentences and questions. In those contexts it is the counterpart of "some", which has to be unstressed "some" (/səm/), meaning for countable nouns "a few, more than one" and for uncountable nouns "a small quantity". (CoGEL, § 5.14, Type (b) (ii))
   This means that for countable nouns, the noun has to be in the plural.

• I've bought some tins of apple jelly. (strictly countable) I've bought some tin of apple jelly.
• I've bought some flour. (strictly uncountable, in the everyday context; notice that this is not true any more for "wine".)

As it is the counterpart of "some", "any" engenders the characteristics of "some" bar that of being proper in the same context, that is, it is applicable to countable nouns and to non countable nouns, and, I think that it is reasonnable that suppose this, the head of the phrase it determines has to be in the plural.

-I haven't bought any tin of apple jelly. (this is not used, the plural of "tin" is necessary.)

This last rather evident contention of mine is corroborated in "CoGEL, § 5.14, Type (b) (ii)", where we can read that the central determiner "any" is a determiner of plural count nouns (countable) and uncount nouns (uncountable). There is no question of singular count nouns.

2. The second meaning of "any" is as an indefinite determiner (CoGEL, table 6.45), this meaning being "it doesn't matter which/what/who". For this second meaning the head can also be singular count.

II

• If the effect of the domiciliary requirement under subparagraph (A) is to render the debtor ineligible for any exemption, the debtor may elect to exempt property that is specified under subsection (d)."

In this sentence, the context of use of "any" is assertive, which is a first point against the interpretation of "any" as the counterpart of "some"; on top of that the head is clearly singular count (no s at the end, the claim in the OP's question that the word connote several cases of exemptions).
It seems, therefore, that there is no ambiguity and that your opposition is right.