alt Hacker NewsActually I would say by the rules of English we cannot conclude any of those multiple choice questions is the absolute case -
A) "The liar has at least one hat." cannot conclude because may have no hats, thus the lie is in the "all my hats"
B) "The liar has only one green hat." cannot conclude because may have 2+ green hats out of a 3+ set.
C) "The liar has no hats." cannot conclude that because he may have hats that are not green.
D) "The liar has at least one green hat." cannot conclude that because he may have no hats, or no green hats.
E) "The liar has no green hats." cannot conclude that because of the "all my" modifier means that he can have some green hats.
This is however different than what is true or not. Concluding from a set of multiple choice questions is not choosing ones that are potentially true, concluding is choosing something that is definitely true. There is not a single statement in that list of questions that is definitely true given the requirements, but all of the questions are potentially true.
on edit: all questions are potentially true, but not all potentially true at the same time of course - some of them lock out the others.
Replies
There are few hard semantic rules in English; it is more a matter of conventional usage and expectations - and when 'all' is used, people usually expect that the sentence is about at least one thing, and probably more. I suspect that this is mainly a matter of omission: in ordinary discourse, there seem to be few occasions for using it when it is definitely the case that the resulting sentence is not referring to anything (or exactly one thing), and if there is doubt about the existence of any referents at all, an alternative phrasing along the lines of 'if there are any X, then...' would be considered the right way to say it.
This is so much so that if you use the 'vacuous all', people will suspect that you have ulterior motives, and are being deliberately obtuse to hide them.
I wonder whether, if we attempted to make explicit all these tacit rules and conventions, we would end up with a consistent logic, and I believe that this looseness of natural languages was the main motivation for formalizing logic, from the enlightenment onward.
As others have noted in the comments, if the liar has no hats, then "All my hats are green" is, in some formal understanding, true. The liar cannot make true statement so must in fact have at least one hat.
That makes A) a good candidate.
But E) can also be true — and in fact both A) and E) can both be correct.