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Actually 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

Actually 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.

mannykannot 2y ago
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.
- bryanrasmussen 2y ago
  given that this is a liar the use of all may be meant to deceive, the main thing here is the use of the word conclude. I doubt, given in what context this was written, that this is a matter of omission.
  - mannykannot 2y ago
    It clearly was not an omission here, and I suspect one of the motivations for this question in the examination was to test the students' use of quantifiers in formal logic, as opposed to according to the tacit rules and conventions of natural language.
- dambi0 2y ago
  I agree in general but I do wonder how unusual using ‘all’ to assume existence is. For example “have you done all your homework?” What we are really asking is whether there is any homework that wasn’t done, not did you do any homework.
  - mannykannot 2y ago
    Fair enough, and this seems to emphasize the notion that being explicit about the tacit rules of natural-language usage is not easy.
JKCalhoun 2y ago
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.
- bryanrasmussen 2y ago
  Thank you for clarifying something that I pointed out in my comment while missing the total point of my comment. on edit: I don't actually think you made my comment "but all of the questions are potentially true." any clearer than I made it, and evidently you seem to think that only A and E are potentially true, which is obviously wrong. on second edit: ah I see that you believe the idea that some people are saying that All my hats are green indicates that the liar must have hats https://news.ycombinator.com/item?id=42369002 https://news.ycombinator.com/item?id=42369002 because evidently the assumption is that the 0 hat has a color green for some reason that is not adequately explained. If that were true then it also follows that if the liar has no hats then all his hats are every color known to humanity. I don't think that is the way it works however.
  - bryanrasmussen 2y ago
    instead of going through and making more and more edits, I finally figured out what the problem is here - there are at least two logics being used here - one set theory, in which case for vacuous truth I guess it's true that all the hats are green if you have an empty set, and the other linguistic logic. In many forms of linguistic logic you need to have an existent hat to have the property green, therefore if you do not have any hats you have no green hats. Which is my preferred method of dealing with this. And obviously there are other forms of logic, as https://iep.utm.edu/liar-paradox/#SH3c https://iep.utm.edu/liar-paradox/#SH3c Is the column that was linked more likely to be interested in linguistic logic games or formal mathematical logic games? I'm guessing the second, in which case I guess it's true we can conclude A - but I inherently dislike vacuous truths when applied to logic, especially language like this where the normal argument that the second parameter "Green" is never evaluated because we do not have the first "A hat" in the empty set seems suspect to me because we know what Green is separate from a hat, we just need to know if there is a hat we can assign Green to.

mannykannot 2y ago

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.

bryanrasmussen 2y ago
given that this is a liar the use of all may be meant to deceive, the main thing here is the use of the word conclude. I doubt, given in what context this was written, that this is a matter of omission.
- mannykannot 2y ago
  It clearly was not an omission here, and I suspect one of the motivations for this question in the examination was to test the students' use of quantifiers in formal logic, as opposed to according to the tacit rules and conventions of natural language.
dambi0 2y ago
I agree in general but I do wonder how unusual using ‘all’ to assume existence is. For example “have you done all your homework?” What we are really asking is whether there is any homework that wasn’t done, not did you do any homework.
- mannykannot 2y ago
  Fair enough, and this seems to emphasize the notion that being explicit about the tacit rules of natural-language usage is not easy.

JKCalhoun 2y ago

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.

bryanrasmussen 2y ago
Thank you for clarifying something that I pointed out in my comment while missing the total point of my comment. on edit: I don't actually think you made my comment "but all of the questions are potentially true." any clearer than I made it, and evidently you seem to think that only A and E are potentially true, which is obviously wrong. on second edit: ah I see that you believe the idea that some people are saying that All my hats are green indicates that the liar must have hats https://news.ycombinator.com/item?id=42369002 https://news.ycombinator.com/item?id=42369002 because evidently the assumption is that the 0 hat has a color green for some reason that is not adequately explained. If that were true then it also follows that if the liar has no hats then all his hats are every color known to humanity. I don't think that is the way it works however.
- bryanrasmussen 2y ago
  instead of going through and making more and more edits, I finally figured out what the problem is here - there are at least two logics being used here - one set theory, in which case for vacuous truth I guess it's true that all the hats are green if you have an empty set, and the other linguistic logic. In many forms of linguistic logic you need to have an existent hat to have the property green, therefore if you do not have any hats you have no green hats. Which is my preferred method of dealing with this. And obviously there are other forms of logic, as https://iep.utm.edu/liar-paradox/#SH3c https://iep.utm.edu/liar-paradox/#SH3c Is the column that was linked more likely to be interested in linguistic logic games or formal mathematical logic games? I'm guessing the second, in which case I guess it's true we can conclude A - but I inherently dislike vacuous truths when applied to logic, especially language like this where the normal argument that the second parameter "Green" is never evaluated because we do not have the first "A hat" in the empty set seems suspect to me because we know what Green is separate from a hat, we just need to know if there is a hat we can assign Green to.