6 ms·

One complication is that in typical English, if you say "All my hats....", you are simultaneously making an existence statement that you have at least one hat..

One complication is that in typical English, if you say "All my hats....", you are simultaneously making an existence statement that you have at least one hat... but the usual formal logic "forall" quantifier does NOT presume existence. Here's a formal proof that "forall" has a "surprise" meaning for those not well-versed in formal logic: https://us.metamath.org/mpeuni/alimp-surprise.html https://us.metamath.org/mpeuni/alimp-surprise.html I propose that when translating such statements to a formal logic, if that's what you really mean, use an "allsome" quantifier as I've described here: https://dwheeler.com/essays/allsome.html https://dwheeler.com/essays/allsome.html It's really easy to forget to include an existence quantifier. Having notation specifically designed to automatically include it can avoid some problems.

Supermancho 2y ago
> All my hats....", you are simultaneously making an existence statement that you have at least one hat It does not. All my unicorns fly. There is no assumption that I have a unicorn. There is an assumption, based on the claim but it is not a fact. The puzzle also assumes that "my" implies there is some ownership (we'll take for granted "my" means "has" for simplicity), which is another quibble that unravels the whole thing. E is correct. I don't see how A comes to be the accepted answer.
- grraaaaahhh 2y ago
  E cannot be correct. "All my hats are green" is still false even when I own a red hat and a green hat.
  - Supermancho 2y ago
    I would agree that's obvious, if not for the original error. The liar doesn't necessarily "have" any hats. Again, the assumption that the liar has hats is incorrect because it's relying on an conversational implication, rather than a specific assertion.
    - grraaaaahhh 2y ago
      Sure, but the question isn't about which statement is possible from the liar's statement, it's about which statement we can conclude from the liar's statement. The liar could be lying because they have no hats. They could be lying because they have a non-green hat. We cannot conclude E because it's possible that E is not correct.
- LudwigNagasena 2y ago
  Would you agree with the following proposition: “if all of my unicorns fly, then some of my unicorns fly”?
  - Supermancho 2y ago
    Yes
    - LudwigNagasena 2y ago
      So if “all my hats” doesn’t imply that I have at least one hat, “some of my hats” doesn’t imply it either; otherwise we wouldn’t be able to derive “some” from “all”. Hence, “some of my hats are green” doesn’t imply that “at least one of my hats is green”. That’s a claim that contradicts both traditional formal logic interpretation and common sense English interpretation.
      - Supermancho 2y ago
        I think the same of your interpretation of some vs all. Some can contain all, just as it contains none. Both some/all imply, but do not assert existence. Claiming it tautologically defies logic is not compelling.
        LudwigNagasena 2y ago
        Well, I think showing that it defies logical inference is quite relevant in the context of that thread being about translating typical English into first order logic to do logical inference.
        Supermancho 2y ago
        I worry about sets and consideration of edge cases. Legal, programmatic, medical. Adhering to a convention that presupposes meaning and claim that interpretation is the only interpretation, cannot be resolved with repetition. I remain unconvinced.
- scoofy 2y ago
  I mean, it's important to remember that the axioms of first-order logic are arbitrary. We could easily argue that the truth value of an empty group is undecidable, and that would better correlate to natural language logic. The fact that we compact these edge cases into arbitrary truth values is just for ease of computing. This is also relevant to the arbitrary choice of the 'inclusive or' as a default over an 'exclusive or', which most people use in natural language.
  - Supermancho 2y ago
    I have to interpret the question at face value, which may equate to natural language logic. I dont know the specific rules of any of these systems, which are obviously particular or wildly different from a layman interpretation. Most of the arguments seem to center around specialized conveniece rules (as you mention), which are eventually equated to the one true way to deconstruct meaning. At least, that is what I got out of this thread.
tantalor 2y ago
> in typical English you are making an existence statement No, you're not.
- scoofy 2y ago
  Oh boy... so I actually wrote a thesis in graduate school on conversational implicature, Paul Grice, and various other theories of implying things. I would actually agree user dwheeler here. Whether or not you agree with Gricean implicature theory (I do not), the point is that making a claim about a group that doesn't exist is absurd. Absurd statements do not convey meaning, and language is a tool for communication, thus it is generally an assumed axiom that statements will have meaning. Here, even when people make borderline nonsensical statements, we assume there is a metaphor or language game involved. So, by making a statement about 'all my hats', if the number of hats you have is zero, then any predication is absurd and the statement is absurd, so given an axiom of not making absurd statements for natural language, you can assume there are at least two hats. Obviously there are no formal rules here, but the functionality of natural language demonstrates that these heuristics exist. https://plato.stanford.edu/entries/implicature/ https://plato.stanford.edu/entries/implicature/
  - tantalor 2y ago
    Granted all that, but we're not really talking about normal everyday English, but a hypothetical conversation with some mythical entity who can only lie, which is not really a capability of humans; even the most pathological liar among us can and will tell the truth. So I'd put all that theory in a drawer somewhere and acknowledge that, when we're talking about logic puzzles, the rules of logic are paramount, not grammar.
    - LudwigNagasena 2y ago
      Sure, if your main use of formal logic is to while away the time doing vacuous puzzles.
    - scoofy 2y ago
      As I said in a previous part of this thread, the rules of logic are as arbitrary (by definition) as they are paramount, and often diverge from natural language logic: https://news.ycombinator.com/item?id=42365222#42368661 https://news.ycombinator.com/item?id=42365222#42368661 --- I mean, it's important to remember that the axioms of first-order logic are arbitrary. We could easily argue that the truth value of an empty group is undecidable, and that would better correlate to natural language logic. The fact that we compact these edge cases into arbitrary truth values is just for ease of computing. This is also relevant to the arbitrary choice of the 'inclusive or' as a default over an 'exclusive or', which most people use in natural language.
tshaddox 2y ago
> One complication is that in typical English, if you say "All my hats....", you are simultaneously making an existence statement that you have at least one hat... but the usual formal logic "forall" quantifier does NOT presume existence. Perhaps. But what if someone asks you "are all your hats green?" Then the interpretation is not so clear.

Supermancho 2y ago

> All my hats....", you are simultaneously making an existence statement that you have at least one hat It does not. All my unicorns fly. There is no assumption that I have a unicorn. There is an assumption, based on the claim but it is not a fact. The puzzle also assumes that "my" implies there is some ownership (we'll take for granted "my" means "has" for simplicity), which is another quibble that unravels the whole thing. E is correct. I don't see how A comes to be the accepted answer.

grraaaaahhh 2y ago
E cannot be correct. "All my hats are green" is still false even when I own a red hat and a green hat.
- Supermancho 2y ago
  I would agree that's obvious, if not for the original error. The liar doesn't necessarily "have" any hats. Again, the assumption that the liar has hats is incorrect because it's relying on an conversational implication, rather than a specific assertion.
  - grraaaaahhh 2y ago
    Sure, but the question isn't about which statement is possible from the liar's statement, it's about which statement we can conclude from the liar's statement. The liar could be lying because they have no hats. They could be lying because they have a non-green hat. We cannot conclude E because it's possible that E is not correct.
LudwigNagasena 2y ago
Would you agree with the following proposition: “if all of my unicorns fly, then some of my unicorns fly”?
- Supermancho 2y ago
  Yes
  - LudwigNagasena 2y ago
    So if “all my hats” doesn’t imply that I have at least one hat, “some of my hats” doesn’t imply it either; otherwise we wouldn’t be able to derive “some” from “all”. Hence, “some of my hats are green” doesn’t imply that “at least one of my hats is green”. That’s a claim that contradicts both traditional formal logic interpretation and common sense English interpretation.
    - Supermancho 2y ago
      I think the same of your interpretation of some vs all. Some can contain all, just as it contains none. Both some/all imply, but do not assert existence. Claiming it tautologically defies logic is not compelling.
      - LudwigNagasena 2y ago
        Well, I think showing that it defies logical inference is quite relevant in the context of that thread being about translating typical English into first order logic to do logical inference.
        Supermancho 2y ago
        I worry about sets and consideration of edge cases. Legal, programmatic, medical. Adhering to a convention that presupposes meaning and claim that interpretation is the only interpretation, cannot be resolved with repetition. I remain unconvinced.
scoofy 2y ago
I mean, it's important to remember that the axioms of first-order logic are arbitrary. We could easily argue that the truth value of an empty group is undecidable, and that would better correlate to natural language logic. The fact that we compact these edge cases into arbitrary truth values is just for ease of computing. This is also relevant to the arbitrary choice of the 'inclusive or' as a default over an 'exclusive or', which most people use in natural language.
- Supermancho 2y ago
  I have to interpret the question at face value, which may equate to natural language logic. I dont know the specific rules of any of these systems, which are obviously particular or wildly different from a layman interpretation. Most of the arguments seem to center around specialized conveniece rules (as you mention), which are eventually equated to the one true way to deconstruct meaning. At least, that is what I got out of this thread.

tantalor 2y ago

> in typical English you are making an existence statement No, you're not.

scoofy 2y ago
Oh boy... so I actually wrote a thesis in graduate school on conversational implicature, Paul Grice, and various other theories of implying things. I would actually agree user dwheeler here. Whether or not you agree with Gricean implicature theory (I do not), the point is that making a claim about a group that doesn't exist is absurd. Absurd statements do not convey meaning, and language is a tool for communication, thus it is generally an assumed axiom that statements will have meaning. Here, even when people make borderline nonsensical statements, we assume there is a metaphor or language game involved. So, by making a statement about 'all my hats', if the number of hats you have is zero, then any predication is absurd and the statement is absurd, so given an axiom of not making absurd statements for natural language, you can assume there are at least two hats. Obviously there are no formal rules here, but the functionality of natural language demonstrates that these heuristics exist. https://plato.stanford.edu/entries/implicature/ https://plato.stanford.edu/entries/implicature/
- tantalor 2y ago
  Granted all that, but we're not really talking about normal everyday English, but a hypothetical conversation with some mythical entity who can only lie, which is not really a capability of humans; even the most pathological liar among us can and will tell the truth. So I'd put all that theory in a drawer somewhere and acknowledge that, when we're talking about logic puzzles, the rules of logic are paramount, not grammar.
  - LudwigNagasena 2y ago
    Sure, if your main use of formal logic is to while away the time doing vacuous puzzles.
  - scoofy 2y ago
    As I said in a previous part of this thread, the rules of logic are as arbitrary (by definition) as they are paramount, and often diverge from natural language logic: https://news.ycombinator.com/item?id=42365222#42368661 https://news.ycombinator.com/item?id=42365222#42368661 --- I mean, it's important to remember that the axioms of first-order logic are arbitrary. We could easily argue that the truth value of an empty group is undecidable, and that would better correlate to natural language logic. The fact that we compact these edge cases into arbitrary truth values is just for ease of computing. This is also relevant to the arbitrary choice of the 'inclusive or' as a default over an 'exclusive or', which most people use in natural language.

tshaddox 2y ago

> One complication is that in typical English, if you say "All my hats....", you are simultaneously making an existence statement that you have at least one hat... but the usual formal logic "forall" quantifier does NOT presume existence. Perhaps. But what if someone asks you "are all your hats green?" Then the interpretation is not so clear.