Melissa wants to date Nick
Melissa has had a crush on Nick for quite some time, and finally found the courage to talk to him. After some good conversation, Melissa asked Nick the following question, oblivious to the fact that he is already dating someone else:
"If I asked you out, would your answer be the same as your answer to this question?"
Assuming he responds truthfully in a yes or no fashion, will Nick go on a date with Melissa?
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6 solutions
Any statement that refers to it's own truth value is meaningless. Here is another question that works the same way "If I were to ask you whether 2+2=5, would your answer be the same as to this question?"
Either I just proved that 2+2=5, or this logic is faulty and the correct answer is "not enough information"
You aren't the first person to fall for this trap. Bertrand Russel caught Goetlib Frege the same way.
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The problem with an assertion like "every statement that refers to its own truth value is meaningless" is that when you attempt to make the prohibition flat-out (as Russel tried at one point) there is a loop where more than one sentence refer to each other:
The sentence after this one is false. The sentence before this one is true.
So, attempt to prohibit all of these: but what about when the loops become more subtle, such that it is impossible to know if there is a loop or not? One can attempt to say all of these cases are meaningless, but any mathematics capable of addition and multiplication is capable of self-contradictory loops.
Hence, mathematicians usually take two routes:
1.) Use the Law of the Excluded Middle such that things are either true or false, so if something is true or false it is a contradiction and there was an unsound assumption or axiom. This is standard mathematical practice and is what Brilliant uses.
2.) More rarely, use something like "inconsistent mathematics" which denies the Law of the Excluded Middle but then requires a new solution, like applying the Law of Explosion as a way of deriving theorems (that is, if something causes all things to be true, that thing is assumed to be false) or using a three-value logic with "true", "false", and "both". (This is very non-standard mathematics and you can assume it is not used in any Brilliant problems.)
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You are confused about the Law of the Excluded Middle. When applied to true and false, is a foundation of boolean logic, but it's more commonly used to talk about sharp boundaries around characteristic than truth values. (e.g. that object is a Foo or it isn't, the middle is excluded). Many other truth models exist, the most famous of which is "fuzzy logic" where truth values range from zero (completely false) to one (completely true). Those different representations of truth are important, but not relevant to the problem with this exercise.
I am disappointed with your comment about standard mathematical practice. In the professional practice mathematics and philosophy of logic, this prohibition is absolute, and your question about subtle loops is generally dismissed as trivial. The lattices of reference are generally quite simple, even in the most sophisticated proofs.
If you allow self-referential truth values, you get paradoxes, as your example shows. Transitive self-reference ("loops," as you call them) like your example is just as bad as direct reference like "This statement is false." If you try to make that statement either true or false, you get a contradiction. It is a worthy and entertaining exercise to play around with paradoxes -- just be sure you are teaching the underlying lesson. I don't think this example does that.
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@Mel Nicholson – Many other truth models exist, the most famous of which is "fuzzy logic" where truth values range from zero (completely false) to one (completely true).
Indeed, and I even mentioned other truth models? I did not mention fuzzy in particular but it does exist. My option (1.) is classical logic. This is what the vast majority of mathematics is built on.
In the professional practice mathematics and philosophy of logic, this prohibition is absolute, and your question about subtle loops is generally dismissed as trivial.
We can do things like produce hierarchy of classes in order to avoid self-reference paradox, but these structures are only established with great care and difficulty. We certainly can't just ignore them, the prohibitions have to be built into the structures themselves. There are many ways to create a set theory in a "wrong" sense.
re: The lattices of reference are quite simple -- I disagree. There are even paradoxes like Yablo's with no apparent self-reference at all, and a complete graph-theoretical classification of reference structures that have paradoxes is a known open problem.
Self-reference is only meaningless in the sense that if your particular set theory statement just doesn't make sense; i.e. you have something contain a member of the "containing a member" symbol. Certainly self-reference can be contradictory, but I have never heard any mathematicians call it "meaningless". (A philosopher or two, but they weren't working with standard mathematics.)
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@Jason Dyer – @Jason Dyer I've literally worked on the math curriculum standards for the state of California, and your statements are not standard by that metric, no matter how many times you claim otherwise.
Yablo's paradox is about diagonalization across an infinite sequence. Priest and Yablo argued at length about whether it was an instance of circular reference. The good news is that we don't need to rehash that argument. There are lots of meaningless statements that have nothing to do with circular reference. I'm saying you made a mistake, not that this is the only possible kind of mistake.
Here is a simple demonstration that your logic doesn't work.
If I were to ask you if your logic was broken, would your answer be the same as the answer to this question?
@Mel Nicholson – Who needs rohipnol when u have the power of logic
This is a very silly question. We are told Nick is dating somebody else. So his likely answer to the request for a date would be 'no'. Creating a convoluted question to secure a date as a logical necessity is ridiculous. Regardless, we need more information because nothing in the question consists of Melissa actually asking for a date. Asking about his hypothetical response isn't the same as asking for a date.
There was no question posed to begin with, therefore, "Not enough information".
I think the answer is not enough info, because the question does not say if Nick was able to express what he wanted to say. I that assumption was explicitly mentioned as "he was able to say what he wanted" I would choose yes.
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It says "assuming he responds truthfully in a yes or no fashion" which covers what you mention.
He could say "Yes, but I already have a girlfriend." That would be as honest as possible. Do they go on a date? I'm betting 'no'.
this question defies logic
But the question actually asks "Assuming he responds truthfully in a yes or no fashion, will Nick GO on a date with Melissa? "
For that, there is not enough information (he is coerced into both giving an answer and the value of that answer) and "truthfully" does not cover that situation.
If the question were "will Nick SAY HE WILL go on a date with Melissa?" then the answer is YES.
He is not even allowed to refuse!
IF the answer is the same, then Nick would say: "yes, it would be the same", thus it would be yes to the date.
IF the answer is not the same, then Nick would say: "no, it wouldn't be the same", thus it would be a yes.
Either way Nick is saying yes to the date!
and they said you couldn't pick up girls with math
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They lied lol
That's true, nerd boys still can't.
The question is simply a nested repeated question, where "negative x negative" = "positive x positive" = positive. However, given the preamble to the question, Nick is unable to answer, whatever his intent:
If Nick does NOT want a date with Melissa, then he cannot "respond truthfully in a yes or no fashion", since both answers lead to the same (undesirable) outcome.
If Nick DOES want to cheat on his current girlfriend, then "yes" or "no" both lead to the same (desirable) outcome, but he cannot choose "truthfully" between yes and no.
Assuming he doesn't say, "I'm gay".
But wouldn't the real response be "What question?". She didn't actually ask a real question. Its like asking the question 'Is the statement "this statement is a lie" true or false?'. I would argue the answer is neither. The statement has no sensible truth value because the referent of "this" is not well defined. Its like doing a mathematical proof where you divide both sides by zero to prove 1=2. (for example 0 1=0 2 so 1=2). Division by zero is undefined so the moment you do it your proof is meaningless. Similarly in the above "this question" is undefined so there is no meaningful true or false answer to the question.
if nick says no to Melissa then he will have to date her, if he says yes he will have to date her. So he really has no choice but to date her.
(Tried to add this as a separate comment, but I'd already answered the question so couldn't - had to comment an existing comment to add this):
The question says that Melissa is "oblivious to the fact that he is already dating someone else". That isn't enough information: perhaps he's dating someone else and has an open relationship, or is otherwise not fully committed to the woman he's dating. The only real answer we can give is "Not enough information ."
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It doesn't matter whether Nick is in a polygamous relationship or not. Right now, Melissa's convoluted question forces Nick to have no choice but to respond with "yes."
If Nick does not want to date her, he cannot answer question honestly which is a condition of the whole problem. Dividing by zero here.
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No division by 0 here.
Yes, Nick doesn't want to date her, but the question is not about whether he wants to date her or not, but it's about whether what option does Nick has to answer Melissa's convoluted question.
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I might agree except for the demand that Nick 'answer honestly'.
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@Richard Desper – Nick's response of "yes" implies that he will go on a date with her (in fact, he doesn't have a choice to say no). Whether he wants to or not is irrelevant here.
The obvious answer is the one not offered, that is to just walk away and not answer the question because she is a jerk.
Melissa is quite the crazy woman.
See, if Nick answers "Yes", then both answers will be same.
So he has to say "Yes" to the date.
But if he says "No", then it does NOT mean that he won't go on the date! This confused me at first as well!!
Nick saying "No" only amounts to the fact that both his answers will not be the same.
Which implies he HAS to go on the date, because his next answer CANNOT be a No, so he's forced to go.
Hopefully his girlfriend will allow him to explain, bwahahaha!
It's a funny problem!
if he says yes then that means that nick would agree to go with her if he says no that means he still agrees
What a confusing but romantic way..
Melissa asked Nick if his response to her request for a date would be the same as his answer to the question she was asking at the time. If Nick responded with a yes, then his answer would be the same and so he would agree to date Melissa. If Nick responded with a no, then his answer would not be the same as a no and so he would still agree to date Melissa since we are under the assumption that he responds in a yes or no fashion.
Because Nick responds in this binary fashion, Melissa is guaranteed a date with him however awkward it may be considering the fact that he is already dating someone else.