What is the logical contradiction that occurs in certain self-referential multiple choice questions?

Question

Consider the following multiple choice question:

If you randomly chose and answer to this question, what is the probability that you are right?

• a) 17%
• b) 0%

Note that when I heard about this question, there where more, and more plausible answers compared to how the question is formulated here. However, I'm only interested in answer b) and thus chose for a) some random probability. I hope most people will agree that a) is not possibly the right answer. If you care to see a slightly different formulation and tons of tons of confused people, simply google the question or check out this link.

So back to the question: If you randomly chose one of the answers, then each of them is chosen with a probability (that depends on the number of answers and is in this case) of 50%.

Suppose that there is a correct answers to this question. Obviously a) is not the right answer. Also b) is not right, since, if there is a right answer the probability to choose it randomly is 50%, which is not the value that is given by the answer b). Therefore, to suppose that there is a right answer leads to a contradiction.

Well then, suppose that there is no right answer. If there is no right answer, the probability to chose the (non existing) right answer is 0%, which is the value that is given by the answer b). So b) is the right answer and again this leads to a contradiction.

It seems that b) is right and wrong at the same time which makes me wonder if it is not the question that is ill posed. But what exactly is not ok with this question?

I understand that the status (right/wrong) of an answer actually depends upon which answer you choose, so that this is somehow a recursive problem. Nevertheless, does this make the question invalid?

Is this a known type of question? And what actually interests me the most: Are there other examples of such questions, ideally not statistical ones?

Answer 1

The main issue here is the explicit self reference at a semantic level: the question talks about semantic properties of itself (the correct answer) which depend on what the question says which in turn depends on these properties. This kind of loops can generate a rich set of related paradoxes that you can find here.

Answer 2

These types of unanswerable questions are somewhat related to the Liar Paradox. Consider:

This statement (S) is false. What is the truth value of S?

• a) S is true
• b) S is false

If I pick a, the statement being true leads to the conclusion that the statement is indeed false (S). If I pick b, the statement being false leads to the conclusion that the statement is indeed true (¬S). So the statement is either both true and false or neither true nor false and the question is unanswerable.

The unsolvability of the Liar Paradox question, however, can be easily rectified.

This statement (S) is false. What is the truth value of S?

• a) S is true
• b) S is false
• c) S true and false
• d) S is neither true nor false

We know that S is not true and S is not false (because we'd run into a paradox) so (implicitly) S cannot be both true and false, therefore the correct answer is d where S is neither true nor false. There are various proofs for this sketch. So the Liar Paradox question is, in fact, solvable in our second form.

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I only introduced the Liar Paradox to show how deep unsolvability can go. Your question, however, is a much simpler breed. Consider E1:

The color of this statement (S) is "red". What is the color of S?

• a) S is blue
• b) S is green

Would this question perplex you as much as your original one? Probably not. The correct color simply isn't a choice. This question is just as unanswerable as yours - no more no less. Why your question seems more interesting is because it makes a key assumption (just like the first incarnation of the Liar Paradox question did - namely, that a statement is either true or false).

Specifically, it makes the assumption that the probability of getting a correct answer is 50%. This is incorrect. The probability is, I would argue, mathematically indeterminate. Consider E2:

The color of this statement (S) is "red". What is the color of S?

• a) S is blue
• b) S is red

Consider E3:

The color of this statement (S) is "red". What is the color of S?

• a) S is red
• b) S is red

E1 has a 0% probability of someone randomly selecting the correct answer. E2 has a 50% probability of someone selecting the correct answer. E3 has a 100% probability of someone selecting the correct answer. Unless it's posited that one (or two, or three, etc.) of the answers is/are correct, your assumption that:

If you randomly chose one of the answers, then each of them is chosen with a probability (that depends on the number of answers and is in this case) of 50%.

...is logically unsound. So lets go back to your original question; if you amend your question and posit that 50% of the time you'll hit a correct answer, the answer to the question is simply 50%. The fact that it's not part of the answer set simply makes the question unanswerable (see E1). If you don't posit that 50% is the magic number, then there is no answer.

The problem with the question is that it's is not specific enough, allowing for up to three solutions (0%, 50%, 100%) - therefore, being mathematically indeterminate. The problem with your reasoning is that you start out on the 50% path, but end up on the 0% when you should, instead, say "my brain tells me 50% should be the correct answer, but I don't see it in the answer set therefore the question has no answer."

Note that there is no paradox here (as many people seem to think on the linked G+ thread). Either way, interesting question :)

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Edit: This answer has sprung up a lot of discussion. I believe I am right for several reasons, one being that what is described here could be described as a form of The Problem of Rational Choice - it's a well defined problem in many philosophy papers, including Isaac Levi's On Indeterminate Probabilities (1974). It's also covered by statisticians in some regard, ex: Indeterminate Probabilities on Finite Sets (Robert F. Nau - 1992).

The Problem of Rational Choice: Given a corpus K_x,t and a credal state B_x,t at t, how should X make decisions between alternative policies from which he must choose one at t?

Not to mention that the question reminds me of Bertrand's paradox in which our good friend Bertie tried to make a clever probability paradox out of the ambiguity of the question. Read E.T. Jaynes' reply in The Well-Posed Problem in which he argued for a principle he defined as "maximum ignorance" (incidentally, I've been using a similar form of the argument). Read more about how it defeats Russel's paradox on Wikipedia and elsewhere.

Answer 3

The question is not referring to the multiple choice answer but rather, the question in itself.

What I mean is:

..answer to this question, what is the probability that you are right? ..answer to THIS question, "THIS QUESTION IS HERE"?

Randomly answer that question of what is the probability that you are right. You could answer ANY probability in the entire world of probabilities. that means 1 out of an infinite amount of solutions is the CORRECT answer. 1/infinity = such a small number that we consider it 0. The answer is 0%, its less philosophy than you think

Answer 4

What is questioned? Questioning is where ourselves, must be considered within process, becoming something (agree or fit to the specific answer, solution or something may be considered fully understood to the specific extent rather than previously). If our question was answered, at that point ourselves (whether through perception, imagining or any possible means) were at the state where something (the answer) are fit into our logical thinking (personally), or fit into any possible pattern personally.

It's like trying to meditate (meditating), it's the same as questioning (in process becoming something). And when we are at the specific extent of meditation level, we may be conditioned to something we consider as progress (an answer) to the specific extent (if we have progress - answer).

Back to the question: "what is wrong with this question?" may be understood as "what is wrong with this progress (this process, this direction) of something?" and similar to this. And the answer is, "if the question is answered which maybe considered there is progress or direction to the specific extent", then there is nothing wrong with this question, in fact what we are facing is the answer". But if the progress of something is not running smoothly because there is an obstacle, then the answer is "there is an obstacle answering something, which may not be fixed, may not be shaped, or may not be done to the specific extent whatever it is as requested by the question.

What is wrong with this question?

• There is nothing wrong with this progress, OR
• There is a progress or direction to the specific extent as predicted (or against) the question, OR
• There is something (whatever it is) unacceptable in specific means related to this question (condition).

Asking something can be understood as: we need help to try to figure it out whether, what is questioned may be directed to the specific of something or not (maybe accomplished, may be fulfilled).

• Asking "what is wrong with this question?" may be understood (reasonably) as "what is wrong with this (unsure) condition?, "how can i be possibly fixed to the specific extent, which is doubted by specific condition (questioning)?", "to what direction of specific condition mentioned by the question?" (to nowhere, no answer, or to something, there is an answer, there is solution)

Understanding something (whether it's a question or not) with no relation whatsoever to appropriate possibilities of reality, is the same as understanding something outside reality. It's out of question!

What is wrong with this question? What is wrong with this condition? What is wrong with this request? Does this condition lead to something? Can you help (response) appropriately to this condition?

If you randomly choose and answer to this question, what is the probability that you are right?

• It has to be like this:

  1. And since, there is no fix assertion for certain condition (questioning), then "this question" may be associated to any possible condition (which can be followed up appropriately, whether point to nowhere - no answer, or point to an answer - solution in any possible means relevantly).

     • The answer is : I don't know how to fulfill this question (request) until you tell me (to be specifically) what your request is. IT ASSERTS "REQUEST SYNCHRONIZING" OR

  2. Questioning is asking for solution (completion to the specific extent) in any possible means relevantly, and answering is giving the fact or solution in any possible means relevantly (whether through giving ability to imagine, or point to nowhere - "helping by asserting there is no answer", or to get something or any possible means relevantly).

     • The answer is : This question (request) may be accomplished (fulfilled) in any possible means relevantly (there is an answer, there is an accomplishment whatever it is). IT ASSERTS (just) POSSIBILITIES OF FULFILLMENT TO THE SPECIFIC EXTENT

The percentage is:

• 50%, if there is the right answer using one of the two answers
• 0% if we don't want to answer this question (since there is no way to solve the request - question, because there is no fix assertion about what the question is, to be specifically)

The problem is, when we are treating a proposition to semantic level, there will be paradox, contradiction or ambiguous. But if we consider a proposition, a statement must be related to reality, then hope we can understand something clearly and lead away from confusion, as a start to put something at the correct placement to avoid confusion.