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I have a simple problem that I and my friend conjured up. Suppose we were given a person, Alice, who claimed that she had demonstrable mastery to some (quantifiable) level L of some subject. Is it always possible to construct a problem p equivalent to one Alice can solve (so it is at most as complicated as L)? If so, could we construct a sequence of problems P that (inevitably?) trip up Alice? Consequently, there would always be a time where Alice couldn't demonstrate her mastery, and would thus be wrong. The final question would therefore be, is Alice telling the truth or not? Is there a certain discipline where she could be telling the truth?

I'm new to philosophy but I hope I formulated the situation well enough. Let me know if I haven't.

  • It sounds like a lot of the terms you're using are a little bit vague. Like, you said "master to some quantifiable level L of some subject" but what does that mean? Especially when you later say "is there a subject where she could be telling the truth", you really need to be more explicit about what kinds of subjects you're talking about and what it means to quantify someone's skill in one of those subjects. The answer to your questions entirely rest on specifics and I don't think there's enough in your question right now to imply any specifics. Are you solely talking about math? – Not_Here Aug 10 '17 at 5:08
  • If we just accept everything you've come up with at face value and don't ask for specifics, then sure lets say for any given subject A,B,C,etc. you can have a skill level of 0,1,2,3,etc If Alice's skill level is L then she can only answer questions that are of level L,L-1,L-2,etc to 0. If the levels go on to infinity then Alice will never be able to solve every problem, assuming at least one problem exists at each level. All of that is fine if we don't ask questions, but do you see how that seems really trivial if you don't put in any specific ideas? What does any of what Ive written apply to? – Not_Here Aug 10 '17 at 5:12
  • The part where you try to trip up Alice is where you lose me. Why do you need to construct a sequence of problems? Just look at her level L and give her a question that is L+1 difficult. If she has skill level L then she can only do L and lower questions. But where do you get the question of her telling the truth, what question are we asking her? You haven't said at any point that Alice asserts anything, so what lie? This all is very confusing without specific examples, we have no idea what "is there a certain discipline" ranges over, every discipline that has ever existed? Who has that list? – Not_Here Aug 10 '17 at 5:18
  • But, finally all of this isn't to discourage you from asking the question, (I do think its interesting, although the answer that I see (in my second comment) seems trivial to show), I'm just trying to articulate some of the difficulties that will arise when trying to give you a specific answer to this question. This site tries to aim for as objective of answers as possible and I think what you've written assumes a little bit too much without giving specifics to be answerable in a completely objective way. I don't think I know what exactly it is you mean by a lot of the terms you're using. – Not_Here Aug 10 '17 at 5:21
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    @Not_Here I think your analysis of this question is in a way an answer to it in its present form, and helpful to the OP, so I suggest converting your comments into an answer. To make it "fully-formed" would pretty much transform it into a new question, to be asked separately. – Conifold Aug 10 '17 at 5:50
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As I pointed out in the comments, I think that there are some points in your question that are left fairly vague and to get as concrete an answer as possible would require them to be made very explicit. However, given this problem and taking it at face value I think that we can answer the problem like this:

Let's assume that there are many different subjects, we can call them subject A, B, C, and so on. Each of these subjects has a list of problems that pertain to them, and this list might be infinite. Furthermore, we are able to categorize each of these problems by the skill level of somebody who wants to solve them. The levels start at 0, where anybody at all could solve them, and then move upwards in increments of 1 (0, 1, 2, 3, 4, ...).

If we know of some person Alice and are told that she has a skill level of L in subject A, that means that she would be able to solve any problem in A that is of the skill level L or lower. This means she can solve the problems in levels L, L-1, L-2, L-3, and so on until we get to 0. As an example, suppose she has skill level 5 in subject A, she would be able to solve any problem that requires a skill level of 5, 4, 3, 2, 1, or 0. So far, this all seems to make sense just based off the idea that we want to quantify someone's ability to solve problems in a specific subject while those problems are also quantified in terms of how hard they are to solve. If we only quantify someone's skill and don't quantify specific problems, I have no idea how you could come up with an answer to that question (how else could you quantify skill level?).

Will there always be a problem that she cannot solve? That depends on two things. First, is there an infinite amount of levels of difficultly with at least one problem at each difficulty in any subject? By that I mean, for example, does A have levels 0, 1, 2, 3, ... to infinity with at least one problem? Secondly, can Alice reach higher levels of skill at a subject if she practices that subject? If there are infinite levels to a subject with infinite problems then Alice will never be able to reach the top, there will always be one more above her that she cannot do yet. That's assuming, of course, she can increase her skill with practice. If Alice cannot increase her skill and there is a subject, B, with at least one level higher, L+1, question than her level, L, in that subject, then she will not ever be able to answer that question because she can never get to a skill level of L+1.

So, in answer to the question "could we give her a sequence of problems that would trip her up?" Well, again if we are defining everything in the way above, we don't need to give her a sequence. We only need to give her one question that is L+1 in a subject where her skill is L: give her a 6 when she is at 5. If you want to give her a sequence of problems that are level L, a sequence of problems that are level 5 in our example, then of course she will be able to answer all of those problems, she has a skill level of 5 and they are all level 5.

However, this is where things get very vague and I think a more specific question is required to understand what is going on. If you are asking about how this works in practice, then sure anybody would eventually get tired of doing problems over and over again, let's say math problems, and eventually they'd get one wrong that they could easily do if they weren't tired. Everyone messes up basic arithmetic every once in a while, especially when they're tired. But again, this is subject specific. Are we talking about math? Science? History? Philosophy? Engineering? There isn't an objective answer to the question "can we give her a sequence of problems that will eventually trip her up" unless we know 1) what kind of problems these are, 2) what subject we're talking about, and 3) are we talking about a real person or some idealized person who has a perfect ability to do problems that are at their skill level. A perfect Alice would never mess up a level L problem, no matter how many we give her. A real life Alice would probably mess up a 0 level problem every once in a while even if she's level 5. It depends on myriad factors about the situation and for that type of answer we need to have a more specific question.

This goes double for the question "is there a subject where she could lie about an answer and get away with it?" That completely depends on what these subjects are. Are you asking about the entire list of anything that can fall under the title of "subject"? That would require the entire list of subjects and I don't think any person has such a list. That question is entirely dependent on the specifics of what you're talking about, the domain of discourse we could call it, and it seems that the domain is fairly vague right now.

On the other hand, if we assume some things and take a mechanical approach to this like we did above, then as long as there is someone who can do a problem at level Q, Alice will never be able to fake a question at level Q because that person could just do the problem themselves and see if she got the right answer. Could Alice lie and convince this person? I do not know, that depends, again, on a myriad of factors like "are these idealized people or real people", "do they have completely rational thoughts", "do they ever make mistakes", and so on.

Hopefully this helps clarify any of the problems you were thinking about, and as was talked about in the comments to your question, if you are inspired to reformulate a new question into a more specific problem there will undoubtedly be a more specific answer.

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What is making it hard to address the question(s), is that two different subject matters are being treated as one. The first subject is "mastery level" proficiency. The second subject is the truth value of Alice's answers.

Addressing the "mastery level" questions: yes, it is possible to construct a series of problems that Alice can solve - they would all be at level L (which has been established Alice can solve). You can now give Alice a problem at level L+1, which she will not be able to solve. What this would show, is that Alice's mastery level is still at level L. So she can in fact, at all times, show her mastery is at level L, even if she failed at the L+1 level!
Whether Alice got the answer at the L+1 level right or wrong, has no effect on her mastery at the previous level (L). Also, whether she is telling the truth or not at the L+1 level, is irrelevant for the L level.
There definitely could be many disciplines were Alice could be telling the truth - but would have "nothing to do" with (irrelevant to) her mastery level!

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