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Topic: Why reasoning-about-reasoning problems are flawed.
Replies: 2   Last Post: Jan 26, 2013 10:50 AM

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quasi

Posts: 10,390
Registered: 7/15/05
Re: Why reasoning-about-reasoning problems are flawed.
Posted: Jan 26, 2013 10:50 AM
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quasi wrote:
>Paul wrote:
>>
>>In what was probably an overly long post, I tried to explain
>>why a genre of maths problems is flawed and I didn't get any
>>responses, so I'm trying again with a shorter post.
>>
>>In this type of problem, there may be two participants A and B
>>where A and B are in the same position. There are two possible
>>scenarios X and Y, and A and B try to deduce whether scenario X
>>or Y pertains.
>>
>>Suppose that if X pertains, this would be readily deducible
>>without any reasoning-about-reasoning psychological logic, and
>>suppose that Y could only be deducible by appealing to the
>>failure of A and B to deduce X.
>>
>>Reasoning-about-reasoning problems claim that A should reason
>>"If X were true, B would be able to deduce it." Similarly B
>>should reason "If X were true, A would be able to deduce it."
>>A then appeals to the lack of a deduction from B. B appeals
>>to the lack of a deduction from A. This lack in deduction
>>is used to deduce that X does not hold and that Y is therefore
>>true.
>>
>>I regard this entire type of problem as being nonsensical,
>>due its failure to make explicit the assumptions the solver
>>should make about the reasoning process.
>>
>>The poser of this genre of puzzle assumes that if X were true,
>>A and B could deduce it immediately. Since the participants
>>have this immediate-deduction facility, why should they not
>>be similarly immediate in following the intended solution to
>>deduce Y? If they only deduce Y after (for example) 5 minutes,
>>then (unless more conditions are given) a contradiction could
>>be said to arise from the fact that the problem wasn't solved
>>(by A and B) in four minutes.

>
>Puzzles typically have lots of implicit assumptions, and often
>this is by design, so as not to make the statement of the puzzle
>too unwieldy.
>
>The issue you describe can easily be fixed by assuming, for
>example, that information and/or decisions are revealed only
>at integer times t = 0, 1, 2, 3 ... and all reasoning by
>participants is done during the time intevals n < t < n+1
>between consecutive integer times.


Thus for the professor puzzle with 2 professors (and one
visiting professor):

At time t=0 the visiting professor makes his announcement
and departs.

In the time interval 0 < t < 1, the 2 professors analyze
the information from time t = 0.

At time t=1, neither professor resigns (not enough
information).

At time t=2, they both resign.

Thus, assuming the weekly meetings occur at consecutive integer
times, and assuming time t = 0 was the time of the last weekly
meeting of the previous year, the 2 remaining professors both
resign at the 2nd weekly meeting of the following year.

With 17 professors (and one visiting professor), the 17
professors all resign at the 17'th weekly meeting after the
visiting professor's announcement.

quasi



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