In this case, I actually mistakenly thought I was offering the first attempt at a solution. I had the ILP tab open for too long, so I didn’t see the page and a half of responses. And I think Arminius’ solution is much better than mine anyway. Radians?
Probably only for James; tabbed because I don’t think anyone else would care (which is not to assume that James still cares):
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As with the Pythagorean Theorem, it will be sufficient to show that a solution follows deductively from the premises.
Here is a new, much simpler syllogism for proving the SR portion of the problem.
- The colors each logician can see are part of the set C of known correct answers to the question, “what color is my headband?” (given)
- A color cannot be deduced from a set of colors (given)
- Therefore, a color cannot be added to C based on the other members of C (from 2, general to specific)
- For the problem to be possible, a logician’s headband color must be a member of C (given)
- Therefore, if the color of a logician’s headband is not in C, then the problem would be impossible (from 3, 4)
- The problem is not impossible (given)
- Therefore, the color of every logician’s headband must be a member of C (from 5,6)
To preempt a possible objection, the second given may be contentious. However, I think it’s true, particularly if you consider the distinction between deductive reasoning and educated guessing. Your examples, such as the color wheel and pattern examples, involve at best educated guessing or scientific induction (as opposed to mathematical induction, which is a deductive method). More generally, colors, even ordered colors, do not bear logical relation to one another. One would not look at circle of people with headbands colored green, red, yellow, green, red, yellow, green, blue, and say that is logically inconsistent for them to be sitting that way. This is true even if the floor were giant color wheel or otherwise implied a pattern with the headbands.
All the work here is done by 2 and 6. I think this is basically the ‘certainty’ argument that Phoneutria offered earlier, but jazzed up with sets and provability statements. It comes down to that you can’t derive a member of a set only given the other members, so you can’t deduce any additional correct answer not already known to be a correct answer for at least one logician.
Tangential: this blog post about common knowledge, including the Blue Eyes problem (cast as the Muddy Children problem) and other considerations.[/tab]