James S Saint wrote:Much like the Blue-Eyes puzzle, if a member cannot prove that a proposed algorithm is the only possible algorithm for what he sees, he cannot use that solution.

This isn't true. If you can prove both pieces of inductive logic you don't need to prove a third piece that says, "and there is nothing that would confound this logic."

For example, in the Blue Eye problem, it's provable that

1) if N blue eyes will leave on day N, then N+1 blue eyes will leave on day N+1

2) 1 blue eyes will leave on day 1.

If those are both true, nothing else needs to be shown (and I phrase this in the conditional because I'm more interested in the general form of the logical argument than I am in actually rehashing the discussion we had of the Blue Eyes problem).

James S Saint wrote:In logic, assumptions can only be made in order to checkout the hypothesis of them being required facts. Assumptions cannot be used until they are proven to not be assumptions, but necessarily true.

I will be pedantic, but logic is nothing if not pedantic: you can make as many assumptions as you like in a logical proof, however you need to eliminate them if you want your proof to show that a conclusion follows from the original premises alone. As in the proof by contradiction, introducing an assumption is a standard logical technique.

James S Saint wrote:Otherwise, you might as well just assume that your color is whatever color you see to your right and just assume that you are right because of it: "Turtles all the way down".

You make a fair point here, but you gloat too much. You have made the bold claim that the Master's statement cannot be true, that the Master is a liar. But again, for that to be true, you need to show that for all possible configurations of logicians, colors, and headbands, there is no configuration that leaves only one logical solution which solves it.

I also think you still underestimate the work that the Master's claim can do. Let's take a toy logic problem with the same premise, "it's not impossible":

A Master and a Student stand in front of a safe that will open if the right number is input on a panel. The Master says to the student, "the number that unlocks this safe is either 8 or it is a number of which it is impossible for a human to conceive. You must deduce the number that unlocks this safe. Don't worry, it's not impossible to solve this problem."

The logician unlocks the safe. How did he do it?

This problem seems trivially solvable, but only because the Master told the student that the problem was not impossible to solve. That's real logical work, and the logician's deduction is perfectly valid. If what would otherwise be an assumption is necessary in order for the premise that the problem is not impossible to be true, it isn't an assumption, it is a corollary.

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