Re: rec.puzzles archive logic/hundred

Canon wrote:
> I don't think formal logic says "you first 'assume' that your system
> has one consistent set of truth values, and if you find a consistent
> set, then you've solved the problem."

Well, probably not. I never actually studied higher-order logic
formally.

>
> Here's an example from http://rec-puzzles.org/new/sol.pl/logic/boxes:
>
> Two boxes are labeled "A" and "B". A sign on box A says "The sign on
> box B is true and the gold is in box A". A sign on box B says "The
> sign on box A is false and the gold is in box A". Assuming there is
> gold in one of the boxes, which box contains the gold?
>
> The only consistent set of truth values that can be assigned is that
> both signs are false, which then forces the conclusion that the gold is
> in box B. Unfortunately, the gold is in box A, so there is something
> wrong with the reasoning. The problem is that just because there is a
> consistent set of truth values does not guarantee that the statements
> are meaningful. Tarski showed that no meaning can be assigned to
> statements that discuss their own truth value. In other words, "Snow
> is white" is true if and only if snow is white. "This sentence is
> true" is neither true nor false; it is meaningless.

I think the problem above is a strange one. You're given two signs, but
no indication that they're actually *true*. In fact, they cannot be. In
further fact, the only consistent solution is that they're both false.
The problem also never says that the signs actually have anything to do
with the real position of the ball -- so, of course, the ball can be in
either box and the *problem* is meaningless.

In other words: the problem with the problem is that the ball's
position is independent of the logical system provided by the two
signs. If it's in position A, the system is a paradox. If it's in
position B, the system is consistently false. But neither of those
cases stop the ball from being in either position.

Now, if the problem had stated that "the ball's position is consistent
with an assignment of logical truth or falsehood to these two
statements" then it would have to be in box B for that statement to be
true.

Back to the statements-on-paper problem, though: this problem
specifically asks you which statements are TRUE and which are FALSE.
The hidden instruction here is that it's asking you which statements
are TRUE and which are FALSE to create a consistent result. Therefore,
if you find a consistent result, you've solved it. If you wanted to,
you could rule out vacuous statements like "this statement is true" by
taking each statement on the paper and showing that it *could*
contradict something. The fact that you can find a consistent result
rules out a paradox. The statements on paper don't refer to any
independent facts about the world, and are self-contained. So again, I
claim that the "plausible" solution *is* the solution, and that the
explanation on the rec-puzzles archive web site is wrong.

Ken

.

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