rec.puzzles archive logic/hundred

>>From http://rec-puzzles.org/sol.pl/logic/hundred

The puzzle is 'A *** of paper has statements numbered from 1 to 100.
Statement n says "exactly n of the statements on this *** are false."
Which statements are true and which are false? What if we replace
"exactly" by "at least"?'

The "solution" explains that the plausible answer to this is not
correct, as the line of thinking assumes that every statement is either
true or false, and ignores paradoxical (ie, "this statement is false")
and vacuous (ie, "this statement is true") outcomes.

It further claims that if these other logical states are allowed, the
problem becomes unsolvable ("nothing whatsoever can be concluded about
which statements are true or false.")

I take issue with this.

In formal logic, from what I can recall, it is a valid method to make a
"truth table." List all the possible combinations of truth values for
your statements, and see which combinations are consistent and which
lead to a contradiction. If all possible combinations lead to a
contradiction, then your system is paradoxical. If all possible
combinations are consistent, then your system is vacuous (or whatever
the term is -- tautological?).

So "meaningless" results come about when you first *assume* that your
system has one consistent set of truth values, and that assumption is
proven wrong. If the assumption *isn't* proven wrong, and you find a
consistent set, then you've solved the problem.

Therefore, I submit that it's not a problem to assume that all
statements have a defined "true" or "false" value. If they don't, then
you will not be able to find a consistent answer to the problem, and
you'll have to conclude that the problem is paradoxical or whatnot
(such as the case where there is only one statement on the pad of
paper). However, if you can find a consistent answer to the problem
(like the "plausible" answer described in the above web page), then you
can rest assured that your assumption was true, and the consistent set
is at least one of the solutions.

In short, the plausible answer, since it is self-consistent and
consistent with the definition of the problem, is a valid solution of
the problem. If truth tabling is good enough for formal logic, it
should be good enough for logic puzzle solvers, no?

Ken

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