Practical Application of the "Liar's Paradox"
jerry kraus
definable in English.
Proof. Suppose it is. Then so is its complement
"False".
Let s be the sentence "This sentence is false" .
Since the phrase "This sentence" refers to s, we
have
s iff "This sentence is false" iff "s is
false" iff not s.
A contradiction.
This concept, presented as a formal logical
presentation of the traditional liar's paradox "if the
liar says he is a liar, then he is telling the truth,"
can be applied directly to counselling psychology and
conflict resolution.
One of the biggest "relationship breakers" is the
statement "I know I'm right!". There is no effective
response, it cuts off all argument or discussion. An
informal presentation of the liar's paradox
effectively proves that there is no absolute truth:
"So, you know you're right?"
"Certainly."
"Then, you know when you're wrong?"
"Of course."
"Then, when you're wrong, you're wrong about it?
"What? No, when I'm wrong I'm right!"
"That doesn't make any sense."
John Jones
jerry kraus wrote:
If you presume erronously, like Godel assumed, that a sentence can
reference 'itself' then you may reach the conclusion you have worked
toward.
LauLuna
Perhaps what you mean is that the truth predicate for English sentences
is not EXPRESSIBLE in English. If so, it is evidently not true;
consider '' 'the snow is white' is true''.
Your proof fails because it assumes that the Liar sentence expresses a
proposition, so that it is possible to reason about it as about any
proposition. This is not the case.
Of course, when one is wrong one cannot know one is. But this does not
imply one cannot know that one is right when one is.
What if your character replied: 'Are you sure you are right when you
say that this does not make any sense'?
Regards
LauLuna
jerry kraus
The concept of mathematical proof itself is, of course, subject to
these same considerations. If my character replied, as you suggest, I
would say "this seems inconsistent, by the rules of logic. But, there
can be no absolute certainty, on any point. That IS my point."
We can no more know that we are right than that we are wrong. Our
knowledge exists in the context of an apparently infinite complex of
interrelated facts in the universe around us, and we can never know to
what extent our convictions are actually consistent with this infinite
complex. We can only guess, on the basis of simplifying assumptions.
Logic is one of those assumptions.
If you can read a bit of French, and are looking for something
completely different, as the Monty Python crew used to say, try my
attempt to generalize these priniciples into a mathematical formulation
of truth for experimental purposes, in science, using summation of sums
of products, at the site below: