For example, taking the operation OR to be defined such that the truth of
statement a OR statement b is [[a]]+[[b]]-[[a]]*[[b]], where [[x]] is the
truth of statement x, and * means multiplication, one can consider the truth
of the statement "This statement is True OR this statement is False" in the
following way:
Let T be the truth value of the statement. Then "This statement is true" has
a truth value of T, and "This statement is false" has a truth value of NOT(T)=
1-T.
The truth of the overall statement is then T + (1-T) - T*(1-T), according to
the definition of OR above. We then have the equation T=T+(1-T)-T*(1-T).
This simplifies to T*T-2T+1=0 => (T-1)(T-1)=0 => T=1, which implies that
the statement is actually True.
As far as I know, traditional logic has no way of addressing the truth of
this statement. One of the axioms of traditional logic is that any statement
is True XOR False, but that doesn't help in this case.
The truth values of other statements, such as "This statement is false", "This
statement is true AND false", and even "This statement has a truth value
greater than 1/2", can similarly be investigated. Interestingly enough, one
result is that the statement "This statement has a truth value greater than
a half" is completely true - i.e. has a truth value of 1. Another interesting
result is that any statement which asserts that it has a particular truth
value actually has a truth value higher than it claims(except "This sentence
is true").
Does anybody know of any further research that has been done in this area? I
am especially interested in fuzzy interpretations of Goedel's theorem.
Thanks,
Ruadhan O'Flanagan <rof@maths.tcd.ie>