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What Paradoxical Statements really are

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Most statement that are not mathematical equations have a truth value depending on time and often even more context. Mathematical logic should be generalized to cater for this.
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  • Added: October, 09th 2011
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  • Tags: paradox, mathematical logic, by gebhard greiter
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What Paradoxical Statements really are
Gebhard Greiter, 2011
The most simple example of a paradoxical statement seems to be
S: This statement S is lying.
If we assume S is lying, then S is not lying; but if we assume S is not lying, then S is lying.
So, what is wrong here?
The problem here lies in what we think a statement is: Under a formal point of view, a state-
ment may be true, may be false, or may be empty. The statement
Alice is pregnant
for example cannot be assigned a truth value (true or false) as long as we do not know of
which person Alice exactly we are speaking. So, at least now, it is an empty statement.
We see: Depending on the context c in which a statement X occurs, the statement's truth
value v(c,X) may be true, false, or undefined. The context in this sense is at least a point in
time, but may also contain definitions of the objects the statement refers to.
Formal logic usually ignores this fact, and so mathematicians see S as a problem. I do not:
Paradoxical statements are statements X for which there is no context c
such that v(c,X) is different from empty.
Now we prove that the statement S given above is paradoxical in the sense of this definition.
If not, a context c would exist, such that v(c,S) is true or false and is solution the equation
v(c,S) = v( v(c,S) == false )
In other words: One element x of { true, false } would be solution of the equation
x = v( x == false )
This equation however is satisfied neither by true nor by false (read == as "is identical to").

What we have learned is: The fact that paradoxical statements exist is telling us that to have
only two truth values (true, false) is not enough; we also need a value undefined.
We also see that mathematical logic should be generalized to
no longer ignore the fact that mapping statements to boolean values is not enough:
Reality is mapping statements context-sensitive to { true, false, undefined }.
The fact that mathematical logic as you find it in text books is ignoring context (especially
time) and knows only two boolean values, may be a consequence of the fact that at least
mathematical statements can only be true or false and are so independent of the point in
time we look at them:
Mathematical laws are valid in every context.
Other statements however do not have this nice feature - maybe not even statements that
describe physical laws: Researchers believing in String Theory are no longer sure that our
universe is the only one, and they tell us that, should other universes exist, the physical laws
valid there may differ from the physical laws valid in our universe.
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