10. Paradox and the Subjective Frame

     Epimenides is the infamous Cretan who declared that "all Cretans are liars". The modern form of this paradox is the sentence:

(1) 'This statement is false'.

The sentence refers to itself and also negates itself. If it is true, then it must be false. If it is false, then what it asserts is true. Now a statement, P, is in effect the assertion that 'P is true'. This constitutes another statement, P1, which in turn is equivalent to the assertion that 'P1 is true'. Since this process can be carried on indefinitely, any statement is equivalent to an infinite recursion of statements about statements about statements... Thus:

'This statement is false' = "'This statement is false' is true" (contradiction).

For assertions that do not self-refer, there is no problem. If the statement is true or false, every recursive version of it is also true or false, accordingly. But for the self-referring statement, the situation is different when the statement is self-negating as well. Each recursive step reverses our verdict concerning its truth. But then consider:

(2) 'This statement cannot be proven in system X'.

Statement (2) is the formal or relativized equivalent of (1), since it is framed in terms of proof-within-a-system rather than in terms of truth. It contains no contradiction so long as the notion of contradiction has an interpreted meaning as "both true and false in some world larger than system X". Thus:

'This statement cannot be proven in system X' = '"This statement cannot be proven in system X' is true" (no contradiction).

But when the concept of contradiction also assumes a relativized definition, then (2) is a contradiction because it and its negation are both provable (derivable from the axioms of system X). Substituting provability-in-X for truth, (2) means:

"'This statement cannot be proven in system X' can be proven in system X".

And substituting 'this statement' for what it refers to, (2) becomes

"'This statement' can be proven in system X".

Therefore if (2) can be proven, then so can its negation, and so (2) is a contradiction. In other words, (2) is not a contradiction so long as there is a method of evaluating truth which lies outside the system. But it is a contradiction when it must be evaluated strictly from within-- in terms of provability rather than truth.[9] We could generalize to say that contradiction or paradox can arise in subjective consciousness, as it can in self-referring statements and logical systems, when one is limited to a closed framework of thought. To transcend the terms of the paradox, it is necessary to expand into a larger system in which the contradictory theorems can be evaluated as "truths". The possibility of self-reference generates paradox in the first place, and also provides the way out.