Monday, March 3, 2014

A Proof of Self-Proof; 6 of 15

        Evaluating Four Quanta

          When we try to prove which of the four quanta are true and which are false, we encounter a logical difficulty. If we assume that truth equals provability, then the quanta yield equations that are either paradoxical or void:
          If    prv x  =  x   for all x   then
          D = not D;  S = not S;  P  =  P;  T  = T
          It’s like dividing by zero; the result is either indefinite or catastrophic. D and S are paradoxes; they do not fit into two-valued Boolean logic. Any Boolean system collapses when it tries to evaluate these sentences.
          Perhaps there are non-Boolean logics, with a place for paradox, which then might evaluate these quanta differently; but that is a topic for another paper. In the context of dualistic logic, the existence of self-doubt and self-shame imply Tarski’s Theorem:

                    Proof is not the same as truth.

          So there may be unprovable truths; or worse, provable falsehoods. So is prv F true or not? Is its negation, poss T, true or not?
          If prv F is true, then the proof system proves anything, including D:
          Prv F           implies         prv D
          But the reverse implication is also valid:
          Prv D
          Implies         prv(D) and prv(prv(D))
          Implies         prv(D and prv(D))
          Implies         prv(D and not D )
          Implies         prv(F)

          Prv F           =        prv D           =        not D           =        S
          So the inconsistency of logic is equivalent to the quantum of self-shame. For logic to collapse would indeed be a shame.
          Negations of the above equations yield:
          Poss T          =        not prv F      =        not S            =        D
          The possibility of truth is indeed dubious. Our reasoning might be void.
          D and S are negatives; which is true? If D is true, then what it says is true, namely that it is not provable, and it would be an unprovable truth. If S is true, then we prove absurdities. Therefore either we do not prove every truth, or we prove absurdities. That is Gödel’s First Incompleteness Theorem:

          Any logical system is either incomplete or inconsistent.

          D is the incompleteness option, S is the inconsistency option. Of the two, D is preferable, but to prove it is to negate it. Thus, Gödel’s Second Incompleteness Theorem:

          If a logical system proves itself consistent, then it is not consistent.

          No logical system that proves it has a model really does have a model! If all is well, then D and poss T are true but unprovable, and S and prv F false but irrefutable.
          The dubious nature of poss T has a catastrophic effect on the self-constructing statement P = poss P.  Self-pride declares itself possible, that is consistent; therefore any logical system that believes P will believe that it has a model. But by Gödel’s Second Incompleteness Theorem, that is inconsistent. To believe P yields absurdity; therefore P is refuted:
          P        =        false
          Taking negatives of that equation yields:
          T        =        not P            =        true
          So the Henkin sentence (T = prv T) is in fact true and provable. That is Löb’s Theorem:

          If a statement asserts nothing more than its own necessity, then it is in fact necessarily true.

          So we have derived these evaluations:
          D  =  poss T  ;   S   =  prv F  ;  P  =  F  ;   T  =  T

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