## 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)

Therefore
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