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:
Implies prv(D) and prv(prv(D))
Implies prv(D and prv(D))
Implies prv(D and not D )
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