*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

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