“Scratch Notes on Quantum Logic, Turing, Goedel, Nash, Cell metabolism, N-tables, B-Tables, parts on Feynman for angular momentum probability area calculation of electrons in cell metabolism”

© William Alexander Patterson IV

All rights reserved

22.11.2012

B-Table

[

]

→ Q

N-Table

[

]

Q →

The distinction shows the difference between a Boolean truth-table and a normative truth-table. An N-

Table harmonizes with everything from Spinoza’s Ethics to von Wright’s Contrary-to-fact-

conditionals, his solutions to the Grelling Paradox, to his work on Modal Logic, to Quine’s Natural

Kinds, to Searle’s Intentionality, to Wittgenstein’s On Certainty, to Boyd’s Modal and Auxiliary

Verbs, to John Austin’s Behabitives and his Plea for Excuses.

The defining element of an N-Table is its proximity to natural laws and to natural law, from Quine’s

Natural Kinds all the way over to Alfred Landé’s Geostatistical Scattering of Particles with his Knife-

Edge filters.

B-tables correspond to Chomsky’s Linguistic Structures, but not to his exposition of von Humboldt in

Cartesian Linguistics. The latter fact is strange. The same strangeness applies to Carnap’s work on

linguistics.

I think this strangeness can be accounted for by looking back to Wittgenstein’s Tractatus.

Wittgenstein’s simple statements about the containment of a in b (‘… is already contained’) states a

logical singularity. W did not at that time see that fact. He did not identify it as such since to do so

would be to admit the strangeness, which was not his aim.

The fact is that a displacement in W’s logical singularity explains the strangeness of those who have

used truth-tables as B-Tables and would never even dream of affirming that they anything whatsoever

to do with N-Tables; that would be taboo.

The best place to see this displacement is in Quantum Logic. That is because the normal divisibility

relation does not hold there, yet the classical mechanics of B-Tables is the medium.

I’ll show this quickly:

3i. Why is it the case that by adding non-commensurability to classical truth-functional (Boolean) logic creates immediate statements of a quantum logic—as illustrated by Tanaka,

1. "W is a quantum world def. (( a, b) (a W & b W ) (aDb bDa)) (( a, b) (a W & b W & aCb))?

2. Because the divisibility relation D cannot be sustained universally for terms a,b without the existential assertion ( a, b) (a W & b W & aCb).

3. Without this existential statement logic could not account for the conditional nature of reality and even grammar.

4. This is a different way of saying that the existential assertion (( a,b) (a W & b W & aCb)), as the qualification that makes the universal assertion (( a,b) (a W & b W ) (aDb bDa)) meaningful for actual states of affairs, is a factual statement of non-locality.

Note that Tanaka is an anti-atomist and subscribes to Whitehead’s "individuals."

Briefly, the so-called ‘classical’ divisibility relation D must accommodate another divisibility

relation, namely C. This is a displacement of D under quantum conditions. It is a displacement of a

logical singularity.

Looking at this in terms of the Propositional Calculus we can look at the ingenious solution that von

Wright gives for the Grelling Paradox, otherwise known as the Heterological Paradox.

3.i.j. C. The classical situation. (Px Px) (Px Px). In generality for all things: ( x) (Px Px). The latter can also be written as: ( x)(Px Px). To express this for any

property, we can quantify with the variable P: (( P)( x) (Px Px)) ( x)(Px Px). The latter can also be written as: ( P)( x)(Px Px).

3.i.j.j. CM. Von Wright's calculus for the modification of the classical situation. The symbols are truth-connectives (, , etc.), an unlimited number of T-symbols, and an unlimited number of P-symbols (Property symbols). An atomic expression of a complex of T-Symbols in quotes, standing immediately to the right of a P-Symbol. A molecular expression is a complex formed by one or several atomic expressions by means of truth-connectives. An expression is an atomic or molecular expression.

3.i.3.j. CM. The axioms are a set of axioms of the propositional calculus (with atomic expressions of the calculus presented instead of propositional variables). The theorems are any expression which may be obtained from an axiom or theorem by (I) substituting a T-Symbol in the axiom or theorem for another T-Symbol throughout, or for a P-Symbol for another P-Symbol throughout, or (II) detachment (modus ponens).

3.i.3.j.k. M. An introduction of the Greek letter does necessitate a modification in the rules as so far stated. The logical space in the calculus is exhausted (collapsed) and the Greek letter benefits from this by the assignment of a new definition to a symbol. The introduction of the Greek letter is a definition of the symbol through the identity of 'X' X'X', where X is a P-Symbol. From this definition we can derive from any theorem of the calculus a new theorem by substituting—not necessarily throughout in this case— for parts of X'X' which occur in the theorem, parts of the form 'X', or vice versa.

3.i.3.j.k.k. M. The modificational definition from the paragraph does indeed demand a new clause to the definition of a theorem. Because a theorem of our calculus is the expression A'A' A'A' (or X'X' X'X'). Where we substitute 'A' for one occurrence of A'A', we obtain the theorem A'A' 'A'. But because of substitutability we may in the last theorem substitute for A throughout, thus obtaining the theorem '' ''. '' 'is a contradiction. Von Wright simply says that we could call this the Heterological Paradox.

3.i.3.j.3.k. M. This substitution was not in any way permitted in the calculus by the rules of C and CM. The rules said that for a P-Symbol in a proven formula, another P-Symbol could be

as a P-Symbol with regard to substitutability. That this substitution occurs, von Wright says, can be

-Symbol of the same kind (type, category) as the P-Symbols" of C and CM.

Again, a displacement of the classical singularity of a proposition, or a P-Symbol and its argument.

We can go further and integrate the two forms of displacement, i.e. the one in Quantum Logic and the one in von Wright’s work on the Grelling Paradox.

3.i.3.j.3.k.l. This allows us to write ( '' '') ( A'A' A'A'). This is not

equivalence relation between the two complexes ( '' '') and ( A'A' A'A'). It is a mapping between the two. It states that we proxy modulation to a classical state of affairs, by a proxy function to a background theory with supremely evolved rules for theorem derivation that do not defy but order the observed phenomena of non-commutating observables.

Tanaka's existential assertion ( a,b) (a W & b W & aCb) could be changed to ( a,b)

(a W & b W & aDb). As formality, ( '' '') f : ( A'A' A'A').

One of the best ways to end this small paper is to try to give an account of logic. This is my account:

Scratch Notes on the logical connectives and the logical operator viz. the Gödel Sentence with a smidgen of Von Wright

1. Logic is natural. It is made up of elements. Logic is not a natural science. It is elementary.

2. Logic is the basis of algebra. Algebra makes the exhibition of geometry possible.

3. Logic’s elements are: PX,

3.1. The quantifiers.

3.2. PX. The propositions and their characteristics, or arguments.

3.3. The logical connectives and the single logical operator.

4. There is no such thing as a lower or higher order logic. There is one unique logical calculus. Its

logical sum is PX,

5. Any element of PX, is syncategorematic except for X.

6. Any element of PX, is not syncategorematic if with X, including X itself. That

contradicts point 5. It simply says that X is neither syncategorematic nor asyncategorematic.

Alone, it stands alone. Any other element that is alone, stands in need of X.

6.1. This is important for ground rules.

6.2. Ground rules are not important for the Gödel-sentence.

6.3. Ground rules are important for the function of the Gödel-sentence. Therefore they are

important for the Gödel-sentence. That contradicts point 6.2. By transmission however 6.3

is normative and therefore not contradictory of 6.2.

PX, ). entails P.P, ξ.P, ξ.ξ, ¬ξ.ξ, ¬P.P], Then Theory.Gödel-sentence IFF World.

I see things as a unified whole. I see all of this as a unified logical whole. I see nothing that can really

destroy my asserted unity except for the roughness of the style of exposition.

Later: A metabolic process cell will be discussed later in terms of QM brain

chemistry-functions, at moment not enough time yet, but it will use Feynman’s

angular momentum probabilities by circling for locating electron and photon location