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Synduction, the Genesis of Memory, and the Habits of the Mind
By: Jerry LR Chandler on: Sat 16 of Feb, 2008 [02:28 UTC] (1421 reads)

For several decades, I investigated the dynamics of chemical induction of biological mutations, how is it that genetic changes are induced by chemical events? Eventually, the critical question became: What is the distinction between a list of number symbols as mathematical numbers as used in arithmetic and a list of number symbols as the atomic numbers, each with a unique chemical identity? In the real number system (RNS), the number symbols are associated with geometric distances; but atomic numbers represent electrical particles, not distances. To describe mutation, new logical operations for changing the relations among nuclei and electrons are necessary. (Jerry LR Chandler, Introduction to the Perplex Number System, Discrete Applied Mathematics, in press.) (abstract continued next window.) See paper at:
http://www.blueberry-brain.org/winterchaos/Chandler%20IPN%20v7.0.pdf external link

abstract continues in next window; synopsis of perplex number system in comment 1.

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What is the nature of the new logical operations? Historically, in the European Universities of the Middle Ages, study was separated into logic, grammar, and rhetoric, the trivium, and the study of arithmetic, geometry, astronomy and music, the quadrivium. I will review the basis of Aristotelian logic and the subsequent morphing of the concept of logic in the 19 th century from the trivium to the quadrivium, citing the works of the modist, Thomas of Erfurt, and Duns Scotus, G. Boole, A. DeMorgan, C. S. Peirce, G. Cantor, and B. Russell. Historically, mathematical and physical logic developed centuries before the empirical order of the chemical table of elements came to be. Consequently, the logic of Boole and DeMorgan was independent of chemical logic. During the 20 th century, chemical logic, the logic of the particular, was created from the atomic numbers and operations on the electrical particles of atoms. Chemical logic is self-reflexive with the dynamics of the embodiment of living systems, the biomedical sciences and cognition. For example, it is well known that specific genetic mutations confirm the relationhoods between atomic numbers and genesis of biological functions made manifest by the sequences of macromolecules and the catalysis of chemical reactions by enzymes.

Synduction is a creative logical operation (Greek, syn, with). In order to apply the synductive operation, it is necessary to briefly sketch the structure of the perplex number system (PNS) as it serves as the reference system for the logical operation. The PNS was constructed by abstracting logical diagrams from the electrical properties of the atomic numbers. The diagrammatic logic of discrete perplex numbers is analogous to the diagrammatic logic of category theory for continuous variables.

Synduction is a symbolic logical operation conjoining diagrams. It extends a spatial pattern of symbols by incorporating a new symbol into the pre-existing pattern. The conjunctive operation creates a new whole from pairings of symbols, preserving the root identity of the precursors. In contrast to the syllogistic deduction of Aristotelian logic, which reduces the number of terms from three to two, the meso-syllogistic operation of synduction increases the number of terms from three to four. The positions of the new relationhoods become part of the internal memory of the new symbol, such as within positions within chemical isomers. Meso-sorites, sequences of meso-syllogisms, can be applied to generate any desired pattern of symbols, any relevant logical diagram.

Mathematical order is an abstract principle that describes a linear list of symbols. Order is necessary for both the material logic of the PNS (identity, what, who) and the spatiotemporal logic of the RNS (when, where, and how much of a generic substance (mass)). As noted above, the logical operations of two number systems invoke different processes for pairing of symbols. In applications to natural systems, the two calculations must be conducted separately. When conducting calculations, precedence must be given to the perplex operations on diagrams prior to the symbolic operations on “points”. The habits of the mind may record an event (in terms of who, what, when, where, and how) concomitantly. Consequently, a system of ordinate logics was developed to identify the nesting of neuronal sub-networks in different situations requiring concomitant dynamic changes (calculations). A meta-system of symbols for ordinate logics will be introduced that specifies the separate abstractions used to describe change.

The concept of self- reflexive properties of the perplex number system within living systems opens new opportunities for the study of the dynamics of emergence of life and mind.

(related article on Perplex Number system in comment 1 (bottom of list)

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fredabraham

Logical Inferences in the Perplex Number System

on: Sat 16 of Feb, 2008 [02:36 UTC]

score: 0.00

by Jerry Chandler, supplement to his post, submitted by fred

The electrical progression and the logical operations of the perplex number system are abstract
generalizations of empirical observations. A distinction between these logical operations (transpositions
and circules) on the natural perplex numbers and the logic of arithmetic operations on artificial real
numbers exists. One aspect of this syntactical distinction is asserted in this section. The roots of this
critical distinction arise from the inferential grammar used to associate logical terms of propositional
statements. Aristotelian syllogisms were the first systematic theory of inferences (8). Porphyry, using,
the ten categories of Aristotle, introduced logical trees as a method of categorizing individuals, species, and
genera. Since the circlets of conjunctive operations on perplex numerals are certainly not trees (figures 3, 4,
5), the question arises as to the quantitative logical structures of categorization of individuals, species and
genus (or genera). The syntax of the circule operation creates new modes of inference from conjunctions
that are different from modes of inference of syllogisms. A necessary property of the circule conjunction is
the emergence of new order relationships such that an unlimited number of perplex numerals can be
conjoined in space (see examples in figure four and five.) This conflicts with the necessity of syllogistic
logic to eliminate one term. After briefly summarizing the nature of a syllogism, the logical forms of the
circule and transposition operations will be developed from two term propositions.
A valid syllogism infers a conclusion from two premises 9. (Boolean logic assigns syllogisms to
arithmetic operations.) As a grammatical object, the propositional structure of a syllogism consists of three
sentences – a major premise, a minor premise and the necessary conclusion. Each premise contains two
terms, one term being common to both premises. The entailments of the logical concepts of all, none and
some are related to one another by the classical square of opposition. For example, in the first square of
opposition (termed Barbara), a valid entailment has the following symbolic form.
“All S is P.”
“All P is M.”
“Therefore all S is M.”
The following semantic example illustrates the essential form of a syllogism.
All men are mammals.
All mammals are animals.
Therefore, all men are animals.
The conclusion omits the middle term, the term mammals. The verbs of the three propositions are
predicates. The absence of the middle term from the conclusion removes the relations explicitly stated in
the first and second propositions. In other words, the relation of the conclusion, S is M, is not necessarily
dependent on the intervening term P. The common term of the syllogistic premises is disassociated from
the conclusion. If syllogistic logic were applied to premises with members of the electrical progression as
terms, the principle of the conservation of matter and preservation of parity would be violated by the
elimination of the middle term.
One essential aspect of the perplex number system (as specified in the parity principle, which is a
restatement of the conservation of matter principle) requires that the inferential operations preserve all
components of perplex numerals of the electrical progression. In addition, the transposition operation and
the circule operation, as asserted in sections 3 and 4, preserved all the terms (units and integers) of the
perplex numerals. Consequently, these operations cannot be based on the deductive syllogism or on the
classical form of predicate logic.
An abstract logical principle is embedded in the routine usage of chemical symbols. Formally, the abstract
inference resembles a syllogism. It differs from a syllogism by preserving the middle term in the
conclusion. This heretofore unstated logical principle is termed a meso-syllogism (L. meso, middle). Like a
syllogism, a meso-syllogism is an inferential term logic, formed from two propositions, each containing
two terms, one common to both premises, and a conclusion. The reasoning of a meso syllogism also
reaches a conclusion from two propositions; however, the middle term is not eliminated. The verbs of the

propositions in meso-syllogisms are not predicates in the traditional sense of predicate logic. Rather, the
verb couples two terms together into a pair; the copula creates a new object that contains all three terms of
the premises. Such a pairing may be thought of as an associative pairing, as a gluing, as an attachment, or
as creating an adjacency relation between terms of the propositions.
Let A, B, and C be symbols of three independent terms. Then, an example of a meso-syllogism is stated in
the following three sentences.
A is adjacent to B.
B is adjacent to C.
Therefore, B is adjacent to both A and C.
Symbolically, we can write the three sentences – the dash signifies adjacency - as
A – B.
B – C.
Therefore, A-B-C.
The semantic and symbolic examples give the logical form of the meso-syllogistic preservation of all three
terms of the two premises. Grammatically, the verb is a copula. Alternatively, the meso - syllogistic
conclusion can be stated as concrete relations implying ordered associations of three terms into an ordered
entity.
Both A and C are attached to B.
B is glued to both A and C.
Both A and C are associated with B.
In all cases, the intent of the propositional statements is to use the copula (verb) to signify the coupling of
components into a new abstract object, a new whole, which inherits the identity of the parts as components
and specifies a sequence for the three terms contained in the conclusion.
The conclusion of a meso-syllogism forms an ordered relationship among the three terms given in the two
premises of the meso-syllogism. This form of associative logic is termed a synduction by analogy of with
deduction, induction and abduction, (Greek, syn -, with.) Symbolically, we can write the associative
relations among the terms as A-B-C or as C-B-A. The order is specified without specifying an origin or a
direction. The inference removes the possibility of concluding that the order of the three terms is B-A-C,
B-C-A, A-C-B or B-C-A.
In the following examples, the exposition extends the associative logic of synduction to the transposition
and circule operations of the perplex numerals of the electrical progression.
The application the synductive logic of a meso-syllogism to a perplex operation starts with the fact the
internal structures of a perplex numerals and numbers exist as inferential pairs of parts. (This is explicitly
stated in the diagrammatic logic.) Each term will be a pair that is glued together from two sub terms. Now,
consider a meso syllogism formed from a pair of premises when both premises contain a pair of sub terms.
That is, the starting terms are a pair of pairs (POP.)
Let the starting terms be A-B and C-D.
We form a proposition by coupling the terms to compose a meso-syllogism:
Sub term B of the term A-B couples to sub term C of the C-D term.
Sub term D of the term C-D couples to sub term A of the A-B term.
Therefore, the cyclic association is created, -A-B-D-C-, or,
A – B
| |
C – D

Analogously, a meso-syllogism can be applied to a pair of perplex numerals.
For example, let
1(a) – O (a) and
1(b) – O (b)
be a pair of the first perplex numeral. (The indices a and b signify only that two distinct representations of
the same perplex numeral are being considered.)
We write the two premises of the meso-syllogism as:
1(a) attaches to O (b).
1(b) attaches to O (a).
Therefore this synduction creates a circlet with alternating terms of units and integers.
O (a) – 1(a)
| |
1(b) – O (b).
Thus, the quaternary circule operation is a meso-syllogism on a pair of perplex numerals where one unit
and the integer are from each member of two independent perplex numerals. Note that this meso-syllogism
has the desired attribute of preserving the parity of the primitive pair of terms. A circlet, as defined in
section 4, is the consequences of the synductive operation between two perplex numerals. In practical
terms, a meso-syllogism on two perplex numerals defines the order of covalent bond formation.
The transposition operation is also dependent on statements that are expressed as copulated propositions.
The propositions are stated in terms of the actions of detaching and attaching. Let the two premises be that
with three terms: perplex numeral (A), perplex numeral (B) and one of the units glued to a perplex numeral
A.
Numeral A detaches one unit.
Numeral B attaches one unit.
Conclusion:
Numeral A and Numeral B have changed by one unit.
The synductive operations of the perplex number system mutate or transform illations between and or
among numerals. In this introduction, only first order (transpositions) and second order (circule) operations
were introduced. Extension to higher orders is obvious. In addition, the empirical observations show the
potential for very high order operations (catalysis).
The order of the synductive logical operations illustrate the deep structure of the perplex number system
lies in the conceptualization of change (mutation) as a consequence of the interior mode of extension of the
electrical progression.

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