Wednesday, August 28, 2024

Part 03: ‘Seldon’s Dialectic Algebras’ Series. Some 2nd-Order-Logic Principles of the Encyclopedia Dialectica 1st Arithmetic for Modeling Dialectics.

 

 



 

 

 

 

 

Part 03: ‘Seldon’s Dialectic Algebras Series.

 

NQ: Some 2nd-Order-Logic Principles

 

of The Encyclopedia Dialectica

 

1st Arithmetic for Modeling Dialectics.

 

 

 

 

 

 

 

Dear Reader,

 

 

 

It is my pleasure, and my honor, as an elected member of the Foundation Encyclopedia Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.

 

This 2nd release in this new such series is posted below [Some E.D. standard edits have been applied, in the version presented below, by the editors of the F.E.D. Special Council for the Encyclopedia, to the direct transcript of our co-founder’s discourse].

 

 

 

Seldon –

 








For more information regarding these Seldonian insights, please see --

 

www.dialectics.info

 

 

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of dialectical artas well as dialectically-illustrated books published by the F.E.D. Press, see

 

https://www.etsy.com/shop/DialecticsMATH

 

 

 

 

 

 

 

 

 

 

 

¡ENJOY!

 

 

 

 

 

 

 

Regards,

 

 

 

Miguel Detonacciones,

 

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

 

 

 

 

 

 

YOU are invited to post your comments on this blog-entry below!

 

 

 

 

 

 

 

 

 

 

 

Tuesday, August 27, 2024

Part 02: ‘Seldon’s Dialectic Algebras’ Series. The Encyclopedia Dialectica first arithmetic/algebra for modeling Dialectic is a NON-STANDARD MODEL of the first-order Peano “Natural Numbers”.













Part 02: ‘Seldon’s Dialectic Algebras Series.

 

The Encyclopedia Dialectica first arithmetic/algebra for modeling Dialectic, N , is a NON-STANDARD MODEL of the first-order Peano Natural Numbers”, .

 

 

 

 

 

 

 

Dear Reader,

 

 

 

It is my pleasure, and my honor, as an elected member of the Foundation Encyclopedia Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.

 

This 2nd release in this new such series is posted below [Some E.D. standard edits have been applied, in the version presented below, by the editors of the F.E.D. Special Council for the Encyclopedia, to the direct transcript of our co-founder’s discourse].

 

 

 

Seldon –

 

We have elsewhere described the Encyclopedia Dialectica NQ arithmetic/algebra for modeling dialectics as an algebraic model of logic – as a contra-Boolean’ arithmetic/algebra modeling dialectical logic.”  

 

“To summarize that description, the NQ arithmetic/algebra opposes Boolean algebraic formal logic’s “fundamental law of thought”, or “law of duality”, which Boole stated algebraically as x2 = x.”

 

“This Boolean law, when its 0 = x – x2 is factored into x(1 - x)  =  0, with (1 - x) representing the Boolean negation of class/category x – thus asserting that any class x, when it “multiplies” the everything class [=1’] less [“minus”] that x itself, equal 0  represents the Aristotelian “law of non-contradiction” in its Boolean form.”

 

Translation: Class x AND class not-x is NOTHING [‘0’].  


Also, the “addition”, or ‘ORing’, of these two “factors” – of x and/plus (1 - x) – asserts the Boolean form of the Aristotelian “law of the excluded middle”:

x  +  (1 – x)   =   1.


Translation: Class x OR class not-x is EVERYTHING [‘1’].

 

“The NQ arithmetic/algebra, it turns out, by contradicting the Boolean “law” x2 = x, constitutes, in particular, an algebraic dialectical logic, whose fundamental theorem is a strong contrary to that Boolean “fundamental law”, and can be stated algebraically as –

 x2 ~<=>  x

because x2  =  x  +  Dx

and Dx ~<=>  x

– wherein the relation-sign ‘~<=>’ signifies a relationship of non-quantitative, i.e., of qualitative – of ontologicalinequality.  That is, the ‘self-product’, x2, produces a net ‘onto-dynamasis’, a new class or kind-category, denoted by Dx.” 

 

“This inequation turns out to mean also that the value of x2, i.e., x + Dx, ‘‘‘diagonally transcends’’’ the “space” in which both x and  Dx inhere, when x denotes a singleton element of that “space”, i.e., a single ‘dialectical meta-number’, so that x2 thus escapes the would-be “closure” of that “number-space”, even though each term within x2 does, separately, inhere in that “space”, and is part of the “closure” of that “space” – both x = x1, and Dx = (Dx)1.” 

 

“The focus of this text, however, is a description of another relation of contrariety involving the NQ arithmetic/algebra.”

 

“In this case, the relation of contrariety is to the “Standard Natural Numbers”, as based upon the four, “first-order logic”, “Peano Postulates”, stated below – the standard core axioms for the first-order “Natural” numbers.”

 

“The phrase “first order logic” means logical assertions which address only the features of individual elements of the axioms-system, i.e., of individual “Natural” numbers, but that do not make any assertions about features of groups of such elements, such as about the qualities of all even numbers, of all odd numbers, of all prime numbers, etc.”

 

“John W. Dawson, Jr., in his 1997 book entitled Logical Dilemmas: The Life and Work of Kurt Gödel, lays out, with careful clarity, the way in which Gödel’s theorems – his Completeness Theorem for first-order logic, and his First Incompleteness Theorem, which applies for first-order logic systems, and also applies for second and higher order logic axiomatic systems of arithmetic – predict the necessary “co-existence” of non-standard models” of the “Natural” numbers, given the “existence” of the standard “Natural” numbers themselves.  He does so as follows –

“Most discussions of Gödel’s proof, … focus on its quasi-paradoxical nature

[K.S.: in that the ‘Gödel Sentence’ “says of itself” that ‘I am not deductively provable from the axioms of the axioms-system within which I have been well-formed’.  Thus, if that assertion is true, then that axioms-system is “incomplete” – contains internal truths which cannot be deduced from its axioms.  If that assertion is false, then that axioms-system in “inconsistent”, or self-contradictory, because a false “theorem” can be deduced – “proven” – from its axions. ].”

“It is illuminating, however, to ignore the proof and ponder the implications of the theorems themselves.”

“It is particularly enlightening to consider together both the completeness and incompleteness theorems and to clarify the terminology, since the names of the two theorems might wrongly be taken to imply their incompatibility.”

“The confusion arises from the two different senses in which the term “complete” is used within logic.”

“In the semantic sense, “complete” means “capable of proving whatever is valid [K.S.: i.e., of proving all propositions that are true within the theory]”, whereas in the syntactic sense it means capable of proving or refuting each sentence [K.S.: i.e., each sentence that is ‘well-formable’ within the theory's syntactic rules] of the theory.”

“Gödel’s completeness theorem states that every…first-order theory, whatever its nonlogical axioms may be, is complete in the former sense: Its theorems coincide with the statements true in all models of its axioms.”

“The incompleteness theorems, on the other hand, show that if formal number theory is consistent [K.S.: if no propositional contradiction is deducible from its axioms], it fails to be complete in the second sense [K.S.: i.e., the theory is "incomplete" because its formal arithmetic cannot deductively “decide” -- prove or refute -- every sentence that is ‘well-formable’ within its first-order logic axioms-system, by means of deduction from its axioms].”

“The incompleteness theorems hold also for higher-order formalizations [K.S: i.e., axiomatizations] of number theory.”

“If only first-order formalizations are considered, then the completeness theorem applies as well, and together they yield not a contradiction, but an interesting conclusion: Any sentence of arithmetic that is undecidable [K.S: i.e., is not deducible from the first-order axioms as either true or false, a truth or a falsity within that theory that is discernible as such by means other than formal deduction] must be true in some models of Peano’s axioms (lest it be formally refutable [K.S.: i.e., refutable in all models of the first-order axioms]) and false in [K.S.: some of the] others (lest it be formally provable [K.S.: i.e., provable in all models of the first-order axioms]).”

“In particular, there must be [K.S.: some] models of first-order Peano arithmetic whose elements [K.S.: i.e., whose “numbers”] do not “behave” the same as the [K.S: “standard”] natural numbers.”

“Such nonstandard models were unforeseen and unintended, but they cannot be ignored, for their existence implies that no first-order axiomatization of number theory can be adequate to the task of deriving as theorems exactly those statements that are true of the [K.S.: standard] natural numbers.

[J. W. Dawson, Logical Dilemmas: The Life and Work of Kurt Gödel, A. K. Peters, Wellesley, MA., 1997, pp. 67-68, bold & underscored & italicized emphases added by K.S.].

 

“The sentences of the original “Natural” Numbers first-order axioms, known as the ‘“Peano-Dedekind Postulates”’, are essentially as follows –

 

P1.  1’ is a “Natural Number.  

Or: ‘1 ÃŽ N’.

 

P2.  The successor of any “Natural Number is also a “Natural Number. 

Or: ‘"n ÃŽ N, s(n) ÃŽ N’.  [s(n)  º  n + 1].

 

P3.   No two distinct “Natural Numbers have the same successor.  

Or: ‘"n, "m ÃŽ N, n ¹ m Þ s(n) ¹ s(m).

 

P4.   There is no “Natural Number that is the predecessor of ‘1’.  

Or: ¬$x ÃŽ N | s(x) = 1.”

 

 

“We may tend typically to think of the “Natural” numbers as cardinal numbers – as representing “pure”, unqualified cardinal quantities: “counts of identical, abstract unit[ie]s”.” 

 

“And, indeed, the full, first and second-order axioms for the “Natural” numbers do encompass both the cardinal aspects and the ordinal aspects of the standard “Natural” numbers.”

 

“However, by inspection of the four Peano sentences above, one can see that the first-order axioms for these “numbers” implicitly emphasize their ordinal number, or ‘order-number’, character; their aspects as ordinal quantities”.”

 

“This means that, as “ordinal numbers”, the “Naturals” represent first, second, third, fourth,…, rather than “counts” such as one unit, two units, three units, four units,… .”

 

“The NQ non-standard, dialectical ‘meta-numbers’, it turns out, pick up on another possible aspect of the ‘‘‘ordinality’’’ emphasized by the first-order Peano-Dedekind axioms: what we call ‘ordinal quality’.” 

 

“Thus, the NQ sequence of ‘meta-numbers’ represent ‘the quality of first-ness, then ‘the quality of second-ness, then ‘the quality of third-ness, and so on.  We will see how these ‘dialectical meta-numbers’ do so below.”

 

“The chart posted below contrasts the first-order axioms of the NQ versus the N – of the non-standard versus “standard” “Naturals”, and their syntax and symbology – as well as their distinct ‘‘‘analytical geometries’’’.”

 

“This chart thus shows what a non-standard” model of the first-order Peano axioms means, in a specific case.” 

 

“The NQ non-standard model of the “Natural” Numbers is implicit in, immanent in, potential in, and hidden in the first-order Peano-Dedekind axioms, as is the N model, the standard model [albeit less hiddenly so, for powerful psychohistorical reasons].  But NQ and N stand, nonetheless, as two different interpretations of the same four, first-order Peano axioms.”

 

“But these two interpretations are not just “different”; they are qualitatively, ‘ideo-ontologically’, dialectically opposed, forming a dialectical, ‘categorial antithesis’.  This is because their elements, their ‘number-kinds’, are opposite in their qualities.  The N kind of numbers are unqualified “pure” quantifiers’.  The NQ kind are unquantified “pure” qualifiers.  To be more precise, the NQ ‘meta-numbers’ kind are, at root, unquantifiable “pure”, ordinal arithmetical qualifiers’.”

 

“With further interpretation, the NQ ‘meta-numbers’ can serve to model ontological categories, kinds-of-things categories, and categorial progressions  progressions or series of such categories, that model categorial dialectics  as instantiated so abundantly and for so long, elsewhere in this blog.

 

 

 

 

 

 

 

For more information regarding these Seldonian insights, please see --

 

www.dialectics.info

 

 

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of dialectical artas well as dialectically-illustrated books published by the F.E.D. Press, see

 

https://www.etsy.com/shop/DialecticsMATH

 

 

 

 

 

 

 

¡ENJOY!

 

 

 

 

 

 

 

Regards,


 


Miguel Detonacciones,

 

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

 

 

 

 

 

 

YOU are invited to post your comments on this blog-entry below!