By this blog-entry let me also wish to all of you a very richly-lived and, thus, a very prosperous 2014 !!!
Below, for your cognitive enjoyment, I have posted an edited -- 'anonymized' -- excerpt from a recent web forum dialogue on F.E.D. dialectics in which I participated.
I have notated my interlocutor's queries by 'Q#.:', and my responses by 'R#.:'.
Please feel free to join in this dialogue here, by posting a query or comment via the Comment Function.
Q1.: Thesis x antithesis = synthesis.
I get that.
But why is thesis x thesis = antithesis?
...it would really be great if you could explain it step by step. ...
R1.: There are two "dialectical functions" that we use to build "dialectical models".
One is called "dyadic" because it expands by twos.
The other is called "triadic", because it expands by threes.
What I cited to you is from the '"convolute"' version of the "dyadic" function.
Its logic is opposite to that of Boole's algebra of logic.
In Boole's algebra of logic, you can never get anywhere new, in terms of the kinds of things that exist.
In Boole's original algebra of logic, you can only address the statical relations of inclusion versus of exclusion between one "class" ['''category'''] versus another, or define one class / category in terms of (an)other(s).
But no "new" classes / categories, "containing" NEW kinds of things -- new kinds of individuals -- can be born into the universe from interactions of the old, already existing kinds of things, or from their '''self-interactions''', and/or from the [self-]interactions of their classes / categories, within that universe[-of-discourse], in the original Boolean algebra model of that universe[-of-discourse] --
thesis x thesis = thesis
1 x 1 = 1
0 x 0 = 0.
This equation, thesis x thesis = thesis, class x class = class, category x category = category, is what Boole called "The Fundamental Law of Thought".
It is certainly NOT "The Fundamental Law of DIALECTICAL thought"!
Boole's "Fundamental Law of Thought" provides only for a "Simple Reproduction" of ideas/categories --
idea x idea = idea.
Dialectical logic needs to model a qualitatively, ontologically "[Self-]Expanding Reproduction" of ideas, and an "[Self-]Expanding Reproduction" of non-idea kinds of things, e.g., external-to-mind, physical things [although, of course, we "grasp" -- "hold" -- these physical things, in mind, and "language" about them, via ideas, via 'mind-things', via '''memes'''. But our ideas of physical things refer to things that we hold to exist "objectively", outside of our minds. Such ideas are not about "things" that exist ONLY in our minds, like, e.g., that idea-only-thing, that idea-unit, which is the unit[y] of the "Natural" Numbers, 1] --
idea x idea = idea + new/better idea.
physical-thing-kind_1 x physical-thing-kind_1 = physical-thing-kind_1 + physical-thing-kind_2.
There is no qualitative change, no ontological change, no ontological gain, per the Boolean model.
The algebra of a dialectical logic needs to model a universe in which new kinds of things regularly come into existence, from out of the interactions and '''self-interactions''' among old kinds of things, i.e., in which ontological revolutions occur -- e.g., the first atomic clouds turned into "molecular clouds" as the quantitative growth and growing density of atoms gave birth to a new kind of things -- molecules -- or, e.g., as the growth of the human societal self-reproductive self-force gave birth to Commodity Barter social relations of production, once the societal "population" and societal "density" of Goods/Use-Values/Gifts exchange social relations of production crossed a critical historic[al] threshold.
Note that 'atoms x atoms = molecules' is not a good model here, because the atoms don't disappear entirely when the molecules first form, and also that 'atoms x atoms = atoms', the Boolean model, is also no good here, because it completely misses out on the first cosmological emergence of the new ontological category/class of "molecules".
We need --
atoms x atoms = atoms + molecules
-- '<<aufheben>>-ing' the Boolean "law", by "conserving" the 'Boolean moment' of 'atoms x atoms = atoms', but also dialectically, determinately "negating", and "elevating" that Boolean "law" by incorporating also the 'contra-Boolean moment' 'atoms x atoms = molecules', to form, as a whole, the Seldonian dialectical "law" of thought, and of physical actuality, namely --
atoms^2 = atoms + molecules
-- so that --
atoms^2 is NOT equal to atoms
-- indeed --
atoms^2 is qualitatively different from, is ontologically different from atoms^1.
Goods/Gifts exchange-relations x Goods/Gifts exchange-relations =
Commodity Barter exchange-relations
-- is not a good model here.
Goods/Gifts exchange-relations did not cease to exist after barter exchange-relations first came into existence -- indeed, Goods/Gifts exchange-relations have continued to exist, in modified forms, all the way up into the capitalist historical present!
We need, instead --
Goods/Gifts exchange-relations x Goods/Gifts exchange-relations =
Goods/Gifts exchange-relations + Commodity Barter exchange-relations.
In the dyadic function, in its fuller, '"evolute"' rendition [I presented only the simplest, '"convolute'', form before] --
thesis x thesis = thesis + antithesis
-- i.e., you keep what you had before, you <<aufheben>>-conserve the "Boolean moment", but you also get, together with the "Boolean moment", a second "moment", something new.
At the critical point of ontological revolution / qualitative change that is modeled by 'thesis x thesis': you get back the "thesis" category, but you also get the "anti-thesis" category with respect to that "thesis" category.
The process I described in my previous post starts with 'thesis x thesis = [thesis +] [first] antithesis', but, thereafter, it becomes 'antithesis x antithesis = antithesis + next antithesis.
That is, in the dyadic function and model, 'antithesis x antithesis = antithesis + new antithesis' means that each synthesis is "outflanked" by a new antithesis, a new antithesis that comes from out of the self-interaction of the previous antithesis, which was transcended by that synthesis.
In a sense, the first antithesis keeps recurring at ever higher levels, in qualitatively, ontologically new forms, superseding each new synthesis, that temporarily transcended each new antithesis, and which, each time, is resolved into/by a new synthesis, only to have that new synthesis outflanked again, from out of the immediately-previous antithesis.
In the triadic function, on the other hand, the fundamental equation is 'synthesis x synthesis = synthesis + new antithesis': it is each new synthesis itself that breaks itself out into a new antithesis, by way of the self-interaction of that synthesis.
Is all of this clear, so far?
If not, please lodge additional queries!