[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 17

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Cybernetics List, Peirce List,

Speaking of algebra, and having seen one example of an algebraic law,
we might as well introduce the axioms of the "primary algebra", once
again deriving their substance and their name from Charles S. Peirce
and G. Spencer Brown, respectively.

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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a(a)` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| Axiom J_1.` ` ` Insert <--- | ---> Delete ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `ab ` ac` ` ` ` ` ` ` b ` c ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` a O ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `((ab)(ac)) ` ` = ` ` a((b)(c)) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| Axiom J_2.` Distribute <--- | ---> Collect` ` ` ` ` ` ` ` |
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The choice of axioms for any formal system is to some degree
a matter of aesthetics, as it is commonly the case that many
different selections of formal rules will serve as axioms to
derive all the rest as theorems.  As it happens, the example
that we noticed first, as simple as it appears, proves to be
provable as a theorem on the grounds of the foregoing axioms.

We might also notice at this point a subtle difference between
the primary arithmetic and the primary algebra with respect to
the grounds of justification that we have naturally if tacitly
adopted for their respective sets of axioms.

The arithmetic axioms were introduced by fiat, in a quasi-apriori fashion,
though of course it is only long prior experience with the practical uses
of comparably developed generations of formal systems that would actually
induce us to such a quasi-primal move.  The algebraic axioms, in contrast,
can be seen to derive their motive and their justice from the observation
and summarization of patterns that are visible in the arithmetic spectrum.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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