Operator Variables in Logical Graphs • Discussion

[Jon Awbrey]

Operator Variables in Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/04/08/operator-variables-in-logical-graphs-discussion-1/

Re: Operator Variables in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-graphs-1/

Re: Academia.edu • Stephen Duplantier
• https://www.academia.edu/community/Lxn1Ww?c=yq1Rxy

SD:
❝The best way for me to read Peirce is as if he was writing poetry.
So if his algebra is poetry — I imagine him approving of the approach
since he taught me abduction in the first place — there is room to wander.
With this, I venture the idea that his “wide field” is a local algebraic
geography far from the tended garden. There, where weeds and wild things
grow and hybridize are the non‑dichotomic mathematics.❞

Stephen,

“Abdeuces Are Wild”, as they say, maybe not today, maybe not tomorrow, but soon …

As far as my own guess, and a lot of my wandering in pursuit of it goes,
I'd venture Peirce's field of vision opens up not so much from dichotomic
to trichotomic domains of value as from dyadic to triadic relations, and
all that with particular significance into the medium of reflection
afforded by triadic sign relations.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Semeiotic
• https://oeis.org/wiki/Semeiotic

Sign Relations
• https://oeis.org/wiki/Sign_relation

Triadic Relations
• https://oeis.org/wiki/Triadic_relation

Regards,

Jon

cc: https://www.academia.edu/community/l7jXzQ
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[Jon Awbrey]

Operator Variables in Logical Graphs • Discussion 2
• https://inquiryintoinquiry.com/2024/04/09/operator-variables-in-logical-graphs-discussion-2/

Re: Operator Variables in Logical Graphs • 1
• https://inquiryintoinquiry.com/2024/04/06/operator-variables-in-logical-graphs-1/

Re: Cybernetics List • Lou Kauffman
• https://groups.google.com/g/cybcom/c/XKT76QI_OnI/m/3u9P2Ir5AgAJ

LK:
❝I am writing to comment that there are some quite interesting
situations that generalize the DeMorgan Duality.

❝One well-known one is this. Let R* denote the real numbers
with a formal symbol @, denoting infinity, adjoined so that:

• @ + @ = @
• @ + 0 = @
• @ + x = @ when x is an ordinary real number
• 1 ÷ @ = 0

❝(Of course you cannot do anything with @ or the system collapses.
One can easily give the constraints.)

❝Define ¬x = 1/x.

• x + y = usual sum otherwise.

❝Define x ∗ y = xy/(x + y) = 1/((1/x) + (1/y)).

❝Then we have x ∗ y = ¬(¬x + ¬y), so that the system (R*, ¬, +, ∗)
satisfies De Morgan duality and it is a Boolean algebra when
restricted to {0, @}.

❝Note also that ¬ fixes 1 and -1. This algebraic system occurs
of course in electrical calculations and also in the properties
of tangles in knot theory, as you can read in the last part of
my included paper “Knot Logic”. I expect there is quite a bit
more about this kind of duality in various (categorical) places.❞

Thanks, Lou,

There's a lot to think about here, so I'll need to study it a while.
Just off hand, the embedding into reals brings up a vague memory of
the very curious way Peirce defines negation in his 1870 “Logic of
Relatives”. I seem to recall it involving a power series, but it's
been a while so I'll have to look it up again.

Regards,

Jon

cc: https://www.academia.edu/community/L2g021
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