Re: Operator Variables in Logical Graphs • 1
Re: Cybernetics List • Lou Kauffman

LK:

I am writing to comment that there are some quite interesting situations that generalize the De Morgan Duality.

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

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

Define

Define

Then we have so that the system satisfies De Morgan duality and it is a Boolean algebra when restricted to

Note also that fixes and   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

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