[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 40

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Cybernetics List, Peirce List,

We had been contemplating the penultimately simple
algebraic expression "(a)" as a name for a set of
arithmetic expressions, namely, (a) = {(), (())},
taking the equality sign in the appropriate sense.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` O ` ` ` ` = ` ` { ` O ` `,` ` O ` } ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Then we asked the corresponding question about the operator "(_)":
The above selection of arithmetic expressions is what it means
to contemplate the absence or presence of the operand "a" in
the algebraic expression "(a)".  But what would it mean to
contemplate the absence or presence of the operator "(_)"
in the algebraic expression "(a)"?

Clearly, a variation between the absence and the presence
of the operator "(_)" in the algebraic expression "(a)"
refers to a variation between the algebraic expressions
"a" and "(a)", respectively, somwhat as pictured here:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ? ` ` ` ` ` ` ` ` ` a ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` O ` ` ` ` = ` ` { ` O ` `,` ` O ` } ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

But how shall we signify such variations in a coherent calculus?

In the days when I scribbled these things on the backs of computer punchcards,
I think that the first thing I tried was drawing big loopy script characters,
placing some inside the loops of others.  Lower case alphas, betas, gammas,
deltas, and so on worked the best, but here in Ascii I will ty to convey
something approaching the same general impression by using p's and q's.

Here is how we might suggest an algebraic expression
of the form "(q)" where the absence or presence of
the operator "(_)" depends on the value of the
algebraic expression "p", the operator "(_)"
being absent whenever p is unmarked and
present when whenever p is marked.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | q | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o---o ` ` ` = ` ` ` { ` q ` , ` (q) ` } ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

It was obvious to me from the very outset that this sort of tactic
would need some work to become a usable calculus, especially when
it became time to feed those punchcards back into the computer.

Jon Awbrey

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