[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 39

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Cybernetics List, Peirce List,

Now that it's come to "logical operator variables" (LOV's),
I can find no grander way to elope than Peirce himself did:

| I shall not further enlarge upon this matter at this point, although the
| conception mentioned opens a wide field;  because it cannot be set in its
| proper light without overstepping the limits of dichotomic mathematics.
|
| C.S. Peirce, 'Collected Papers', CP 4.306

And that goes treble for me.  The further exploration of operator laws and
operator variables, touching as it does on the ground that was classically
called "second intentional logic", like the man said, "opens a wide field",
one that we may have hopes of revisiting in a less whirlwind touristy way
one of these days.  For now I will tend to that corner of the field where
this particular variety of logical graphs grows, communing with a few of
the ways that operative variations and operative themes sprout therein.

To begin with a concrete case that's as easy as possible,
let's examine this extremely simple algebraic expression:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` O ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

In this context the variable name "a" appears as an "operand name".
In functional terms, "a" is called an "argument name", but we are
probably well advised to avoid the confusing connotations of the
word "argument" here, as it also refers in logical discussions
to a more or less specific pattern of reasoning.  In syntactic
terms, this same "a" would be classified as a "terminal sign".

As we've already discussed, the algebraic variable name indicates
the contemplated absence or presence of any arithmetic expression
taking its place in the surrounding template, which expression is
proxied well enough by its value, of which values we know but two.
Thus, the given algebraic expression varies between these choices:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` O ` ` ` ` , ` ` ` ` O ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

This selection of arithmetic expressions what it means to
contemplate the absence or presence of the operand "a" in
the algebraic expression "(a)".  But what does it mean to
contemplate the absence or presence of the operator "(_)"
in the algebraic expression "(a)"?

I will take up that question next time.

Jon Awbrey

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