[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 42

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Cybernetics List, Peirce List,

A funny thing just happened.  Let's see if we can tell where.
We started with the algebraic expression "(a)", in which the
operand "a" suggests the contemplated absence or presence of
any arithmetic expression or its value, then we contemplated
the absence of presence of the operator "(_)" in "(a)" to be
indicated by a cross or a space, respectively, for the value
of a newly introduced variable, "b", placed in a new slot of
a newly extended operator form, as suggested by this picture:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ----------o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` a ` ` | b | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The funny thing that just happened -- it's an acquired sense of humor --
is that our contemplation of an operator variable just as quickly got
converted into the contemplation of a newly introjected but otherwise
quite ordinary operand variable, albeit within a newly-fanged formula.
In its interpretation for logic, the new form of operation that forms
here may be viewed as an extension of ordinary negation, specifically,
a negation of the first variable that is "controlled" by the value of
the second variable.  Thus, we may regard this development as marking
a form of "controlled reflection", or a form of "reflective control".
By way of an inline syntax for this, I will employ the form "(a, b)".

Writing out a formal operation table yields the following summary:

o-------------------o-------------------o-------------------o
| ` ` ` ` a ` ` ` ` | ` ` ` ` b ` ` ` ` | ` ` `(a , b)` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` blank ` ` ` | ` ` ` blank ` ` ` | ` ` ` cross ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` blank ` ` ` | ` ` ` cross ` ` ` | ` ` ` blank ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` cross ` ` ` | ` ` ` blank ` ` ` | ` ` ` blank ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` cross ` ` ` | ` ` ` cross ` ` ` | ` ` ` cross ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o

En, for which blank = false and cross = true, calls this "equivalence".
Ex, for which blank = true and cross = false, calls this "distinction".

Jon Awbrey

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