[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 54

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Arisbe List, Cybernetics List,

If you will excuse the bits of autobio-graphical anec.age,
it will help me to reconstruct the steps that I actually
took in my thinking as I worked through these problems
about logical graphs late in the last millennium.

By 1980 my logical graphs were becoming too large and complex
to keep within the bounds of 2-dimensional manifolds of paper,
and so I started to think once again, with extreme reluctance --
given earlier traumatic experiences trying to use Fortran and
a CDC 3600 mainframe to do my chem and physics lab work in an
era when "turn-around time" was counted in days not microsecs --
of representing logical graphs and logical transformations in
the computer medium.

By a bit of serendipity that still amazes me, it happened that my earlier work
on Peirce's use of operator variables, that led in its turn to my discovery of
the cactus language, also turned out to provide workable solutions for several
problems that arose in the process of trying to find efficient implementations
for logical graphs and their logical transformations.

For a salient example, consider the existential graph for "p <=> q", namely:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` o-----------------o ` ` ` o-----------------o ` ` ` `
` ` ` ` | ` ` ` o-------o | ` ` ` | ` ` ` o-------o | ` ` ` `
` ` ` ` | ` ` ` | ` ` ` | | ` ` ` | ` ` ` | ` ` ` | | ` ` ` `
` ` ` ` | ` p ` | ` q ` | | ` ` ` | ` q ` | ` p ` | | ` ` ` `
` ` ` ` | ` ` ` | ` ` ` | | ` ` ` | ` ` ` | ` ` ` | | ` ` ` `
` ` ` ` | ` ` ` o-------o | ` ` ` | ` ` ` o-------o | ` ` ` `
` ` ` ` o-----------------o ` ` ` o-----------------o ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

This can be read as "not p without q and not q without p",
or what's the same, "[p => q] and [q => p]", or "p <=> q".

Graphing the topological dual form, one obtains this rooted tree:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` q o ` o p ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` p o ` o q ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` O ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` `(p (q))`(q (p))` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Now it is not the sort of thing that I ever noticed until it came time
to program a theorem prover for logical graphs at Peirce's alpha level,
but expressions like these, that mention each variable twice simply in
order to express a basic 2-variate operator, are extremely inefficient
forms of representation, and their use is enough to bog down a routine
logical modeler or an automatic theorem prover in a slough of despond.

However, the cactus graph expression for equivalence works much better:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` p o---o q ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` O ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ((p , q)) ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The cactus language syntax also improves the "reflective capacities"
of the logical calculus, in particular, it facilitates our ability
to use the calculus to reflect on the process of proof, that is,
the process of establishing equivalences between expressions.

Jon Awbrey

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