Inquiry Into Inquiry

Re: Laws of Form • Lyle Anderson
Re: Brading, K., Castellani, E., and Teh, N., (2017), “Symmetry and Symmetry Breaking”, The Stanford Encyclopedia of Philosophy (Winter 2017), Edward N. Zalta (ed.).  Online.

Dear Lyle,

Thanks for the link to the article on symmetry and symmetry breaking.  I did once take a Master’s in Mathematics, specializing in combinatorics, graph theory, and group theory.  When it comes to the bearing of symmetry groups on logical graphs and the calculus of indications, it will take careful attention to the details of the relationship between the two interpretations singled out by Peirce and Spencer Brown.

Both Peirce and Spencer Brown recognized the relevant duality, if they differed in what they found most convenient to use in their development and exposition, and most of us will emphasize one interpretation or the other as a matter of taste or facility in a chosen application, so it requires a bit of effort to keep the underlying unity in focus.  I recently made another try at taking a more balanced view, drawing up a series of tables in parallel columns the way one commonly does with dual theorems in projective geometry, so I will shortly share more of that work.

Resources

• Duality Indicating Unity
• C.S. Peirce • Logic of Number
• C.S. Peirce • Syllabus • Selection 1

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All other sciences without exception depend upon the principles of mathematics;  and mathematics borrows nothing from them but hints.

C.S. Peirce • “Logic of Number”

A principal intention of this essay is to separate what are known as algebras of logic from the subject of logic, and to re-align them with mathematics.

G. Spencer Brown • Laws of Form

The duality between entitative and existential interpretations of logical graphs tells us something important about the relation between logic and mathematics.  It tells us the mathematical forms giving structure to reasoning are deeper and more abstract at once than their logical interpretations.

A formal duality points to a more encompassing unity, founding a calculus of forms whose expressions can be read in alternate ways by switching the meanings assigned to a pair of primitive terms.  Spencer Brown’s mathematical approach to Laws of Form and the whole of Peirce’s work on the mathematics of logic shows both thinkers were deeply aware of this principle.

Peirce explored a variety of dualities in logic which he treated on analogy with the dualities in projective geometry.  This gave rise to formal systems where the initial constants, and thus their geometric and graph-theoretic representations, had no uniquely fixed meanings but could be given dual interpretations in logic.

It was in this context that Peirce’s systems of logical graphs developed, issuing in dual interpretations of the same formal axioms which Peirce referred to as entitative graphs and existential graphs, respectively.  He developed only the existential interpretation to any great extent, since the extension from propositional to relational calculus appeared more natural in that case, but whether there is any logical or mathematical reason for the symmetry to break at that point is a good question for further research.

Resources

• Duality Indicating Unity
• C.S. Peirce • Logic of Number
• C.S. Peirce • Syllabus • Selection 1

References

• Peirce, C.S., [Logic of Number — Le Fevre] (MS 229), in Carolyn Eisele (ed., 1976),
  The New Elements of Mathematics by Charles S. Peirce, vol. 2, 592–595.  Excerpt.
• Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin, London, UK.

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Re: Animated Logical Graphs • 55
Re: Laws of Form • William Bricken

WB:

Weird how we’ve been doing this for so many years!  I look forward to what you have to say.  Dunno if you’ve seen this, may be of interest.

We built some stuff similar to logic graphs, we called distinction networks (d-nets), in the deep past.  Here’s some implementation details (1995) for asynchronous d-net computation.  Ran it first on an Intel Hypercube with 16 nodes (ugh, course-grain parallelism — a technical abstract (1987) at “The Losp Parallel Deduction Engine” (PDF)) and eventually migrated to a distributed network architecture in which each node was an independent operating system, more for the convenience of doing VR than for the elegance of fine-grain logic parallelism.

Distinction Networks

Abstract.  Intelligent systems can be modeled by organizationally closed networks of interacting agents.  An interesting step in the evolution from agents to systems of agents is to approach logic itself as a system of autonomous elementary processes called distinctions.  Distinction networks are directed acyclic graphs in which links represent logical implication and nodes are autonomous agents which act in response to changes in their local environment of connectivity.  Asynchronous communication of local decisions produces global computational results without global coordination.  Biological/environmental programming uses environmental semantics, spatial syntax, and boundary transformation to produce strongly parallel logical deduction.

Reference

• Bricken, W. (July 1995), “Distinction Networks” (PDF).

Dear William,

Thanks for the readings.  Maybe I’ve just got McCulloch on the brain right now but the things I’m reading in several groups lately keep flashing me back to themes from his work.  What you wrote on distinction networks took me back to the beginnings of my interest in AI, especially as approached from logical directions.  There’s a couple of posts on my blog where I made an effort to point up what I regard as critical issues.  I’ll reshare those next and see if I can throw more light on what’s at stake.

cc: Cybernetics (1) (2) • Laws of Form • FB | Logical Graphs • Ontolog Forum (1) (2)
• Structural Modeling (1) (2) • Systems Science (1) (2)

Re: Laws of Form • William Bricken

WB:

Kauffman’s 2001 piece on “Peirce” (title is “The Mathematics of Charles Sanders Peirce” [PDF]) is IMO fundamental to this discussion.

Here’s a brief excerpt from a piece I did in 2005:
“Boundary Logic and Alpha Existential Graphs (AEG)”

4.3  LoF and Alpha Graphs Compared

AEG applies the diagrammatic structure of enclosure specifically to logic. The representations of LoF and AEG are isomorphic, while the systems of transformation rules are remarkably close to being the same.  …

Dear William,

Many thanks for your excerpt.  It highlights many of the most critical points in comparing the systems of Peirce and Spencer Brown so I’ll take up the topic of duality first.  Over the years I’ve always found that to be one of the stickier wickets in the whole field.  I’ll discuss it in my Animated Logical Graphs series as that’s where I’ve recently redoubled my efforts to explain the issue and why it’s important.

Regards,

Jon

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Re: Peter Cameron • Doing Research
Re: Gil Kalai • Chomskian Linguistics

Oneirocritical Interlude

Speaking of dreams, the night before last I had a dream where I was listening to a lecturer and something he said brought to mind a logical formula having the form which I visualized as the Peircean logical graph shown below.

I knew I had seen something the day before prompting that fragment and a search through my browser history turned up Gil Kalai’s post on Chomskian Linguistics where I’d read the phrase “anti anti anti missile missile missile missile”.

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