## Charles Sanders Peirce, George Spencer Brown, and Me • 16

GZ:
I’m quite confused on why people are interested in Laws of Form.
What is LOF trying to do? Is it just rewriting logic or is there
something more fundamental. e.g. a universal algebraic system?
What does GSB has to do with DNA, or DNA computing?
What does Lou’s work in topology has to do with GSB?
What does GSB’s theory has to do with knot theory?
What does GSB’s theory has to do with quaternions?
How can GSB’s theory be used for designing circuits?
What’s wrong with Frege?

Dear Gary,

I am deep in the middle of other work right now, but here’s a smattering of resources relevant to the relation between Peirce’s logical graphs and Spencer Brown’s calculus of indications, at least as far as the core subjects of boolean functions and propositional calculus are concerned.

As far as the extension to relations and quantification, I start from where Peirce started in 1870 and follow up several of his more radical ideas, ones he himself did not fully develop.  That is what I’m doing on the 1870 Logic of Relatives thread.

Regards,

Jon

## Peirce’s 1870 “Logic of Relatives” • Discussion 5

PZ:
I’m studying imprecise probabilities which initially works as an extension in Boole’s Laws of Thoughts.  It seems like Boole was solving a set of algebraic equations for probabilities where some of the probabilities do not have precise values therefore need to be bounded.  Has anyone studied Boole’s algebraic system of probabilities?  Is Peirce extending Boole’s algebraic system in his Logic of Relatives?

Dear Peiyuan,

Issues related to the ones you mention will come up in the Selections and Commentary I’m posting on Peirce’s 1870 Logic of Relatives, the full title of which, “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, is sufficient hint of the author’s intent, namely, to extend the correspondence Boole discovered between the calculus of propositions and the statistics of simple events to a correspondence between the calculus of relations and the statistics of complex events, contingency matrices, higher order correlations, and ultimately the full range of information theory.

But it will take a while to develop all that …

Regards,
Jon

## Peirce’s 1870 “Logic of Relatives” • Discussion 4

HE:
What’s your point, it’s obviously too graphical, but perhaps you are driving at something else.  Explain?

Dear Henning,

My aim here is to survey the source from which radiates all our most enlightening graphical systems of logic — from Peirce’s own entitative and existential graphs, to Spencer Brown’s calculus of indications, to John Sowa’s conceptual graphs.  The first glimmerings of that evolution go further back than widely appreciated, being especially well marked in Peirce’s 1870 “Logic of Relatives”, as I hope to make clear in time.

Regards,
Jon

## Peirce’s 1870 “Logic of Relatives” • Discussion 3

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

Re: Peirce’s 1870 “Logic of Relatives” • Overview
Re: Laws of Form • James Bowery (1) (2)

Dear James,

I am pleased to see you engaging the material on Peirce’s Logic of Relatives.  For my part I’ll need to lay out several more Selections before the major themes of Peirce’s essay begin to emerge from the supporting but sometimes distracting details.

In the meantime two clues to the Big Picture can be gleaned from the paired epigraphs I put up in lights at the top of the post.  For what we have here is a return to the thrilling days of yesteryear when the mathematics of logic was still mathematics, shortly before Frege (maybe unwittingly) and Russell (in a way less wittingly) detoured it down the linguistic U‑turn to nominalism.

Regards,
Jon

## Triadic Relations • Discussion 3

Re: Conceptual GraphsEdwina Taborsky

ET:
A few comments on your outline of the Sign.  I think one has to be careful not to set up a Saussurian linguistic dyad.  …

Dear Edwina,

I copied your comments to a draft page and will take them up in the fullness of time, but a few remarks by way of general orientation to relations, triadic relations, sign relations, and sign transformations, partly prompted by the earlier discussion of complex systems, may be useful at this point.

One does not come to terms with systems of any complexity — adaptive, anticipatory, intelligent systems, and those with a capacity to support scientific inquiry, whether as autonomous agents or assistive utilities — without the use of mathematical models to negotiate the gap between our naturally evolved linguistic capacities and the just barely scrutable realities manifesting in phenomena.

Peirce’s quest to understand how science works takes its first big steps with his lectures on the Logic of Science at Harvard and the Lowell Institute (1865–1866), where he traces the bearings of deduction, induction, and hypothesis on the conduct of scientific inquiry.  There Peirce makes a good beginning by taking up Boole’s functional recasting of logic, a major advance over traditional logic rooted in the paradigms of historical grammars.  But developing a minimal adequate mathematical basis for the logic of science will take drilling down to a deeper core.

The mathematics we need to build models of inquiry as a sign-relational process appears for the first time in history with Peirce’s early work, especially his 1870 Logic of Relatives.  It has its sources in the mathematical realism of Leibniz and De Morgan, the functional logic of Boole, and the algebraic research of Peirce’s own father, Benjamin Peirce, whose Linear Associative Algebra Charles edited for publication in the American Journal of Mathematics (1881).

My own contributions to this pursuit I’ve collected over the years under the heading of Inquiry Driven Systems, portions of which I’ve shared here and there across the Web for lo! this whole millennium in progress.  A few resources along those lines are listed below.

### References

• Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284.  AbstractOnline.
• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action • The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.
ArchiveJournal.  Online (doc) (pdf).
• Awbrey, S.M., and Awbrey, J.L. (1991), “An Architecture for Inquiry • Building Computer Platforms for Discovery”, Proceedings of the Eighth International Conference on Technology and Education, Toronto, Canada, 874–875.  Online.
• Awbrey, J.L., and Awbrey, S.M. (1990), “Exploring Research Data Interactively. Theme One : A Program of Inquiry”, Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training, Society for Applied Learning Technology, Washington, DC, 9–15.  Online.

### Resources

cc: Category TheoryConceptual Graphs • Cybernetics (1) (2) • Ontolog (1) (2)
cc: Peirce List (1) (2) • Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation TheoryLaws of Form

## Survey of Semiotics, Semiosis, Sign Relations • 2

This is a Survey of blog and wiki resources on the theory of signs, variously known as semeiotic or semiotics, and the actions referred to as semiosis which transform signs among themselves in relation to their objects, all as based on C.S. Peirce’s concept of triadic sign relations.

### Blog Series

• Sign Relations, Triadic Relations, Relation Theory • (1)
• C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • (1)(2)

### References

• Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.
• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  ArchiveJournal.  Online (doc) (pdf).
| | 1 Comment

## Genus, Species, Pie Charts, Radio Buttons • Discussion 5

Dear John,

Once we grasp the utility of minimal negation operators for partitioning a universe of discourse into several regions and any region into further parts, there are quite a few directions we might explore as far as our next steps go.

One thing I always did when I reached a new level of understanding about any logical issue was to see if I could actualize the insight in whatever programming projects I was working on at the time.  Conversely and recursively the trials of doing that would often force me to modify my initial understanding in the direction of what works in brass tacks practice.

The use of cactus graphs to implement minimal negation operators made its way into the Theme One Program I worked on all through the 1980s and the applications I made of it went into the work I did for a master’s in psych.  At any rate, I can finally answer the “what next” question by pointing to one of the exercises I set for the logical reasoning module of that program, as described in the following excerpt from its User Guide.

• Theme One Guide • Molly’s World (pdf)

The writing there is a little rough by my current standards, so I’ll work on revising it over the next few days.

Regards,

Jon

## Genus, Species, Pie Charts, Radio Buttons • Discussion 4

JM:
I feel as though you have posted these same diagrams many times, and it is always portrayed as clearing the ground for something else.  But the something else never arrives!  I would be really interested to know what the next step is in your ideas.

Dear John,

Thanks for the question.  Bruce Schuman mentioned radio button logic and I jumped on it “like a duck on a June bug” — as they say in several southern States I know — because that very thing marks an important first step in the application of minimal negation operators to represent finite domains of values, contextual individuals, genus and species, partitions, and so on.  But some of the comments I got next gave me pause and made me feel I should go back and clarify a few points.

I wasn’t sure, but I got the sense Bruce was reading the cactus graphs I posted as an order of hierarchical, ontological, or taxonomic diagrams.  What they really amount to are the abstract, human-viewable renditions of linked data structures or “pointer” data structures in computer memory.  I explained the transformation from planar forms of enclosure to their topological dual trees to the pointer structures in one of the articles on logical graphs I wrote for Wikipedia and later for Google’s now-defunct Knol.  People can find a version of that on the following page of my blog.

## Genus, Species, Pie Charts, Radio Buttons • Discussion 3

Last time I alluded to the general problem of relating a variety of formal languages to a shared domain of formal objects, taking six notations for the boolean functions on two variables as a simple but critical illustration of the larger task.  This time we’ll take up a subtler example of cross-calculus communication, where the same syntactic forms bear different logical interpretations.

In each of the Tables below —

• Column 1 shows a conventional name $f_{i}$ and a venn diagram for each of the sixteen boolean functions on two variables.
• Column 2 shows the logical graph canonically representing the boolean function in Column 1 under the entitative interpretation.  This is the interpretation C.S. Peirce used in his earlier work on entitative graphs and the one Spencer Brown used in his book Laws of Form.
• Column 3 shows the logical graph canonically representing the boolean function in Column 1 under the existential interpretation.  This is the interpretation C.S. Peirce used in his later work on existential graphs.

$\text{Boolean Functions and Logical Graphs on Two Variables} \stackrel{_\bullet}{} \text{Index Order}$

$\text{Boolean Functions and Logical Graphs on Two Variables} \stackrel{_\bullet}{} \text{Orbit Order}$

### Resources

• Logical Graphs, Iconicity, Interpretation • (1)(2)
• Minimal Negation Operators • (1)(2)(3)(4)

## Genus, Species, Pie Charts, Radio Buttons • Discussion 2

A problem we often encounter is the need to relate a variety of formal languages to the same domain of formal objects.  In our present engagement we are using languages not only to describe but further to compute with the objects in question and so we call our languages so many diverse calculi.

When it comes to propositional calculi, a couple of Tables may be useful at this point and also for future reference.  They present two arrangements of the sixteen boolean functions on two variables, collating their truth tables with their expressions in several systems of notation, including the parenthetical versions of cactus expressions, here read under the existential interpretation.  They appear as the first two Tables on the following page.

### Differential Logic and Dynamic Systems • Appendices

The copies I posted to my blog will probably load faster.

### Differential Logic • 8

Table A1.  Propositional Forms on Two Variables • Index Order

### Differential Logic • 9

Table A2.  Propositional Forms on Two Variables • Orbit Order