## Charles Sanders Peirce, George Spencer Brown, and Me • 5

Here are blog and wiki versions of an article I wrote on Peirce’s Law, an axiom or theorem (depending on your choice of logical basis) that distinguishes classical from intuitionistic propositional calculus.  Aside from its pivotal logical status it affords a nice illustration of several important features of logical graphs in the style of Peirce and Spencer Brown.

## Charles Sanders Peirce, George Spencer Brown, and Me • 4

Two things that had a big impact on my studies of Peirce and Spencer Brown over the years were my parallel studies in mathematics and computer science.  In the overlap between those areas came courses in logic, mathematical linguistics, and the theory of formal languages, grammars, and automata.  My intellectual wanderings over a nine-year undergraduate career would take me through a cycle of majors from math and physics, to communication, psychology, philosophy, and a cross-cultural liberal arts program, then back to grad school in mathematics.  The puzzles that Peirce and Spencer Brown beset my brain with were a big part of what drove me back to math, as I could see I had no chance to work them out without learning a lot more algebra, logic, and topology than I had learned till then.

To be continued …

## Charles Sanders Peirce, George Spencer Brown, and Me • 3

There are a number of “difficulties at the beginning” that arise here.  I’ve been trying to get to the point where I can respond to James Bowery’s initial comments and also to questions about the relation between Spencer Brown’s imaginary logical values and the development of differential logic.

The larger issue I see at this point has to do with the relationship between the arithmetic and the algebra of logical graphs.  Peirce came close to the point of discovering that relationship several times in his later work on existential graphs (ExG) but never quite pushed it through to full realization.  It was left to Spencer Brown to bring it to light.

The relationship between Primary Arithmetic and Primary Algebra is discussed in the following article:

The other issue has to do with my using a different J1 than Spencer Brown.  I believe I even called it J1′ in the early days but eventually lost the prime as time went by.  The best I can recall, it had something to do with negotiating between the systems of CSP and GSB, but I think I stuck with the variant because it sorts two types of change, modifying structure and moving variables, into different bins.

• Image Files

See also the discussions at the following locations:

## The Difference That Makes A Difference That Peirce Makes : 18

From a functional point of view it was a step backward when we passed from Peirce’s $\sum$ and $\prod$ to the current convention of $\exists$ and $\forall$ for logical quantifiers.  There’s a rough indication of what I mean at the following location:

Just a reminder to get back to this later …

## Charles Sanders Peirce, George Spencer Brown, and Me • 2

I’m making an effort to present this material in a more gradual and logical order than I’ve ever managed to do before.  There are issues about the relationship between episodic and semantic memory that are giving me trouble as I try to remember how I came to look at things the way I do … but never mind that now.  I’ll eventually get around to explaining the forces that drove me to generalize the forms of logical graphs from trees to cacti, as graph theorists call them, and how that made the transition to differential logic so much easier than it would have been otherwise, but I think it would be better now to begin at the beginning with the common core of forms introduced by CSP and GSB.

Here’s a couple of articles I wrote up for that purpose:

There are versions of those articles at several other places on the web that may be better formatted or more convenient for discussion:

One big issue that comes up at the beginning is the question of “duality”.  Both C.S. Peirce and Spencer Brown understood they were dealing with a very abstract calculus, one that could be interpreted for the purposes of ordinary propositional logic in two different ways.  Peirce called the two different ways of interpreting the abstract graphs his entitative and existential graphs.  He started out with a system of graphs he chose to read in the entitative manner but switched over to the existential choice as he developed his logical graphs beyond the purely propositional level.  Spencer Brown elected to emphasize the entitative reading in his main exposition but he was very clear in the terminology he used that the forms and transformations themselves are independent of their interpretations.

Table 1 at either of the locations linked below has columns for the graph-theoretic forms and the parenthesis-string forms of several basic expressions, reading them under the existential interpretation.

• Table 1. Syntax and Semantics of a Calculus for Propositional Logic • (a)(b)

The Tables linked below serve to compare the existential and entitative interpretations of logical graphs by providing translations into familiar notations and English paraphrases for a few of the most basic and commonly occurring forms.

## Charles Sanders Peirce, George Spencer Brown, and Me • 1

It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away.  I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph-theoretic data structures formed on their ideas to good purpose in the mechanics of its propositional reasoning engine.  I thought it might contribute to a number of ongoing discussions if I could articulate what I think I learned from that experience.

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

James Bowery left a comment on my blog and opened a thread in the Yahoo! group devoted to discussing the mathematics of George Spencer Brown’s Laws of Form.  I’ve been meaning to join that discussion as soon as I could work up the time and concentration to think about it … at long last I think I can do that now.  I’ll use the above heading on this blog to post any bits from my side of the conversation that I think might serve a wider audience.

It’s been a long time since I joined a new discussion group so I thought I’d start by posting a bit of the old-fashioned self-intro.