A wider field of investigation opens up at this point, having to do with the diversity of interactions among the languages we use, and systems of signs in general, to the thoughts that stream through our heads, to the universes we talk and think upon, from Plato’s Heaven to Gaia’s Green Earth to the Tumbling Galaxies Beyond.

The complexities that come into play when we consider a domain of Signs, a domain of Ideas, and a domain of Objects all wound up in relationship to one another is what Peirce’s “semiotics” or theory of sign relations is all about. Viewing the enterprise of logic within the broader frame of semiotics not only gives us more insight into its means and ends but affords us more “elbow room” for carrying out its operations.

To make a long short, we don’t have to “escape language” because we don’t live inside any language or system of signs, even if we get so confused sometimes as to think we do. We live in that wider world of reality and only use languages and other systems of signs to describe what little we can of it.

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For me, the heart of the matter is what is the purpose of logic and what is the purpose of math and what is their relationship?

There are ways of thinking about semiotic situations that seem to violate the initial conditions of logic that don’t reduce our brains to jelly from the getgo, and C.S. Peirce, following oddly enough on Aristotle, is a thinker who thought quite a bit about the issue.

I’m determined to keep my gnosis close to the grinstone for the time being, but here is a smattering of old notes that give a hint as to Peirce’s way of approaching the question:

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A statement that implies both and is called a false statement, and anyone can prove anything at all from a false statement, as we all too frequently observe on the political front these days.

There is however a reasonable way of handling boundaries, for instance, as illustrated by the circumference of a region in a venn diagram, and that is by means of differential logic. I’ve been tortoising my way toward the goal line of explaining all that, and it’s going a bit slow, but there is a gentle introduction at the other end of the link below, if you wish to achilles ahead.

There’s also a Facebook page devoted to the subject, for anyone who uses that medium:

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The formal system of logical graphs is defined by a foursome of formal equations, called *initials* when regarded purely formally, in abstraction from potential interpretations, called *axioms* when interpreted as logical equivalences. There are two *arithmetic initials* and two *algebraic initials*, as follows:

Spencer Brown uses a different formal equation for his first algebraic initial — where I use he uses For the moment, let’s refer to my as and his as and use that notation to examine the relationship between the two systems.

It is easy to see that the two systems are equivalent, since we have the following proof of by way of and

a a o---o | @ =======J1a {delete} o---o | @ =======I2 {cancel} @ =======QED J1b

In choosing between systems I am less concerned with small differences in the lengths of proofs than I am with other factors. It is difficult for me to remember all the reasons for decisions I made forty or fifty years ago — as a general rule, Peirce’s way of looking at the relation between mathematics and logic has long been a big influence on my thinking and the other main impact is accountable to the nuts and bolts requirements of computational representation.

But looking at the choice with present eyes, I think I continue to prefer the system over the alternative simply for the fact that it treats two different types of operation separately, namely, changes in graphical structure versus changes in the placement of variables.

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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.

- Peirce’s Law • This Blog • InterSciWiki • MyWikiBiz • Wikiversity

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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 …*

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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.

See also the discussions at the following locations:

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

☞ Functional Logic : Higher Order Propositions

Just a reminder to get back to this later …

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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:

- Logical Graphs (OEIS Wiki)
- Logical Graphs (Wikiversity)
- Logical Graphs (Inquiry Blog) • (1) • (2)

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.

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.

- Table A. Existential Interpretation
- Table B. Entitative Interpretation
- Table C. Dualing Interpretations

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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.

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