Inquiry Into Inquiry

Re: Laws of Form

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 algebra and the arithmetic of logical graphs.  Peirce came right up to the threshold of discovering that relationship several times in his later work on existential graphs 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.

• Logical Graphs
• Primary Arithmetic as Semiotic System
• Primary Algebra as Pattern Calculus

The other issue has to do with my using a different than Spencer Brown.  I believe I even called it in the early days but eventually lost the prime as time went by.  As far as I can remember, it initially had to do with negotiating between the systems of C.S. Peirce and Spencer Brown but I think I stuck with the variant because it sorts the types of change — modifying structure and moving variables — into different bins.

• Image Files
• This Blog • I₁ • I₂ • J₁ • J₂
• Oeis Wiki • I₁ • I₂ • J₁ • J₂

See also the discussions at the following locations.

• Logical Graphs • Formal Development
• Propositional Equation Reasoning Systems

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science

Re: Laws of Form

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 for that purpose:

• Logical Graphs
• Propositional Equation Reasoning Systems

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

• Logical Graphs (Wikiversity)
• Logical Graphs (Inquiry Blog) • (1) • (2)

One big issue arising 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 which 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 opted to interpret 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.

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

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science

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, converting 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.

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science