Inquiry Into Inquiry

Re: Boundary Logic

A wider field of investigation opens up at this point, spanning the diversity of interactions among languages we use, and systems of signs in general, to the thoughts ever streaming 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 in 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 story 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.

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

Re: Boundary Logic

For me, the heart of the matter is “what is the purpose of logic and what is the purpose of mathematics and what is their relationship?”

There are semiotic situations which appear to violate the initial conditions of logic but there are ways of approaching them without reducing our brains to jelly from the getgo.  Charles S. Peirce, following on Aristotle’s negotiation of the boundary between logic and rhetoric, developed his theory of triadic sign relations in large part to manage just those sorts of situations.

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

C.S. Peirce on “General” and “Vague”

• Collected Papers • CP 5.448
• Foundations of Mathematics (FOM) • (1) • (2) • (3)

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

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’s a gentle introduction at the other end of the link below, if you wish to achilles ahead.

• Differential Propositional Calculus • Part 1

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

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, and called axioms when interpreted as logical equivalences.  There are two arithmetic initials and two algebraic initials, as shown below.

Arithmetic Initials

Algebraic Initials

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 it treats two different types of operation separately, namely, changes in graphical structure versus changes in the placement of variables.

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

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) which 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 • OEIS Wiki • InterSciWiki • MyWikiBiz • Wikiversity

Resource

• Propositions As Types Analogy

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

Two things impacting 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 Peirce and Spencer Brown beset my brain with were a big part of what drove me back to math, since I could see I had no chance of resolving them without learning a lot more algebra, logic, and topology than I had learned till then.

Resources

• Prospects for Inquiry Driven Systems • Bibliography
• Mathematical Notes • Philosophical Notes • Source Materials

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

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