## Survey of Theme One Program • 3

This is a Survey of blog and wiki posts relating to the Theme One Program I worked on all through the 1980s.  The aim was to develop fundamental algorithms and data structures for integrating empirical learning with logical reasoning.  I had earlier developed separate programs for basic components of those tasks, namely, 2-level formal language learning and propositional constraint satisfaction, the latter using an extension of C.S. Peirce’s logical graphs as a syntax for propositional logic.  Thus arose the question of how well it might be possible to get “empiricist” and “rationalist” modi operandi to cooperate.  The ultimate vision was the design of an Automated Research Tool able to double as a medium for Inquiry Driven Education.

### References

• Awbrey, S.M., and Awbrey, J.L. (May 1991), “An Architecture for Inquiry • Building Computer Platforms for Discovery”, Proceedings of the Eighth International Conference on Technology and Education, Toronto, Canada, pp. 874–875.  Online.
• Awbrey, J.L., and Awbrey, S.M. (January 1991), “Exploring Research Data Interactively • Developing a Computer Architecture for Inquiry”, Poster presented at the Annual Sigma Xi Research Forum, University of Texas Medical Branch, Galveston, TX.
• Awbrey, J.L., and Awbrey, S.M. (August 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, pp. 9–15.  Online.

## Animated Logical Graphs • 38

Three examples of propositional proofs in logical graphs using equational inference rules can be found at the following location.

Animated proofs of the three examples, along with links to detailed descriptions, are shown below.

• Two-Thirds Majority Function • (1)(2)(3)

## Survey of Animated Logical Graphs • 3

This is one of several Survey posts I’ll be drafting from time to time, starting with minimal stubs and collecting links to the better variations on persistent themes I’ve worked on over the years.  After that I’ll look to organizing and revising the assembled material with an eye toward developing more polished articles.

## Animated Logical Graphs • 37

Another dimension of proof style has to do with how much information is kept or lost as the argument develops.  For the moment let’s focus on classical deductive reasoning at the propositional level.  Then we can distinguish between equational inferences, which keep all the information represented by the input propositions, and implicational inferences, which permit information to be lost as the proof proceeds.

Information-Preserving vs. Information-Reducing Inferences
Implicit in Peirce’s systems of logical graphs is the ability to use equational inferences.  Spencer Brown drew this out and turned it to great advantage in his revival of Peirce’s graphical forms.  As it affects “logical flow” this allows for bi-directional or reversible flows, you might even say a “logical equilibrium” between two states of information.

It is probably obvious when we stop to think about it, but seldom remarked, that all the more familiar inference rules, like modus ponens and resolution or transitivity, entail in general a loss of information as we traverse their arrows or turnstiles.

For example, the usual form of modus ponens takes us from knowing $p$ and $p \Rightarrow q$ to knowing $q$ but in fact we know more, we actually know $p \land q.$  With that in mind we can formulate two variants of modus ponens, one reducing and one preserving the actual state of information, as shown in the following figure.

There’s more discussion of this topic at the following location.

To be continued …

## Animated Logical Graphs • 36

Dear Dick,

You asked, “Is this measure, the logical flow of a proof, of any interest?”

I was not sure how you define the measure of flow in a proof — it seemed to have something to do with the number of implication arrows in the argument structure?

But this does bring up interesting issues of “proof style” …

Propositional calculus as a formal language and boolean functions as an object domain form an instructive microcosm for many issues of logic writ large.  The relation between proof theory and model theory is one of those issues, the status of propositional logic as a special case notwithstanding.

Folks who pursue the CSP–GSB line of development in graphical syntax for propositional calculus are especially likely to notice the following dimensions of proof style.

Formal Duality
This goes back to Peirce’s discovery of the “amphecks” as sole sufficient primitives for propositional calculus and the duality between Both Not (nnor) and Not Both (nand).  The same duality is present in Peirce’s graphical systems for propositional calculus.  It is analogous to the duality in projective geometry and it means we are always proving two theorems for the price of one.  That’s a reduction in complexity — it raises the question of how many such group-theoretic reductions we can find.

To be continued …

## Animated Logical Graphs • 35

The smoothest way I know to do propositional calculus is by using minimal negation operators as primitives, parsing propositional formulas into (painted and rooted) cactus graphs, and using the appropriate extension of the axiom set from Charles S. Peirce’s logical graphs and G. Spencer Brown’s laws of form.  There’s a quick link here:

## Mathematical Method • Discussion 7

Dear Alex,

You raised the following point:

AS:
One important usage of a sign is as an element of a language,
especially a formal one, i.e. with a formal grammar.

For context you cited a standard definition of a formal language with a formal grammar (Aho and Ullman 1972).

Viewed from the standpoint of pragmatic semiotics, where a sign relation $L$ is a structure of the form $L \subseteq O \times S \times I,$ we are starting out on pretty much the same page, since I’m always thinking of a sign $s$ as an element of a sign domain $S$ and I’m mainly interested in cases where the sign domain $S$ is a formal language with a formal grammar along the lines defined above.

That brings us to your question, “What is the grammar of Peirce’s language?”, which I will take up next time.

### References

• Barwise, J. (1977), “An Introduction to First-Order Logic”, pp. 5–46 in Barwise, J. (1977, ed.), Handbook of Mathematical Logic, Elsevier (North Holland), Amsterdam.
• Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science

## Mathematical Method • Discussion 6

Dear Alex,

Thanks for the very apt segue from Jon Barwise —

Modern mathematics might be described as the science of abstract objects, be they real numbers, functions, surfaces, algebraic structures or whatever.  Mathematical logic adds a new dimension to this science by paying attention to the language used in mathematics, to the ways abstract objects are defined, and to the laws of logic which govern us as we reason about these objects.  The logician undertakes this study with the hope of understanding the phenomena of mathematical experience and eventually contributing to mathematics, both in terms of important results that arise out of the subject itself (Gödel’s Second Incompleteness Theorem is the most famous example) and in terms of applications to other branches of mathematics.  (Barwise p. 6)

When it comes to mathematics as the science of abstract objects I have my personal favorite classes among its abstract gardens and zoos.  One order of particular interest in the great chain of abstract being descends from the family of mathematical relations to the genus of triadic relations to the species of triadic sign relations.

By a curious turn, but no real surprise when we stop to think about it, sign relations, with their object, sign, and interpretant sign domains, come into being whenever we reflect on the systems of signs we use to describe any universe of objects, abstract or otherwise, and thus they are just the tickets we need to enter that “new dimension” of mathematical logic.

### References

• Barwise, J. (1977), “An Introduction to First-Order Logic”, pp. 5–46 in Barwise, J. (1977, ed.), Handbook of Mathematical Logic, Elsevier (North Holland), Amsterdam.
• Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science

## Animated Logical Graphs • 34

Dear John,

I can’t imagine why anyone would bother with Peirce’s logic if it’s just Frege and Russell in a different syntax, which has been the opinion I usually get from FOL fans.  But the fact is Peirce’s 1870 “Logic of Relatives” is already far in advance of anything we’d see again for a century, in principle in most places, in practice in many others, chock full of revolutionary ideas, not all of which he developed fully in subsequent work.  Although I studied the 1870 Logic from early on I did not realize how far ahead of its time it was until I began reading approaches to logic from category-theoretic and computation-theoretic angles in the 1970s and 1980s.  An indication of Peirce’s innovations can be found in the series of selections and commentary I started on the 1870 Logic of Relatives.

Here’s the work in progress so far on the OEIS Wiki.

Here’s the overview for a parallel series of blog posts.

## Mathematical Method • Discussion 5

Dear Paul,

“How We Think” is a topic for the descriptive science of psychology, and its ways are legion beyond definitive or exhaustive description.

“How We Ought To Think” if we wish to succeed at specified purposes is a topic for the normative science of logic, lumping together for the moment the evolving varieties of informal, formal, mathematical, and technologically augmented methods.

They’re all good questions and I see no reason not to pursue them all, aside from the limitations of our brief lives, but we have to keep the spectrum of different aims sorted.

John Dewey wrote the book How We Think in 1910.  Peirce had earlier summed up his “non-psychological conception of logic” in the pithy motto “Logic has nothing to do with how we think” and this led some scholars to suspect Dewey’s title was aimed as a poke in Peirce’s ribs.  But the book itself is a How-To guide devoted to improving our capacity for learning and reasoning, what we’d call today instruction in critical thinking.

All that is prologue to Vannevar Bush’s 1945 article, “As We May Think”, projecting the ways technology may amplify our capacity for inquiry going forward into the future.  I think this is where we came in …

### Reference

• Eisele, C. (1982), “Mathematical Methodology in the Thought of Charles S. Peirce”, Historia Mathematica 9, pp. 333–341.  OnlinePDF.

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science