Inquiry Into Inquiry

Re: Mathematical Duality in Logical Graphs • 1
Re: Laws of Form • Lyle Anderson
Re: Brading, K., Castellani, E., and Teh, N., (2017), “Symmetry and Symmetry Breaking”, The Stanford Encyclopedia of Philosophy (Winter 2017), Edward N. Zalta (ed.).  Online.

Dear Lyle,

Thanks for the link to the article on symmetry and symmetry breaking.  I did once take a Master’s in Mathematics, specializing in combinatorics, graph theory, and group theory.  When it comes to the bearing of symmetry groups on logical graphs and the calculus of indications, it will take careful attention to the details of the relationship between the two interpretations singled out by Peirce and Spencer Brown.

Both Peirce and Spencer Brown recognized the relevant duality, if they differed in what they found most convenient to use in their development and exposition, and most of us will emphasize one interpretation or the other as a matter of facility or taste in a chosen application, so it requires a bit of effort to keep the underlying unity in focus.  I recently made another try at taking a more balanced view, drawing up a series of tables in parallel columns the way one commonly does with dual theorems in projective geometry, so I will shortly share more of that work.

Resources

• Logic Syllabus
• Logical Graphs • First Impressions
• Logical Graphs • Formal Development

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

All other sciences without exception depend upon the principles of mathematics;  and mathematics borrows nothing from them but hints.

C.S. Peirce • “Logic of Number”

A principal intention of this essay is to separate what are known as algebras of logic from the subject of logic, and to re‑align them with mathematics.

G. Spencer Brown • Laws of Form

The duality between entitative and existential interpretations of logical graphs tells us something important about the relation between logic and mathematics.  It tells us the mathematical forms giving structure to reasoning are deeper and more abstract at once than their logical interpretations.

A formal duality points to a more encompassing unity, founding a calculus of forms whose expressions can be read in alternate ways by switching the meanings assigned to a pair of primitive terms.  Spencer Brown’s mathematical approach to Laws of Form and the whole of Peirce’s work on the mathematics of logic shows both thinkers were deeply aware of this principle.

Peirce explored a variety of dualities in logic which he treated on analogy with the dualities in projective geometry.  This gave rise to formal systems where the initial constants, and thus their geometric and graph‑theoretic representations, had no uniquely fixed meanings but could be given dual interpretations in logic.

It was in this context that Peirce’s systems of logical graphs developed, issuing in dual interpretations of the same formal axioms which Peirce referred to as entitative graphs and existential graphs, respectively.  He developed only the existential interpretation to any great extent, since the extension from propositional to relational calculus appeared more natural in that case, but whether there is any logical or mathematical reason for the symmetry to break at that point is a good question for further research.

Resources

• Duality Indicating Unity
• C.S. Peirce • Logic of Number
• C.S. Peirce • Syllabus • Selection 1

References

• Peirce, C.S., [Logic of Number — Le Fevre] (MS 229), in Carolyn Eisele (ed., 1976),
  The New Elements of Mathematics by Charles S. Peirce, vol. 2, 592–595.  Excerpt.
• Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin, London, UK.

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Re: Interpretive Duality in Logical Graphs • 6

The last of our six ways of looking at interpretive duality is arrived at by taking the previous Table of Logical Graphs and Venn Diagrams and sorting it in Orbit Order.

Resources

• Logic Syllabus
• Logical Graphs • First Impressions
• Logical Graphs • Formal Development
• Information = Comprehension × Extension

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Re: Interpretive Duality in Logical Graphs • 2

Dualities are symmetries of order two and symmetries bear on complexity by reducing its measure in proportion to their order.  The inverse relationship between symmetry and the usual dissymmetries from dispersion and diversity to entropy and uncertainty is governed in cybernetics by the Law of Requisite Variety, the medium of whose exchange C.S. Peirce invested in the formula Information = Comprehension × Extension.

The duality between entitative and existential interpretations of logical graphs is one example of a mathematical symmetry but it’s not unusual to find symmetries within symmetries and it’s always rewarding to find them where they exist.  To that end let’s take up our Table of Venn Diagrams and Logical Graphs on Two Variables and sort the rows to bring together diagrams and graphs having similar shapes.  What defines their similarity is the action of a mathematical group whose operations transform the elements of each class among one another but intermingle no dissimilar elements.  In the jargon of transformation groups those classes are called orbits.  We find the sixteen rows partition into seven orbits, as shown below.

Resources

• Logic Syllabus
• Logical Graphs • First Impressions
• Logical Graphs • Formal Development
• Information = Comprehension × Extension

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Re: Interpretive Duality in Logical Graphs • 2

A more graphic picture of interpretive duality is given by the next Table, showing how logical graphs map to venn diagrams under entitative and existential interpretations.  Column 1 shows the logical graphs for the sixteen boolean functions on two variables.  Column 2 shows the venn diagrams associated with the entitative interpretation and Column 3 shows the venn diagrams associated with the existential interpretation.

Resources

• Logic Syllabus
• Logical Graphs • First Impressions
• Logical Graphs • Formal Development

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Re: Interpretive Duality in Logical Graphs • 1

Another way of looking at interpretive duality in logical graphs is given by the following Table, showing how logical graphs denote the sixteen boolean functions on two variables under entitative and existential interpretations, respectively.

Resources

• Logic Syllabus
• Logical Graphs • First Impressions
• Logical Graphs • Formal Development

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Re: Interpretive Duality in Logical Graphs • (1) • (2) • (3)

Last time we took up Peirce’s law, and saw how it might be expressed in two different ways, under the entitative and existential interpretations, respectively.  The next thing to do is see how our choice of interpretation bears on the patterns of proof we might find.  A sense of the possibilities may be gotten by displaying the two styles of proof in parallel columns, as shown below.

For convenience, the formal axioms and a few theorems of frequent use are linked below.

• Axiom I₁ • Condense/Protract
• Axiom I₂ • Cancel/Elicit
• Axiom J₁ • Delete/Insert
• Axiom J₂ • Collect/Distribute
• C₁ • Double Negation Theorem • Reflect/Reflect
• C₂ • Generation Theorem • Regenerate/Degenerate
• C₃ • Dominant Form Theorem • Quit/Quip

Resources

• Logical Graphs • First Impressions
• Logical Graphs • Formal Development

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Re: Interpretive Duality in Logical Graphs • (1) • (2)

For a sense of how the choice of interpretation bears on cases beyond the bare minimum complexity let us start with the familiar example of Peirce’s law, commonly expressed in the following form.

The following two formal equations show how Peirce’s law may be expressed in terms of logical graphs, operating under the entitative and existential interpretations, respectively.

Resources

• Peirce’s Law • (1) • (2) • (3) • (4) • (5) • (6) • (7)

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Re: Interpretive Duality in Logical Graphs • 1

A logical concept represented by a boolean variable has its extension, the cases it covers in a designated universe of discourse, and its comprehension (or intension), the properties it implies in a designated hierarchy of predicates.

The formulas and graphs tabulated in the previous post are well‑adapted to articulate the syntactic and intensional aspects of propositional logic.  But their very tailoring to those tasks tends to slight the extensional and therefore empirical applications of logic.  Venn diagrams, despite their unwieldiness as the number of logical dimensions increases, are indispensable in providing the visual intuition with a solid grounding in the extensions of logical concepts.  All that makes it worthwhile to reset our table of boolean functions on two variables to include the corresponding venn diagrams.

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

The duality between Entitative and Existential interpretations of logical graphs is a good example of a mathematical symmetry, in this case a symmetry of order two.  Symmetries of this and higher orders give us conceptual handles on excess complexity in the manifold of sensuous impressions, making it well worth the effort to seek them out and grasp them where we find them.

Both Peirce and Spencer Brown understood the significance of the mathematical unity underlying the dual interpretation of logical graphs.  Peirce began with the Entitative option and later switched to the Existential choice while Spencer Brown exercised the Entitative option in his Laws of Form.

In that vein, here’s a Rosetta Stone to give us a grounding in the relationship between boolean functions and our two readings of logical graphs.

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science