Mathematical Duality in Logical Graphs

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Jon Awbrey

May 3, 2024, 3:30:32 PMMay 3
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Mathematical Duality in Logical Graphs • 1

“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.

• Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin, London, UK.



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