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May 3, 2024, 3:30:32 PMMay 3

to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG

Mathematical Duality in Logical Graphs • 1

• https://inquiryintoinquiry.com/2024/05/03/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”

All,

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

• https://inquiryintoinquiry.com/2013/01/31/duality-indicating-unity-1/

C.S. Peirce • Logic of Number

• https://inquiryintoinquiry.com/2012/09/01/c-s-peirce-logic-of-number-ms-229/

C.S. Peirce • Syllabus • Selection 1

• https://inquiryintoinquiry.com/2014/08/24/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.

Regards,

Jon

cc: https://www.academia.edu/community/LbAn0D

• https://inquiryintoinquiry.com/2024/05/03/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”

All,

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

• https://inquiryintoinquiry.com/2013/01/31/duality-indicating-unity-1/

C.S. Peirce • Logic of Number

• https://inquiryintoinquiry.com/2012/09/01/c-s-peirce-logic-of-number-ms-229/

C.S. Peirce • Syllabus • Selection 1

• https://inquiryintoinquiry.com/2014/08/24/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.

Regards,

Jon

cc: https://www.academia.edu/community/LbAn0D

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