#minimalnegationoperators — Public Fediverse posts

Transformations of Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/05/22/transformations-of-logical-graphs-discussion-1/

Re: Laws of Form
• https://groups.io/g/lawsofform/topic/transformations_of_logical/105927945

Mauro Bertani
• https://groups.io/g/lawsofform/message/3204

Dear Mauro,

The couple of pages linked below give the clearest and quickest introduction I've been able to manage so far when it comes to the elements of logical graphs, at least, in the way I've come to understand them. The first page gives a lot of detail by way of motivation and computational implementation, so you could easily put that off till you feel a need for it. The second page lays out the precise axioms or initials I use — the first algebraic axiom varies a bit from Spencer Brown for a better fit with C.S. Peirce — and also shows the parallels between the dual interpretations.

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Additional Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Transformations of Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/05/22/transformations-of-logical-graphs-discussion-1/

Re: Laws of Form
• https://groups.io/g/lawsofform/topic/transformations_of_logical/105927945

Mauro Bertani
• https://groups.io/g/lawsofform/message/3204

Dear Mauro,

The couple of pages linked below give the clearest and quickest introduction I've been able to manage so far when it comes to the elements of logical graphs, at least, in the way I've come to understand them. The first page gives a lot of detail by way of motivation and computational implementation, so you could easily put that off till you feel a need for it. The second page lays out the precise axioms or initials I use — the first algebraic axiom varies a bit from Spencer Brown for a better fit with C.S. Peirce — and also shows the parallels between the dual interpretations.

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Additional Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Transformations of Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/05/22/transformations-of-logical-graphs-discussion-1/

Re: Laws of Form
• https://groups.io/g/lawsofform/topic/transformations_of_logical/105927945

Mauro Bertani
• https://groups.io/g/lawsofform/message/3204

Dear Mauro,

The couple of pages linked below give the clearest and quickest introduction I've been able to manage so far when it comes to the elements of logical graphs, at least, in the way I've come to understand them. The first page gives a lot of detail by way of motivation and computational implementation, so you could easily put that off till you feel a need for it. The second page lays out the precise axioms or initials I use — the first algebraic axiom varies a bit from Spencer Brown for a better fit with C.S. Peirce — and also shows the parallels between the dual interpretations.

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Additional Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Transformations of Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/05/22/transformations-of-logical-graphs-discussion-1/

Re: Laws of Form
• https://groups.io/g/lawsofform/topic/transformations_of_logical/105927945

Mauro Bertani
• https://groups.io/g/lawsofform/message/3204

Dear Mauro,

The couple of pages linked below give the clearest and quickest introduction I've been able to manage so far when it comes to the elements of logical graphs, at least, in the way I've come to understand them. The first page gives a lot of detail by way of motivation and computational implementation, so you could easily put that off till you feel a need for it. The second page lays out the precise axioms or initials I use — the first algebraic axiom varies a bit from Spencer Brown for a better fit with C.S. Peirce — and also shows the parallels between the dual interpretations.

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Additional Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Transformations of Logical Graphs • Discussion 1
• https://inquiryintoinquiry.com/2024/05/22/transformations-of-logical-graphs-discussion-1/

Re: Laws of Form
• https://groups.io/g/lawsofform/topic/transformations_of_logical/105927945

Mauro Bertani
• https://groups.io/g/lawsofform/message/3204

Dear Mauro,

The couple of pages linked below give the clearest and quickest introduction I've been able to manage so far when it comes to the elements of logical graphs, at least, in the way I've come to understand them. The first page gives a lot of detail by way of motivation and computational implementation, so you could easily put that off till you feel a need for it. The second page lays out the precise axioms or initials I use — the first algebraic axiom varies a bit from Spencer Brown for a better fit with C.S. Peirce — and also shows the parallels between the dual interpretations.

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Additional Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2024/01/26/survey-of-semiotics-semiosis-sign-relations-5/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • Discussion 2.2
• https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-2/

What you say about deriving arithmetic, algebra, group theory, and all the rest from the calculus of indications may well be true, but it remains to be shown if so, and that's aways down the road from here.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • Discussion 2.2
• https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-2/

What you say about deriving arithmetic, algebra, group theory, and all the rest from the calculus of indications may well be true, but it remains to be shown if so, and that's aways down the road from here.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • Discussion 2.2
• https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-2/

What you say about deriving arithmetic, algebra, group theory, and all the rest from the calculus of indications may well be true, but it remains to be shown if so, and that's aways down the road from here.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • Discussion 2.2
• https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-2/

What you say about deriving arithmetic, algebra, group theory, and all the rest from the calculus of indications may well be true, but it remains to be shown if so, and that's aways down the road from here.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • Discussion 2.2
• https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-2/

What you say about deriving arithmetic, algebra, group theory, and all the rest from the calculus of indications may well be true, but it remains to be shown if so, and that's aways down the road from here.

Resources —

Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.2
• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.2
• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.2
• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.2
• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.2
• https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.1
• 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”

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.1
• 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”

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.1
• 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”

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.1
• 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”

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

Mathematical Duality in Logical Graphs • 1.1
• 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”

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.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
#CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

@hochstenbach @josd

A program I worked on all through the 80s implemented a propositional modeler based on #Peirce's #LogicalGraphs, improving the efficiency of the Alpha level through the use of #MinimalNegationOperators. There's a collection of articles, blog posts, and group discussions about that linked on the following page.

#ThemeOneProgram • #SurveyPage
• https://inquiryintoinquiry.com/2022/06/12/survey-of-theme-one-program-4/

@albertcardona @NicoleCRust @complexsystems @cogneurophys @PessoaBrain @SussilloDavid @carlosbrody @Neurograce @neuralreckoning @tyrell_turing @DrYohanJohn @cian @WiringtheBrain @tdverstynen @neuralengine

I'll just tag this for later when I've had a chance to collect my thoughts.

#WinnerTakeAll #Grossberg #CompetitionCooperation
#McClelland #Rumelhart #ParallelDistributedProcessing
#CactusLanguage #LogicalGraphs #MinimalNegationOperators

#ThemeOneProgram • #JetsAndSharks 2
• https://inquiryintoinquiry.com/2022/08/30/theme-one-program-jets-and-sharks-2/

As we saw last time, Theme One reads the text file shown below and constructs a #CactusGraph #DataStructure in computer memory. The cactus graph represents a single #LogicalFormula in #PropositionalCalculus which embodies all the #LogicalConstraints defining the Jets and Sharks #DataBase.

Jets and Sharks • Log File
• https://inquiryintoinquiry.files.wordpress.com/2022/08/theme-one-guide-e280a2-jets-and-sharks-e280a2-log-file.png

#Logic #LogicalGraphs #Peirce
#CactusSyntax #LogicalCacti
#MinimalNegationOperators

#ThemeOneProgram • #JetsAndSharks 1.5
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

Reference —

#McClelland, J.L. (2015),
Explorations in #ParallelDistributedProcessing :
A Handbook of Models, Programs, and Exercises, 2nd ed. (draft),
#StanfordParallelDistributedProcessingLab.
• https://web.stanford.edu/group/pdplab/
#PDPHandbook Online —
• https://web.stanford.edu/group/pdplab/pdphandbook/
Section 2.3 —
• https://web.stanford.edu/group/pdplab/pdphandbook/handbookch3#x7-320002.3
Figure 2.1 —
• https://web.stanford.edu/group/pdplab/pdphandbook/jetsandsharkstable.png

#Logic #LogicalGraphs #Peirce
#CactusSyntax #LogicalCacti
#MinimalNegationOperators

#ThemeOneProgram • #JetsAndSharks 1.4
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

Displayed below is the text expression of a #TraversalString which Theme One parses into a #CactusGraph #DataStructure in computer memory. The cactus graph represents a single #LogicalFormula in #PropositionalCalculus which embodies all the #LogicalConstraints defining the Jets & Sharks #DataBase.

Log File
• https://inquiryintoinquiry.files.wordpress.com/2022/08/theme-one-guide-e280a2-jets-and-sharks-e280a2-log-file.png

#Logic #LogicalGraphs #Peirce
#CactusSyntax #LogicalCacti
#MinimalNegationOperators

#ThemeOneProgram • #JetsAndSharks 1.3
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

The manner of representation may be illustrated by transcribing a well-known example from the #ParallelDistributedProcessing literature (#McClelland and #Rumelhart 1988) and working through a couple of the associated exercises as translated into #LogicalGraphs.

#Logic #Peirce #Semiotics #Semiosis
#Grossberg #CompetitionCooperation
#GraphTheory #ModelTheory #ProofTheory
#LogicalCacti #MinimalNegationOperators

#ThemeOneProgram • #JetsAndSharks 1.2
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

One way to do this is to interpret the blank or #UnmarkedState as the #RestingState of a #NeuralPool, the bound or #MarkedState as its #ActivatedState, and to represent a mutually inhibitory pool of #Neurons \(a,b,c\) by the proposition \(\texttt{(}a\texttt{,}b\texttt{,}c\texttt{)}.\)

#Logic #LogicalGraphs #Peirce
#Grossberg #McClelland #Rumelhart
#GraphTheory #ModelTheory #ProofTheory
#LogicalCacti #MinimalNegationOperators

#ThemeOneProgram • #JetsAndSharks 1.1
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

Example 5. Jets and Sharks

The #PropositionalCalculus based on #MinimalNegationOperators can be interpreted in a way resembling the logic of #ActivationStates and #CompetitionConstraints in one class of #NeuralNetwork models.

#Logic #LogicalGraphs
#Peirce #Semiotics #Semiosis
#Grossberg #McClelland #Rumelhart
#ParallelDistributedProcessing #PDP
#GraphTheory #ModelTheory #ProofTheory