Monthly Archives: October 2023

Logical Graphs • Interpretive Duality 2

Posted on October 29, 2023 by Jon Awbrey

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 … Continue reading →

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Interpretive Duality, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 4 Comments

Logical Graphs • Interpretive Duality 1

Posted on October 26, 2023 by Jon Awbrey

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 … Continue reading →

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Interpretive Duality, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 4 Comments

Peirce’s Law • 7

Posted on October 25, 2023 by Jon Awbrey

Equational Form (concl.) The following animation replays the steps of the proof. Reference Peirce, Charles Sanders (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7 (1885), 180–202.  Reprinted (CP 3.359–403), (CE 5, 162–190). … Continue reading →

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 5 Comments

Peirce’s Law • 6

Posted on October 24, 2023 by Jon Awbrey

Equational Form (cont.) Using the axioms and theorems listed in the entries on logical graphs, the equational form of Peirce’s law may be proved in the following manner. Reference Peirce, Charles Sanders (1885), “On the Algebra of Logic : A … Continue reading →

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 5 Comments

Peirce’s Law • 5

Posted on October 23, 2023 by Jon Awbrey

Equational Form A stronger form of Peirce’s law also holds, in which the final implication is observed to be reversible, resulting in the following equivalence. The converse implication is clear enough on general principles, since holds for any proposition Representing … Continue reading →

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 5 Comments

Peirce’s Law • 4

Posted on October 22, 2023 by Jon Awbrey

Proof Animation The following animation replays the steps of the proof. Reference Peirce, Charles Sanders (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7 (1885), 180–202.  Reprinted (CP 3.359–403), (CE 5, 162–190). Resources … Continue reading →

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 5 Comments

Peirce’s Law • 3

Posted on October 21, 2023 by Jon Awbrey

Graphical Proof Using the axiom set given in the articles on logical graphs, Peirce’s law may be proved in the following manner. Reference Peirce, Charles Sanders (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, … Continue reading →

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 5 Comments

Peirce’s Law • 2

Posted on October 20, 2023 by Jon Awbrey

Graphical Representation Representing propositions in the language of logical graphs, and operating under the existential interpretation, Peirce’s law is expressed by means of the following formal equivalence or logical equation. Reference Peirce, Charles Sanders (1885), “On the Algebra of Logic … Continue reading →

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 5 Comments

Peirce’s Law • 1

Posted on October 19, 2023 by Jon Awbrey

A Curious Truth of Classical Logic Peirce’s law is a propositional calculus formula which states a non‑obvious truth of classical logic and affords a novel way of defining classical propositional calculus. Introduction Peirce’s law is commonly expressed in the following … Continue reading →

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 5 Comments

Peirce’s Law

Posted on October 18, 2023 by Jon Awbrey

A Curious Truth of Classical Logic Peirce’s law is a propositional calculus formula which states a non‑obvious truth of classical logic and affords a novel way of defining classical propositional calculus. Introduction Peirce’s law is commonly expressed in the following … Continue reading →

Posted in C.S. Peirce, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce's Law, Proof Theory, Propositional Calculus, Propositions As Types Analogy, Semiotics, Spencer Brown, Visualization | Tagged , , , , , , , , , , , , , | 5 Comments