Inquiry Into Inquiry

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 Contribution to the Philosophy of Notation”, American Journal of Mathematics 7 (1885), 180–202. Reprinted (CP 3.359–403), (CE 5, 162–190).

Resources

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

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 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 | 9 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.

$((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p$

The converse implication $p \Rightarrow ((p \Rightarrow q) \Rightarrow p)$ is clear enough on general principles, since $p \Rightarrow (r \Rightarrow p)$ holds for any proposition $r.$

Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce’s law is expressed by the following equation.

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

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

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

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

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”, American Journal of Mathematics 7 (1885), 180–202. Reprinted (CP 3.359–403), (CE 5, 162–190).

Resources

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

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 : 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

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

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 form.

$((p \Rightarrow q) \Rightarrow p) \Rightarrow p$

Peirce’s law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system one chooses for classical propositional calculus determines whether Peirce’s law is taken as an axiom or proven as a theorem.

History

Here is Peirce’s own statement and proof of the law:

A fifth icon is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:

$\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.$

This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent $x$ being false while its antecedent $(x \,-\!\!\!< y) \,-\!\!\!< x$ is true. If this is true, either its consequent, $x,$ is true, when the whole formula would be true, or its antecedent $x \,-\!\!\!< y$ is false. But in the last case the antecedent of $x \,-\!\!\!< y,$ that is $x,$ must be true. (Peirce, CP 3.384).

Peirce goes on to point out an immediate application of the law:

From the formula just given, we at once get:

$\{ (x \,-\!\!\!< y) \,-\!\!\!< \alpha \} \,-\!\!\!< x,$

where the $\alpha$ is used in such a sense that $(x \,-\!\!\!< y) \,-\!\!\!< \alpha$ means that from $(x \,-\!\!\!< y)$ every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of $x$ follows the truth of $x.$ (Peirce, CP 3.384).

Note. Peirce uses the sign of illation $``-\!\!\!<"$ for implication. In one place he explains $``-\!\!\!<"$ as a variant of the sign $``\le"$ for less than or equal to; in another place he suggests that $A \,-\!\!\!< B$ is an iconic way of representing a state of affairs where $A,$ in every way that it can be, is $B.$

References

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).
Peirce, Charles Sanders (1931–1935, 1958), Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph).
Peirce, Charles Sanders (1981–), Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page).

Resources

Logic Syllabus • Logical Graphs • Peirce’s Law
Metamath Proof Explorer • Peirce’s Axiom
Charles Sanders Peirce • Bibliography

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

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 form.

$((p \Rightarrow q) \Rightarrow p) \Rightarrow p$

History

Here is Peirce’s own statement and proof of the law:

A fifth icon is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:

$\{ (x \,-\!\!\!< y) \,-\!\!\!< x \} \,-\!\!\!< x.$

Peirce goes on to point out an immediate application of the law:

From the formula just given, we at once get:

$\{ (x \,-\!\!\!< y) \,-\!\!\!< \alpha \} \,-\!\!\!< x,$

Graphical Representation

Graphical Proof

Using the axiom set given in the articles on logical graphs, Peirce’s law may be proved in the following manner.

The following animation replays the steps of the proof.

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.

$((p \Rightarrow q) \Rightarrow p) \Leftrightarrow p$

The converse implication $p \Rightarrow ((p \Rightarrow q) \Rightarrow p)$ is clear enough on general principles, since $p \Rightarrow (r \Rightarrow p)$ holds for any proposition $r.$

Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce’s law is expressed by the following equation.

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.

The following animation replays the steps of the proof.

References

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).
Peirce, Charles Sanders (1931–1935, 1958), Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as (CP volume.paragraph).
Peirce, Charles Sanders (1981–), Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN. Cited as (CE volume, page).

Resources

Logic Syllabus • Logical Graphs • Peirce’s Law
Metamath Proof Explorer • Peirce’s Axiom
Charles Sanders Peirce • Bibliography

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

Survey of Animated Logical Graphs • 6

Posted on October 16, 2023 by Jon Awbrey

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph-theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

Beginnings

Logical Graphs • First Impressions
Logical Graphs • Formal Development

Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)

Elements

Examples

Peirce’s Law

This Blog • OEIS Wiki
Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Praeclarum Theorema

This Blog • OEIS Wiki
Blog Series • (1) • (2) • (3)

Proof Animations

Excursions

Applications

Blog Series

Animated Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21) • (22) • (23) • (24) • (25) • (26) • (27) • (28) • (29) • (30) • (31) • (32) • (33) • (34) • (35) • (36) • (37) • (38) • (39) • (40) • (41) • (42) • (43) • (44) • (45) • (46) • (47) • (48) • (49) • (50) • (51) • (52) • (53) • (54) • (55) • (56) • (57) • (58) • (59) • (60) • (61) • (62) • (63) • (64) • (65) • (66) • (67) • (68) • (69) • (70) • (71) • (72) • (73) • (74) • (75) • (76) • (77) • (78) • (79) • (80) • (81)

Logic Syllabus

Discussions • (1) • (2)

Logical Graphs

Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Logical Graphs • Interpretive Duality • (1) • (2) • (3) • (4)

Logical Graphs, Iconicity, Interpretation • (1) • (2)

Logical Graphs, Truth Tables, Venn Diagrams • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Minimal Negation Operators • (1) • (2) • (3) • (4)

Discussions • (1) • (2)

Genus, Species, Pie Charts, Radio Buttons • (1)

Discussions • (1) • (2) • (3) • (4) • (5)

Anamnesis

CSP, GSB, & Me • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16)

cc: FB | Logical Graphs • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Computational Complexity, Constraint Satisfaction Problems, Differential Logic, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Computational Complexity, Constraint Satisfaction Problems, Differential Logic, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | 1 Comment

Praeclarum Theorema • 3

Posted on October 15, 2023 by Jon Awbrey

Re: Praeclarum Theorema • (1) • (2)

The steps of the proof are replayed in the following animation.

Reference

Leibniz, Gottfried W. (1679–1686?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.

Readings

Jon Awbrey • Propositional Equation Reasoning Systems
John F. Sowa • Peirce’s Rules of Inference

Resources

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

Posted in Animata, C.S. Peirce, Cactus Graphs, Deduction, Equational Inference, Form, Laws of Form, Leibniz, Logic, Logical Graphs, Mathematics, Model Theory, Painted Cacti, Praeclarum Theorema, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Relation Theory, Semiotics, Spencer Brown, Visualization | Tagged Animata, C.S. Peirce, Cactus Graphs, Deduction, Equational Inference, Form, Laws of Form, Leibniz, Logic, Logical Graphs, Mathematics, Model Theory, Painted Cacti, Praeclarum Theorema, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Relation Theory, Semiotics, Spencer Brown, Visualization | 6 Comments

Praeclarum Theorema • 2

Posted on October 14, 2023 by Jon Awbrey

Re: Praeclarum Theorema • 1

And here’s a neat proof of that nice theorem —

Reference

Leibniz, Gottfried W. (1679–1686?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.

Readings

Jon Awbrey • Propositional Equation Reasoning Systems
John F. Sowa • Peirce’s Rules of Inference

Resources

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

	Differential Logic •… on Survey of Animated Logical Gra…
	Differential Logic •… on Survey of Differential Logic •…
	Differential Logic •… on Logic Syllabus
	Survey of Differenti… on Differential Logic • 4
	Differential Logic •… on Survey of Animated Logical Gra…
	Differential Logic •… on Survey of Differential Logic •…
	Differential Logic •… on Logic Syllabus
	Survey of Differenti… on Differential Logic • 3
	Survey of Differenti… on Differential Logic • 2
	Survey of Differenti… on Differential Logic • 1

Equational Form (cont.)

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