Inquiry Into Inquiry

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

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

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

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.

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

Praeclarum Theorema • 1

Posted on October 13, 2023 by Jon Awbrey

The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus noted and named by G.W. Leibniz, who stated and proved it in the following manner.

If a is b and d is c, then ad will be bc.
This is a fine theorem, which is proved in this way:
a is b, therefore ad is bd (by what precedes),
d is c, therefore bd is bc (again by what precedes),
ad is bd, and bd is bc, therefore ad is bc. Q.E.D.

— Leibniz • Logical Papers, p. 41.

Expressed in contemporary logical notation, the theorem may be written as follows.

$((a \Rightarrow b) \land (d \Rightarrow c)) \Rightarrow ((a \land d) \Rightarrow (b \land c))$

Expressed as a logical graph under the existential interpretation, the theorem takes the shape of the following formal equivalence or propositional equation.

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

Logical Graphs • Discussion 9

Posted on October 12, 2023 by Jon Awbrey

Re: Logical Graphs • Formal Development
Re: Laws of Form • Lyle Anderson

LA:: The Gestalt Switch from parenthesis to graphs is stimulating. There are probably things in Laws of Form that we didn’t see because we were blinded by the crosses.

Lyle,

That has been my experience. Viewing a space of mathematical objects from a new angle and changing the basis of representation can bring out new and surprising aspects of their form and even expand the field of view to novel directions of generalization.

One of the first things I learned in the early years of computing with logical graphs is how essential it is to “slip the surly bonds” of the planar embedding and work with free trees in a space of their own.

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

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Propositional Equation Reasoning Systems, Relation Theory, Semiotics, Sign Relations, Spencer Brown, Topology, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Propositional Equation Reasoning Systems, Relation Theory, Semiotics, Sign Relations, Spencer Brown, Topology, Visualization | 5 Comments

Praeclarum Theorema

Posted on October 5, 2023 by Jon Awbrey

Introduction

The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus noted and named by G.W. Leibniz, who stated and proved it in the following manner.

— Leibniz • Logical Papers, p. 41.

Expressed in contemporary logical notation, the theorem may be written as follows.

$((a \Rightarrow b) \land (d \Rightarrow c)) \Rightarrow ((a \land d) \Rightarrow (b \land c))$

Expressed as a logical graph under the existential interpretation, the theorem takes the shape of the following formal equivalence or propositional equation.

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

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

Logical Graphs • Discussion 8

Posted on October 2, 2023 by Jon Awbrey

Re: Logical Graphs • Formal Development
Re: Laws of Form • Alex Shkotin

Hi Alex,

I got my first brush with graph theory in a course on the Foundations of Mathematics Frank Harary taught at the University of Michigan in 1970. Frank was the don, founder, mover, and shaker of what we affectionately called the “MiGhTy” school of graph theory, spawned at U of M, Michigan State, Illinois, Indiana, and eventually spreading to other hotbeds of research in the Midwest and beyond. Later I took my first graduate course in graph theory from Ed Palmer at Michigan State, using Harary’s Graph Theory as the text of choice.

Definitions of graphs vary in style and substance in accord with the level of abstraction required by a particular approach or application. The following is a classic formulation, one which covers the essential ideas in a very short space, and one whose elegance and power I’ve come to appreciate more and more as time goes by.

A graph $G$ consists of a finite nonempty set $V = V(G)$ of $p$ points together with a prescribed set $X$ of $q$ unordered pairs of distinct points of $V.$ Each pair $x = \{ u, v \}$ of points in $X$ is a line of $G,$ and $x$ is said to join $u$ and $v.$ We write $x = uv$ and say that $u$ and $v$ are adjacent points (sometimes denoted $u ~\mathrm{adj}~ v$ ); point $u$ and line $x$ are incident with each other, as are $v$ and $x.$ If two distinct lines $x$ and $y$ are incident with a common point, then they are adjacent lines. A graph with $p$ points and $q$ lines is called a $(p, q)$ graph. The $(1, 0)$ graph is trivial. (Harary, Graph Theory, p. 9).

I’ll be hewing fairly close to that definition and terminology, though most graph theorists are used to the more common variations, like nodes instead of points and edges instead of lines — except for the notion of painted graphs where I had to invent a new term due to the fact that labels and colors were already taken for other uses.

References

Harary, F. (1969), Graph Theory, Addison-Wesley, Reading, MA.
Harary, F., and Palmer, E.M. (1973), Graphical Enumeration, Academic Press, New York, NY.
Palmer, E.M. (1985), Graphical Evolution : An Introduction to the Theory of Random Graphs, John Wiley and Sons, New York, NY.

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

Logical Graphs • Discussion 7

Posted on October 1, 2023 by Jon Awbrey

Re: Logical Graphs • Formal Development
Re: Laws of Form • Alex Shkotin

AS:: When we look at undirected graph it is usual, before describing a rules of graph transformation, to describe exactly what kind of graphs we are working with —

Hi Alex,

I am traveling this week, with limited internet. There’s a quickie glossary under the heading “Painted And Rooted Cacti” on the following blog page.

Theme One Program • Exposition 2

Regards,
Jon

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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