Peirce's Law

[Jon Awbrey]

Peirce's Law • 1
• https://inquiryintoinquiry.com/2023/10/19/peirces-law-1/

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 ⇒ q) ⇒ p) ⇒ 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) ‒< α} ‒< x,

❝where the α is used in such a sense that (x ‒< y) ‒< α 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 “≤” 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
• https://oeis.org/wiki/Logic_Syllabus

Logical Graphs
• https://oeis.org/wiki/Logical_Graphs

Peirce's Law
• https://oeis.org/wiki/Peirce%27s_law

Metamath Proof Explorer
• https://us.metamath.org/

Peirce's Axiom
• https://us.metamath.org/mpeuni/peirce.html

Regards,

Jon

cc: https://www.academia.edu/community/V1grBl

[Jon Awbrey]

Peirce's Law • 2
• https://inquiryintoinquiry.com/2023/10/20/peirces-law-2/

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.

Peirce's Law
• https://inquiryintoinquiry.files.wordpress.com/2023/10/peirces-law-3.0.png

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 —

Logical Graphs
• https://oeis.org/wiki/Logical_Graphs

Peirce's Law
• https://oeis.org/wiki/Peirce%27s_law

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/

Propositional Equation Reasoning Systems
• https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

cc: https://www.academia.edu/community/l8o69L

[Jon Awbrey]

Peirce's Law • 3
• https://inquiryintoinquiry.com/2023/10/21/peirces-law-3/

Graphical Proof —

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

Peirce's Law • Proof
• https://inquiryintoinquiry.files.wordpress.com/2023/10/peirces-law-e280a2-proof-3.0.png

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 —

Logical Graphs
• https://oeis.org/wiki/Logical_Graphs

Peirce's Law
• https://oeis.org/wiki/Peirce%27s_law

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/

Propositional Equation Reasoning Systems
• https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

cc: https://www.academia.edu/community/lz7Nj5

[Jon Awbrey]

Peirce's Law • 4
• https://inquiryintoinquiry.com/2023/10/22/peirces-law-4/

Proof Animation —

The following animation replays the steps of the proof.

Peirce's Law • Proof Animation
• https://inquiryintoinquiry.files.wordpress.com/2012/01/peirces-law-2-0-animation.gif

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 —

Logical Graphs
• https://oeis.org/wiki/Logical_Graphs

Peirce's Law
• https://oeis.org/wiki/Peirce%27s_law

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/

Propositional Equation Reasoning Systems
• https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

cc: https://www.academia.edu/community/L2kXEl

[Jon Awbrey]

Peirce's Law • 5
• https://inquiryintoinquiry.com/2023/10/23/peirces-law-5/

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 ⇒ q) ⇒ p) ⇔ p

The converse implication p ⇒ ((p ⇒ q) ⇒ p) is clear enough on
general principles, since p ⇒ (r ⇒ 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.

Peirce's Law : Strong Form
• https://inquiryintoinquiry.files.wordpress.com/2023/10/peirces-law-strong-form-3.0.png

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 —

Logical Graphs
• https://oeis.org/wiki/Logical_Graphs

Peirce's Law
• https://oeis.org/wiki/Peirce%27s_law

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/

Propositional Equation Reasoning Systems
• https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

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

[Jon Awbrey]

Peirce's Law • 6
• https://inquiryintoinquiry.com/2023/10/24/peirces-law-6/

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.

Peirce's Law : Strong Form • Proof
• https://inquiryintoinquiry.files.wordpress.com/2023/10/peirces-law-strong-form-e280a2-proof-3.0.png

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 —

Logical Graphs
• https://oeis.org/wiki/Logical_Graphs

Peirce's Law
• https://oeis.org/wiki/Peirce%27s_law

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/

Propositional Equation Reasoning Systems
• https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

cc: https://www.academia.edu/community/5kqx9L

[Jon Awbrey]

Peirce's Law • 7
• https://inquiryintoinquiry.com/2023/10/25/peirces-law-7/

Equational Form (concl.)

The following animation replays the steps of the proof.

Peirce's Law : Strong Form • Proof Animation
• https://inquiryintoinquiry.files.wordpress.com/2012/01/peirces-law-strong-form-2-0-animation.gif

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 —

Logical Graphs
• https://oeis.org/wiki/Logical_Graphs

Peirce's Law
• https://oeis.org/wiki/Peirce%27s_law

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/

Propositional Equation Reasoning Systems
• https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

cc: https://www.academia.edu/community/V3pxwL