Difference between revisions of "Peirce's law"

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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
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'''Peirce's law''' is a formula in [[propositional calculus]] that is commonly expressed in the following form:
 
'''Peirce's law''' is a formula in [[propositional calculus]] that is commonly expressed in the following form:
  
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==Graphical proof==
 
==Graphical proof==
  
Under the existential interpretation of Peirce's [[logical graph]]s, Peirce's law is represented by means of the following formal equivalence or logical equation.
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Under the existential interpretation of Peirce's [[logical graphs]], Peirce's law is represented by means of the following formal equivalence or logical equation.
  
 
{| align="center" border="0" cellpadding="10" cellspacing="0"
 
{| align="center" border="0" cellpadding="10" cellspacing="0"
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===Focal nodes===
 
===Focal nodes===
  
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* [[Inquiry Live]]
 
* [[Inquiry Live]]
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* [[Logic Live]]
 
* [[Logic Live]]
{{col-end}}
 
  
 
===Peer nodes===
 
===Peer nodes===
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* [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law @ InterSciWiki]
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* [http://mywikibiz.com/Peirce's_law Peirce's Law @ MyWikiBiz]
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* [http://ref.subwiki.org/wiki/Peirce's_law Peirce's Law @ Subject Wikis]
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* [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Wikiversity Beta]
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===Logical operators===
  
 
{{col-begin}}
 
{{col-begin}}
 
{{col-break}}
 
{{col-break}}
* [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law @ Beta Wikiversity]
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* [[Exclusive disjunction]]
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* [[Logical conjunction]]
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* [[Logical disjunction]]
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* [[Logical equality]]
 
{{col-break}}
 
{{col-break}}
* [http://mywikibiz.com/Peirce's_law Peirce's Law @ MyWikiBiz]
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* [[Logical implication]]
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* [[Logical NAND]]
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* [[Logical NNOR]]
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* [[Logical negation|Negation]]
 
{{col-end}}
 
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* [[Boolean function]]
 
* [[Boolean function]]
 
* [[Boolean-valued function]]
 
* [[Boolean-valued function]]
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* [[Differential logic]]
 
{{col-break}}
 
{{col-break}}
 
* [[Logical graph]]
 
* [[Logical graph]]
* [[Logical matrix]]
 
 
* [[Minimal negation operator]]
 
* [[Minimal negation operator]]
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* [[Multigrade operator]]
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* [[Parametric operator]]
 
* [[Peirce's law]]
 
* [[Peirce's law]]
 
{{col-break}}
 
{{col-break}}
 
* [[Propositional calculus]]
 
* [[Propositional calculus]]
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* [[Sole sufficient operator]]
 
* [[Truth table]]
 
* [[Truth table]]
 
* [[Universe of discourse]]
 
* [[Universe of discourse]]
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{{col-end}}
 
{{col-end}}
  
===Related articles===
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===Relational concepts===
  
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
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{{col-begin}}
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{{col-break}}
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* [[Continuous predicate]]
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* [[Hypostatic abstraction]]
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* [[Logic of relatives]]
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* [[Logical matrix]]
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* [[Relation (mathematics)|Relation]]
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* [[Relation composition]]
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* [[Relation construction]]
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* [[Relation reduction]]
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{{col-break}}
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* [[Relation theory]]
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* [[Relative term]]
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* [[Sign relation]]
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* [[Triadic relation]]
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{{col-end}}
  
* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;]
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===Information, Inquiry===
  
* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;]
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{{col-begin}}
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{{col-break}}
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* [[Inquiry]]
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* [[Dynamics of inquiry]]
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* [[Semeiotic]]
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* [[Logic of information]]
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{{col-break}}
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* [[Descriptive science]]
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* [[Normative science]]
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{{col-break}}
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* [[Pragmatic maxim]]
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* [[Truth theory]]
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{{col-end}}
  
==Document history==
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===Related articles===
 
 
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
 
  
 
{{col-begin}}
 
{{col-begin}}
 
{{col-break}}
 
{{col-break}}
* [http://mywikibiz.com/Peirce's_law Peirce's Law], [http://mywikibiz.com/ MyWikiBiz]
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
* [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://beta.wikiversity.org/ Beta Wikiversity]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
* [http://planetmath.org/encyclopedia/PeircesLaw.html Peirce's Law], [http://planetmath.org/ PlanetMath]
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* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
 
{{col-break}}
 
{{col-break}}
* [http://www.proofwiki.org/wiki/Peirce's_Law Peirce's Law], [http://www.proofwiki.org/ ProofWiki]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
* [http://knol.google.com/k/jon-awbrey/peirce-s-law/3fkwvf69kridz/10 Peirce's Law], [http://knol.google.com/ Google Knol]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
* [http://www.getwiki.net/-Peirce%92s_Law Peirce's Law], [http://www.getwiki.net/ GetWiki]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
 
{{col-break}}
 
{{col-break}}
* [http://www.wikinfo.org/index.php?title=Peirce%92s_law&oldid=29173 Peirce's Law], [http://www.wikinfo.org/ Wikinfo]
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* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
* [http://www.textop.org/wiki/index.php?title=Peirce's_law Peirce's Law], [http://www.textop.org/wiki/ Textop Wiki]
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* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
* [http://en.wikipedia.org/w/index.php?title=Peirce%27s_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia]
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* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
 
{{col-end}}
 
{{col-end}}
  
<br><sharethis />
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==Document history==
 +
 
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Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
 +
 
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law], [http://intersci.ss.uci.edu/ InterSciWiki]
 +
* [http://mywikibiz.com/Peirce's_law Peirce's Law], [http://mywikibiz.com/ MyWikiBiz]
 +
* [http://planetmath.org/PeircesLaw Peirce's Law], [http://planetmath.org/ PlanetMath]
 +
* [http://wikinfo.org/w/index.php/Peirce's_law Peirce's Law], [http://www.wikinfo.org/w/ Wikinfo]
 +
* [http://en.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://en.wikiversity.org/ Wikiversity]
 +
* [http://beta.wikiversity.org/wiki/Peirce's_law Peirce's Law], [http://beta.wikiversity.org/ Wikiversity Beta]
 +
* [http://en.wikipedia.org/w/index.php?title=Peirce's_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia]
  
 
[[Category:Artificial Intelligence]]
 
[[Category:Artificial Intelligence]]
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[[Category:Charles Sanders Peirce]]
 
[[Category:Combinatorics]]
 
[[Category:Combinatorics]]
 
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[[Category:History of Logic]]
 
[[Category:History of Logic]]
 
[[Category:History of Mathematics]]
 
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[[Category:Knowledge Representation]]
 
[[Category:Knowledge Representation]]
 
[[Category:Logic]]
 
[[Category:Logic]]

Latest revision as of 04:14, 18 November 2015

This page belongs to resource collections on Logic and Inquiry.

Peirce's law is a formula in propositional calculus that 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 that 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:

\(\{ (x \,-\!\!\!< y) \,-\!\!\!< a \} \,-\!\!\!< x,\)

where the \(a\!\) is used in such a sense that \((x \,-\!\!\!< y) \,-\!\!\!< a\) 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 \(B.\!\)

Graphical proof

Under the existential interpretation of Peirce's logical graphs, Peirce's law is represented by means of the following formal equivalence or logical equation.

Peirce's Law 1.0 Splash Page.png (1)

Proof. Using the axiom set given in the entry for logical graphs, Peirce's law may be proved in the following manner.

Peirce's Law 1.0 Marquee Title.png
Peirce's Law 1.0 Storyboard 1.png
Equational Inference Band Collect p.png
Peirce's Law 1.0 Storyboard 2.png
Equational Inference Band Quit ((q)).png
Peirce's Law 1.0 Storyboard 3.png
Equational Inference Band Cancel (( )).png
Peirce's Law 1.0 Storyboard 4.png
Equational Inference Band Delete p.png
Peirce's Law 1.0 Storyboard 5.png
Equational Inference Band Cancel (( )).png
Peirce's Law 1.0 Storyboard 6.png
Equational Inference Marquee QED.png
(2)

The following animation replays the steps of the proof.

Peirce's Law 2.0 Animation.gif
(3)

Equational form

A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible:

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

Proof 1

Given what precedes, it remains to show that:

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

But this is immediate, since \(p \Rightarrow (r \Rightarrow p)\) for any proposition \(r.\!\)

Proof 2

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 1.0 Splash Page.png (4)

Using the axioms and theorems listed in the article on logical graphs, the equational form of Peirce's law may be proved in the following manner:

Peirce's Law Strong Form 1.0 Marquee Title.png
Peirce's Law Strong Form 1.0 Storyboard 1.png
Equational Inference Rule Collect p.png
Peirce's Law Strong Form 1.0 Storyboard 2.png
Equational Inference Rule Quit ((q)).png
Peirce's Law Strong Form 1.0 Storyboard 3.png
Equational Inference Rule Cancel (( )).png
Peirce's Law Strong Form 1.0 Storyboard 4.png
Equational Inference Marquee QED.png
(5)

The following animation replays the steps of the proof.

Peirce's Law Strong Form 2.0 Animation.gif
(6)

Bibliography

  • 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).

Syllabus

Focal nodes

Peer nodes

Logical operators

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Related topics

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Relational concepts

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Information, Inquiry

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Related articles

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Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.