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Peirce's law (view source)

Revision as of 04:14, 18 November 2015

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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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</center>
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<p>where the <math>a\!</math> is used in such a sense that <math>(x \,-\!\!\!< y) \prec a</math> means that from <math>(x \,-\!\!\!< y)</math> every proposition follows.  With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math>  (Peirce, CP&nbsp;3.384).</p>
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<p>where the <math>a\!</math> is used in such a sense that <math>(x \,-\!\!\!< y) \,-\!\!\!< a</math> means that from <math>(x \,-\!\!\!< y)</math> every proposition follows.  With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of <math>x\!</math> follows the truth of <math>x.\!</math>  (Peirce, CP&nbsp;3.384).</p>
 
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|}
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==Graphical proof==
 
==Graphical proof==
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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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| [[Image:Peirce's_Law_Figure_1.jpg|500px]] || (1)
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| [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (1)
 
|}
 
|}
    
'''Proof.'''  Using the axiom set given in the entry for [[logical graphs]], Peirce's law may be proved in the following manner.
 
'''Proof.'''  Using the axiom set given in the entry for [[logical graphs]], Peirce's law may be proved in the following manner.
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{| align="center" border="0" cellpadding="10" cellspacing="0"
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{| align="center" cellpadding="8"
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| [[Image:Peirce's_Law_Figure_2.jpg|500px]] || (2)
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|
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
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|-
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| [[Image:Peirce's Law 1.0 Marquee Title.png|500px]]
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|-
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| [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]]
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|-
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| [[Image:Equational Inference Band Collect p.png|500px]]
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|-
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| [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]]
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|-
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| [[Image:Equational Inference Band Quit ((q)).png|500px]]
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|-
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| [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]]
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|-
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| [[Image:Equational Inference Band Cancel (( )).png|500px]]
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|-
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| [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]]
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|-
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| [[Image:Equational Inference Band Delete p.png|500px]]
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|-
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| [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]]
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|-
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| [[Image:Equational Inference Band Cancel (( )).png|500px]]
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|-
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| [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]]
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|-
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| [[Image:Equational Inference Marquee QED.png|500px]]
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|}
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| (2)
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|}
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The following animation replays the steps of the proof.
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{| align="center" cellpadding="8"
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|
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
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|-
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| [[Image:Peirce's Law 2.0 Animation.gif]]
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|}
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| (3)
 
|}
 
|}
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{| align="center" border="0" cellpadding="10" cellspacing="0"
 
{| align="center" border="0" cellpadding="10" cellspacing="0"
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| [[Image:Peirce's_Law_Figure_3.jpg|500px]] || (3)
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| [[Image:Peirce's Law Strong Form 1.0 Splash Page.png|500px]] || (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:
 
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:
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{| align="center" border="0" cellpadding="10" cellspacing="0"
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{| align="center" cellpadding="8"
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| [[Image:Peirce's_Law_Figure_4.jpg|500px]] || (4)
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|
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
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|-
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| [[Image:Peirce's Law Strong Form 1.0 Marquee Title.png|500px]]
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|-
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| [[Image:Peirce's Law Strong Form 1.0 Storyboard 1.png|500px]]
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|-
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| [[Image:Equational Inference Rule Collect p.png|500px]]
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|-
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| [[Image:Peirce's Law Strong Form 1.0 Storyboard 2.png|500px]]
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|-
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| [[Image:Equational Inference Rule Quit ((q)).png|500px]]
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|-
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| [[Image:Peirce's Law Strong Form 1.0 Storyboard 3.png|500px]]
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|-
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| [[Image:Equational Inference Rule Cancel (( )).png|500px]]
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|-
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| [[Image:Peirce's Law Strong Form 1.0 Storyboard 4.png|500px]]
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|-
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| [[Image:Equational Inference Marquee QED.png|500px]]
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|}
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| (5)
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|}
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The following animation replays the steps of the proof.
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{| align="center" cellpadding="8"
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|
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
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|-
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| [[Image:Peirce's Law Strong Form 2.0 Animation.gif]]
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|}
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| (6)
 
|}
 
|}
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* Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN.  Cited as (CE&nbsp;volume,&nbsp;page).
 
* Peirce, Charles Sanders (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN.  Cited as (CE&nbsp;volume,&nbsp;page).
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==See also==
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==Syllabus==
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===Focal nodes===
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* [[Charles Sanders Peirce]]
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* [[Inquiry Live]]
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* [[Logic Live]]
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===Peer nodes===
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* [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_law Peirce's Law @ InterSciWiki]
 +
* [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===
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{{col-begin}}
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{{col-break}}
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* [[Exclusive disjunction]]
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* [[Logical conjunction]]
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* [[Logical disjunction]]
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* [[Logical equality]]
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{{col-break}}
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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]]
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{{col-end}}
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===Related topics===
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{{col-begin}}
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{{col-break}}
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* [[Ampheck]]
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* [[Boolean domain]]
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* [[Boolean function]]
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* [[Boolean-valued function]]
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* [[Differential logic]]
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{{col-break}}
 
* [[Logical graph]]
 
* [[Logical graph]]
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* [[Minimal negation operator]]
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* [[Multigrade operator]]
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* [[Parametric operator]]
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* [[Peirce's law]]
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{{col-break}}
 
* [[Propositional calculus]]
 
* [[Propositional calculus]]
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* [[Sole sufficient operator]]
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* [[Truth table]]
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* [[Universe of discourse]]
 
* [[Zeroth order logic]]
 
* [[Zeroth order logic]]
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{{col-end}}
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===Relational concepts===
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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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{{col-break}}
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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}}
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===Information, Inquiry===
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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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{{col-break}}
 +
* [[Semeiotic]]
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* [[Logic of information]]
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{{col-break}}
 +
* [[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}}
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===Related articles===
 +
 +
{{col-begin}}
 +
{{col-break}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
 +
{{col-break}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
 +
{{col-break}}
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
 +
{{col-end}}
 +
 +
==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.
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 +
* [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]
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* [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]
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* [http://en.wikipedia.org/w/index.php?title=Peirce's_law&oldid=60606482 Peirce's Law], [http://en.wikipedia.org/ Wikipedia]
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[[Category:Artificial Intelligence]]
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[[Category:Charles Sanders Peirce]]
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[[Category:Combinatorics]]
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[[Category:Computer Science]]
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[[Category:Cybernetics]]
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[[Category:Equational Reasoning]]
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[[Category:Formal Languages]]
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[[Category:Formal Systems]]
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[[Category:Graph Theory]]
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[[Category:History of Logic]]
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[[Category:History of Mathematics]]
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[[Category:Inquiry]]
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[[Category:Knowledge Representation]]
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[[Category:Logic]]
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[[Category:Logical Graphs]]
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[[Category:Mathematics]]
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[[Category:Philosophy]]
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[[Category:Semiotics]]
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[[Category:Visualization]]
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