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A two-place [[Logical operation|logical operation]], usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719601.png" />, specified by the following [[Truth table|truth table]]:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719602.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719603.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719604.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">T</td> <td colname="2" style="background-color:white;" colspan="1">T</td> <td colname="3" style="background-color:white;" colspan="1">F</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">T</td> <td colname="2" style="background-color:white;" colspan="1">F</td> <td colname="3" style="background-color:white;" colspan="1">F</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">F</td> <td colname="2" style="background-color:white;" colspan="1">T</td> <td colname="3" style="background-color:white;" colspan="1">F</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">F</td> <td colname="2" style="background-color:white;" colspan="1">F</td> <td colname="3" style="background-color:white;" colspan="1">T</td> </tr> </tbody> </table>
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A two-place [[Logical operation|logical operation]], usually denoted by $\downarrow$, specified by the following [[Truth table|truth table]]:
  
</td></tr> </table>
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<center>
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{| border="1" class="wikitable" style="text-align:center; width:300px;"
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|$A$||$B$||$A\downarrow B$
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|-
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|$T$||$T$||$F$
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|-
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|$T$||$F$||$F$
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|-
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|$F$||$T$||$F$
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|-
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|$F$||$F$||$T$
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|}
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</center>
  
Therefore, the statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719605.png" /> denotes  "neither A nor B" . Peirce's arrow has the property that all logical operations can be expressed in terms of it. For example, the statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719606.png" /> (the [[Negation|negation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719607.png" />) is equivalent to the statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719608.png" />, while the [[Conjunction|conjunction]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p0719609.png" /> of two statements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p07196010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p07196011.png" /> is expressed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p07196012.png" /> and the disjunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p07196013.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p07196014.png" />. This arrow was introduced by C. Peirce.
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Therefore, the statement $A\downarrow B$ denotes  "neither A nor B". Peirce's arrow has the property that all logical operations can be expressed in terms of it. For example, the statement $\neg A$ (the [[Negation|negation]] of $A$) is equivalent to the statement $A\downarrow A$, while the [[Conjunction|conjunction]] $A\&B$ of two statements $A$ and $B$ is expressed as $(A\downarrow A)\downarrow(B\downarrow B)$ and the disjunction $A\lor B$ is equivalent to $(A\downarrow B)\downarrow(A\downarrow B)$. This arrow was introduced by C. Peirce.
  
  
  
 
====Comments====
 
====Comments====
A more familiar two-place logical operation in terms of which all others can be expressed is the so-called [[Sheffer stroke|Sheffer stroke]]  "AB" : either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p07196015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071960/p07196016.png" /> but not both. The Peirce arrow and the Sheffer stroke are each other negations.
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A more familiar two-place logical operation in terms of which all others can be expressed is the so-called [[Sheffer stroke|Sheffer stroke]]  "AB": either $A$ or $B$ but not both. The Peirce arrow and the Sheffer stroke are each other negations.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1950)  pp. 139</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1950)  pp. 139</TD></TR></table>

Revision as of 12:22, 12 August 2014

A two-place logical operation, usually denoted by $\downarrow$, specified by the following truth table:

$A$ $B$ $A\downarrow B$
$T$ $T$ $F$
$T$ $F$ $F$
$F$ $T$ $F$
$F$ $F$ $T$

Therefore, the statement $A\downarrow B$ denotes "neither A nor B". Peirce's arrow has the property that all logical operations can be expressed in terms of it. For example, the statement $\neg A$ (the negation of $A$) is equivalent to the statement $A\downarrow A$, while the conjunction $A\&B$ of two statements $A$ and $B$ is expressed as $(A\downarrow A)\downarrow(B\downarrow B)$ and the disjunction $A\lor B$ is equivalent to $(A\downarrow B)\downarrow(A\downarrow B)$. This arrow was introduced by C. Peirce.


Comments

A more familiar two-place logical operation in terms of which all others can be expressed is the so-called Sheffer stroke "AB": either $A$ or $B$ but not both. The Peirce arrow and the Sheffer stroke are each other negations.

References

[a1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1950) pp. 139
How to Cite This Entry:
Peirce arrow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peirce_arrow&oldid=32867
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article