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''Sheffer bar''
 
''Sheffer bar''
  
A [[Logical operation|logical operation]], usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848601.png" />, given 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/s/s084/s084860/s0848602.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848603.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848604.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">T</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">T</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 [[Logical operation|logical operation]], usually denoted by $|$, given 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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|$A$||$B$||$A|B$
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|-
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|$T$||$T$||$F$
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|-
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|$T$||$F$||$T$
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|-
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|$F$||$T$||$T$
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|-
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|$F$||$F$||$T$
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|}
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</center>
  
Thus, the assertion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848605.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848607.png" /> are incompatible, i.e. are not true simultaneously. All other logical operations can be expressed by the Sheffer stroke. For example, the assertion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848608.png" /> (the [[Negation|negation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s0848609.png" />) is equivalent to the assertion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486010.png" />; the [[Disjunction|disjunction]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486011.png" /> of two assertions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486013.png" /> is expressed as:
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Thus, the assertion $A|B$ means that $A$ and $B$ are incompatible, i.e. are not true simultaneously. All other logical operations can be expressed by the Sheffer stroke. For example, the assertion $\neg A$ (the [[Negation|negation]] of $A$) is equivalent to the assertion $A|A$; the [[Disjunction|disjunction]] $A\lor B$ of two assertions $A$ and $B$ is expressed as:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486014.png" /></td> </tr></table>
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$$(A|A)|(B|B).$$
  
The [[Conjunction|conjunction]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486015.png" /> and the [[Implication|implication]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486016.png" /> are expressed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084860/s08486018.png" />, respectively. Sheffer's stroke was first considered by H. Sheffer.
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The [[Conjunction|conjunction]] $A\&B$ and the [[Implication|implication]] $A\to B$ are expressed as $(A|B)|(A|B)$ and $A|(B|B)$, respectively. Sheffer's stroke was first considered by H. Sheffer.
  
 
====References====
 
====References====

Revision as of 12:13, 12 August 2014

Sheffer bar

A logical operation, usually denoted by $|$, given by the following truth table:

$A$ $B$ $A|B$
$T$ $T$ $F$
$T$ $F$ $T$
$F$ $T$ $T$
$F$ $F$ $T$

Thus, the assertion $A|B$ means that $A$ and $B$ are incompatible, i.e. are not true simultaneously. All other logical operations can be expressed by the Sheffer stroke. For example, the assertion $\neg A$ (the negation of $A$) is equivalent to the assertion $A|A$; the disjunction $A\lor B$ of two assertions $A$ and $B$ is expressed as:

$$(A|A)|(B|B).$$

The conjunction $A\&B$ and the implication $A\to B$ are expressed as $(A|B)|(A|B)$ and $A|(B|B)$, respectively. Sheffer's stroke was first considered by H. Sheffer.

References

[1] H.M. Sheffer, "A set of five independent postulates for Boolean algebras, with applications to logical constants" Trans. Amer. Math. Soc. , 14 (1913) pp. 481–488


Comments

The Sheffer stroke operation is also called alternative denial.

References

[a1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1950) pp. 139
[a2] W. Marek, J. Onyszkiewicz, "Elements of logic and the foundations of mathematics in problems" , Reidel & PWN (1982) pp. 4
How to Cite This Entry:
Sheffer stroke. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sheffer_stroke&oldid=32866
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article