View source for Sheffer stroke

{{TEX|done}}{{MSC|03B05}}

''Sheffer bar''

A [[Logical operation|logical operation]], usually denoted by $|$, given by the following [[Truth table|truth table]]:

<center>
{| border="1" class="wikitable" style="text-align:center; width:300px;"
|$A$||$B$||$A|B$
|-
|$T$||$T$||$F$
|-
|$T$||$F$||$T$
|-
|$F$||$T$||$T$
|-
|$F$||$F$||$T$
|}
</center>

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:

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

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
The Sheffer stroke operation is also called alternative denial.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S.C. Kleene,   "Introduction to metamathematics" , North-Holland  (1950)  pp. 139</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Marek,   J. Onyszkiewicz,   "Elements of logic and the foundations of mathematics in problems" , Reidel &amp; PWN  (1982)  pp. 4</TD></TR>
</table>