View source for Peirce arrow

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A two-place [[Logical operation|logical operation]], usually denoted by $\downarrow$, specified by the following [[Truth table|truth table]]:

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

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====
A more familiar two-place logical operation in terms of which all others can be expressed is the so-called [[Sheffer stroke|Sheffer stroke]]  "$A|B$": not both $A$ and $B$.

====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>