Peirce arrow

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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.


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


[a1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1950) pp. 139
How to Cite This Entry:
Peirce arrow. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article