# Peirce arrow

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 "$A|B$": not both $A$ and $B$.

#### 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=32868