# Sheffer stroke

2010 Mathematics Subject Classification: *Primary:* 03B05 [MSN][ZBL]

*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:**

Alternative denial.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Alternative_denial&oldid=35097