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The alternative denial is a truth-functional operator in logic which is equivalent to the English wording “not both“ and”. It is used to deny a conjunction, stating that at least one of the connected propositions is false.

For example, if one of P and Q are false, or if P and Q cannot both be true, we may say formally:

**P|Q**

This may be read as “not both P and Q” or “P and Q are mutually exclusive”.

In symbolic logic, the vertical line symbol ( | ) is used to indicate an alternative denial. This mark is known as a **Sheffer stroke**, after the logician Henry M. Sheffer, who defined the operator for use in Boolean algebra. Sometimes, the capitalized term NAND (meaning “not and”), or the upwards arrow symbol ( ↑ ) is used, in contrast to the NOR operation, demonstrated by the Peirce arrow ( ↓ ).

Some, or even most, logical systems do not include the | symbol in their formal language. In these systems, an alternative denial must be formulated using other operators. The following is an alternative expression of an altenrative denial:

**¬(P ∧ Q)**

The following table illustrates the possible truth values of P|Q, given each possible valuation of its terms, P and Q. Note that P|Q does not remove the possibility that both P and Q are false, only that P and Q are both true.

P | Q | P|Q |
---|---|---|

T | T | F |

T | F | T |

F | T | T |

F | F | T |

It is actually the case that a system of logic could be created using *only* the alternative denial, and it is for this realization that Henry Sheffer is credited. The following table shows equivalent formulas for other operators using only the Sheffer stroke:

Operator | Sheffer stroke equivalent |
---|---|

Negation (¬) | P|P |

Conjunction (∧) | (P|Q)|(P|Q) |

Disjunction (∨) | (P|P)|(Q|Q) |

Conditional (→) | P|(P|Q) |