# Logical NOR

(Redirected from Ampheck)
"Peirce arrow" redirects here. It is not to be confused with Pierce-Arrow, an automobile manufacturer.
Venn diagram of $~A \downarrow B$
(nor part in red)

In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form (p NOR q) is true precisely when neither p nor q is true—i.e. when both of p and q are false. In grammar, nor is a coordinating conjunction.

The NOR operator is also known as Peirce's arrowCharles Sanders Peirce introduced the symbol ↓ for it,[1] and demonstrated that the logical NOR is completely expressible: by combining uses of the logical NOR it is possible to express any logical operation on two variables. Thus, as with its dual, the NAND operator (a.k.a. the Sheffer stroke — symbolized as either | or /), NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete). It is also known as Quine's dagger (his symbol was †), the ampheck (from Greek αμφηκης, cutting both ways; compare amphi-) by Peirce,[2] or "neither-nor".

One way of expressing p NOR q is $\overline{p \lor q}$, where the symbol $\or$ signifies OR and the bar signifies the negation of the expression under it: in essence, simply $\neg(p \lor q)$. Other ways of expressing p NOR q are Xpq, and $\overline{p + q}$.

The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, was constructed entirely using NOR gates with three inputs.[3]

## Definition

The NOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false. In other words, it produces a value of false if and only if at least one operand is true.

### Truth table

The truth table of A NOR B (also written as A ↓ B) is as follows:

 INPUT OUTPUT A B A NOR B 0 0 1 0 1 0 1 0 0 1 1 0

## Properties

Logical NOR does not possess any of the five qualities (truth-preserving, false-preserving, linear, monotonic, self-dual) required to be absent from at least one member of a set of functionally complete operators. Thus, the set containing only NOR suffices as a complete set.

## Introduction, elimination, and equivalencies

NOR has the interesting feature that all other logical operators can be expressed by interlaced NOR operations. The logical NAND operator also has this ability.

The logical NOR $\downarrow$ is the negation of the disjunction:

 $P \downarrow Q$ $\Leftrightarrow$ $\neg (P \or Q)$ $\Leftrightarrow$ $\neg$

Expressed in terms of NOR $\downarrow$, the usual operators of propositional logic are:

 $\neg P$ $\Leftrightarrow$ $P \downarrow P$ $\neg$ $\Leftrightarrow$

 $P \rightarrow Q$ $\Leftrightarrow$ $\Big( (P \downarrow P) \downarrow Q \Big)$ $\downarrow$ $\Big( (P \downarrow P) \downarrow Q \Big)$ $\Leftrightarrow$ $\downarrow$

 $P \and Q$ $\Leftrightarrow$ $(P \downarrow P)$ $\downarrow$ $(Q \downarrow Q)$ $\Leftrightarrow$ $\downarrow$

 $P \or Q$ $\Leftrightarrow$ $(P \downarrow Q)$ $\downarrow$ $(P \downarrow Q)$ $\Leftrightarrow$ $\downarrow$