Logical NOR

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 arrow—Charles Sanders Peirce introduced the symbol ↓ for it, 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 ) by Peirce, or neither-nor.

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

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