Logical NOR

Alternative notations and names

Peirce is the first to show the functional completeness of non-disjunction while he doesn't publish his result. Peirce used ${\displaystyle \overline{⋏}}$ for non-conjunction and ${\displaystyle ⋏}$ for non-disjunction (in fact, what Peirce himself used is ${\displaystyle ⋏}$ and he didn't introduce ${\displaystyle \overline{⋏}}$ while Peirce's editors made such disambiguated use). Peirce called ${\displaystyle ⋏}$ the ampheck (from Ancient Greek ἀμφήκης, amphēkēs, "cutting both ways").

In 1911, Stamm was the first to publish a description of both non-conjunction (using ${\displaystyle \sim }$, the Stamm hook), and non-disjunction (using ${\displaystyle \ast }$, the Stamm star), and showed their functional completeness. Note that most uses in logical notation of ${\displaystyle \sim }$ use this for negation.

In 1913, Sheffer described non-disjunction and showed its functional completeness. Sheffer used ${\displaystyle \mid }$ for non-conjunction, and ${\displaystyle \wedge }$ for non-disjunction.

In 1935, Donald L. Webb described non-disjunction for ${\displaystyle n}$-valued logic, and use ${\displaystyle \mid }$ for the operator. So some people call it Webb operator, Webb operation or Webb function.

In 1940, Quine also described non-disjunction and use ${\displaystyle \downarrow }$ for the operator. So some people call the operator Peirce arrow or Quine dagger.

In 1944, Church also described non-disjunction and use ${\displaystyle \overline{\vee }}$ for the operator.

In 1954, Bocheński used ${\displaystyle X}$ in ${\displaystyle Xpq}$ for non-disjunction in Polish notation.

APL uses a glyph ⍱ that combines a ∨ with a ~.

Properties

NOR is commutative but not associative, which means that ${\displaystyle P\downarrow Q\leftrightarrow Q\downarrow P}$ but ${\displaystyle (P\downarrow Q)\downarrow R\nleftrightarrow P\downarrow (Q\downarrow R)}$.

Functional completeness

The logical NOR, taken by itself, is a functionally complete set of connectives. This can be proved by first showing, with a truth table, that ${\displaystyle \mathrm{\neg }A}$ is truth-functionally equivalent to ${\displaystyle A\downarrow A}$. Then, since ${\displaystyle A\downarrow B}$ is truth-functionally equivalent to ${\displaystyle \mathrm{\neg }(A\vee B)}$, and ${\displaystyle A\vee B}$ is equivalent to ${\displaystyle \mathrm{\neg }(\mathrm{\neg }A\wedge \mathrm{\neg }B)}$, the logical NOR suffices to define the set of connectives ${\displaystyle \{\wedge ,\vee ,\mathrm{\neg }\}}$, which is shown to be truth-functionally complete by the Disjunctive Normal Form Theorem.

This may also be seen from the fact that 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.

Other Boolean operations in terms of the logical NOR

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.

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