Logical NOR

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Logical NOR
Venn diagram of Logical NOR
Definition[math]\displaystyle{ \overline{x + y} }[/math]
Truth table[math]\displaystyle{ (0001) }[/math]
Logic gateNOR ANSI.svg
Normal forms
Disjunctive[math]\displaystyle{ \overline{x} \cdot \overline{y} }[/math]
Conjunctive[math]\displaystyle{ \overline{x} \cdot \overline{y} }[/math]
Zhegalkin polynomial[math]\displaystyle{ 1 \oplus x \oplus y \oplus xy }[/math]
white;">Post's lattices

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. It is logically equivalent to [math]\displaystyle{ \neg(p \lor q) }[/math] and [math]\displaystyle{ \neg p \land \neg q }[/math], where the symbol [math]\displaystyle{ \neg }[/math] signifies logical negation, [math]\displaystyle{ \lor }[/math] signifies OR, and [math]\displaystyle{ \land }[/math] signifies AND. In grammar, neithernor are a pair of correlative coordinating conjunctions.

The NOR operator is also known as Peirce's arrow. Peirce, in unpublished manuscripts, first considered it as a logical operator, and showed that it can express logical NOT, AND, and OR. Stamm,[2] Sheffer,[3] and Nicod[4] were the first to discuss it in print. Quine introduced the symbol [math]\displaystyle{ \downarrow }[/math] for it.[5] As with its dual, the NAND operator (a.k.a. the Sheffer stroke—symbolized as either [math]\displaystyle{ \uparrow }[/math], [math]\displaystyle{ \mid }[/math] or [math]\displaystyle{ / }[/math]), NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete). Other terms for the NOR operator include Quine's dagger, the ampheck (from Ancient Greek ἀμφήκης, amphēkēs, "cutting both ways") used by Peirce,[6] and neither-nor. Other ways of notating [math]\displaystyle{ P \downarrow Q }[/math] include, P NOR Q, and "Xpq" (in Bocheński notation).

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.[7]


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 [math]\displaystyle{ P \downarrow Q }[/math] (also written as P NOR Q) is as follows:

[math]\displaystyle{ P }[/math] [math]\displaystyle{ Q }[/math] [math]\displaystyle{ P \downarrow Q }[/math]
True True False
True False False
False True False
False False True

Logical Equivalences

The logical NOR [math]\displaystyle{ \downarrow }[/math] is the negation of the disjunction:

[math]\displaystyle{ P \downarrow Q }[/math]     [math]\displaystyle{ \Leftrightarrow }[/math]     [math]\displaystyle{ \neg (P \lor Q) }[/math]
Venn1000.svg     [math]\displaystyle{ \Leftrightarrow }[/math]     [math]\displaystyle{ \neg }[/math] Venn0111.svg


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.

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 [math]\displaystyle{ \downarrow }[/math], the usual operators of propositional logic are:

[math]\displaystyle{ \neg P }[/math]     [math]\displaystyle{ \Leftrightarrow }[/math]     [math]\displaystyle{ P \downarrow P }[/math]
[math]\displaystyle{ \neg }[/math] Venn01.svg     [math]\displaystyle{ \Leftrightarrow }[/math]     Venn10.svg
[math]\displaystyle{ P \rightarrow Q }[/math]     [math]\displaystyle{ \Leftrightarrow }[/math]     [math]\displaystyle{ \Big( (P \downarrow P) \downarrow Q \Big) }[/math] [math]\displaystyle{ \downarrow }[/math] [math]\displaystyle{ \Big( (P \downarrow P) \downarrow Q \Big) }[/math]
Venn1011.svg     [math]\displaystyle{ \Leftrightarrow }[/math]     Venn0100.svg [math]\displaystyle{ \downarrow }[/math] Venn0100.svg
[math]\displaystyle{ P \land Q }[/math]     [math]\displaystyle{ \Leftrightarrow }[/math]     [math]\displaystyle{ (P \downarrow P) }[/math] [math]\displaystyle{ \downarrow }[/math] [math]\displaystyle{ (Q \downarrow Q) }[/math]
Venn0001.svg     [math]\displaystyle{ \Leftrightarrow }[/math]     Venn1010.svg [math]\displaystyle{ \downarrow }[/math] Venn1100.svg
[math]\displaystyle{ P \lor Q }[/math]     [math]\displaystyle{ \Leftrightarrow }[/math]     [math]\displaystyle{ (P \downarrow Q) }[/math] [math]\displaystyle{ \downarrow }[/math] [math]\displaystyle{ (P \downarrow Q) }[/math]
Venn0111.svg     [math]\displaystyle{ \Leftrightarrow }[/math]     Venn1000.svg [math]\displaystyle{ \downarrow }[/math] Venn1000.svg

See also


  1. Brent, Joseph (1998), Charles Sanders Peirce: A Life, 2nd edition, Bloomington and Indianapolis: Indiana University Press (catalog page); also NetLibrary.
  2. Stamm, Edward (1911). "Beitrag zur Algebra der Logik". Monatshefte für Mathematik und Physik 22 (1): 137–149. doi:10.1007/BF01742795. 
  3. Sheffer, Henry Maurice (1913). "A set of five independent postulates for Boolean algebras, with application to logical constants". Transactions of the American Mathematical Society 14 (4): 481–488. doi:10.1090/S0002-9947-1913-1500960-1. 
  4. Nicod, Jean G. P. (1917). "A reduction in the number of the primitive propositions of logic". Proceedings of the Cambridge Philosophical Society, Mathematical and Physical Sciences 19: 32–41. 
  5. Quine, Willard Van Orman (1940). Mathematical logic (1 ed.). New York: W. W. Norton & Co.. http://archive.org/details/mathematicallogi00quin. 
  6. C.S. Peirce, CP 4.264
  7. Hall, Eldon C. (1996), Journey to the Moon: The History of the Apollo Guidance Computer, Reston, Virginia, USA: AIAA, p. 196, ISBN 1-56347-185-X 

External links

de:Peirce-Funktion it:Algebra di Boole#NOR