Logical NOR

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Short description: Binary operation that is true if and only if both operands are false

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, non-disjunction, 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 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.

Non-disjunction is usually denoted as [math]\displaystyle{ \downarrow }[/math] or [math]\displaystyle{ \overline{\vee} }[/math] or [math]\displaystyle{ X }[/math] (prefix) or [math]\displaystyle{ \operatorname{NOR} }[/math].

As with its dual, the NAND operator (also known as 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).

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.[2]


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] 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

Alternative notations and names

Peirce is the first to show the functional completeness of non-disjunction while he doesn't publish his result.[3][4] Peirce used [math]\displaystyle{ \overline{\curlywedge} }[/math] for non-conjunction and [math]\displaystyle{ \curlywedge }[/math] for non-disjunction (in fact, what Peirce himself used is [math]\displaystyle{ \curlywedge }[/math] and he didn't introduce [math]\displaystyle{ \overline{\curlywedge} }[/math] while Peirce's editors made such disambiguated use).[4] Peirce called [math]\displaystyle{ \curlywedge }[/math] as ampheck (from Ancient Greek ἀμφήκης, amphēkēs, "cutting both ways").[4]

In 1911, Stamm (pl) was the first to publish a description of both non-conjunction (using [math]\displaystyle{ \sim }[/math], the Stamm hook), and non-disjunction (using [math]\displaystyle{ * }[/math], the Stamm star), and showed their functional completeness.[5]Zach, R. (18 February 2023). "Sheffer stroke before Sheffer: Edward Stamm". https://richardzach.org/2023/02/sheffer-stroke-before-sheffer-edward-stamm/. </ref> Note that most uses in logical notation of [math]\displaystyle{ \sim }[/math] use this for negation.

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

In 1935, Webb described non-disjunction for [math]\displaystyle{ n }[/math]-valued logic, and use [math]\displaystyle{ \mid }[/math] for the operator. So some people call it Webb operator,[6] Webb operation[7] or Webb function.[8]

In 1940, Quine also described non-disjunction and use [math]\displaystyle{ \downarrow }[/math] for the operator.[9] So some people call the operator Peirce arrow or Quine dagger.

In 1944, Church also described non-disjunction and use [math]\displaystyle{ \overline{\vee} }[/math] for the operator.[10]

In 1954, Bocheński used [math]\displaystyle{ X }[/math] in [math]\displaystyle{ Xpq }[/math] for non-disjunction in Polish notation.[11]


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. Journey to the Moon: The History of the Apollo Guidance Computer. Reston, Virginia, USA: American Institute of Aeronautics and Astronautics. 1996. p. 196. ISBN 1-56347-185-X. 
  3. Peirce, C. S. (1933). "A Boolian Algebra with One Constant". in Hartshorne, C.; Weiss, P.. Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 13–18. 
  4. 4.0 4.1 4.2 Peirce, C. S. (1933). "The Simplest Mathematics". in Hartshorne, C.; Weiss, P.. Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 189–262. 
  5. "Beitrag zur Algebra der Logik" (in de). Monatshefte für Mathematik und Physik 22 (1): 137–149. 1911. doi:10.1007/BF01742795. 
  6. "Generation of any n-valued logic by one binary operation". Proceedings of the National Academy of Sciences (USA: National Academy of Sciences) 21 (5): 252. May 1935. doi:10.1073/pnas.21.5.252. Bibcode1935PNAS...21..252W. 
  7. "1.10 Venjunctive Properties (Basic Formulae)". written at Riga, Latvia. Asynchronous Operators of Sequential Logic: Venjunction & Sequention — Digital Circuits Analysis and Design. Lecture Notes in Electrical Engineering (LNEE). 101 (1st ed.). Berlin / Heidelberg, Germany: Springer-Verlag. 2011. p. 20. doi:10.1007/978-3-642-21611-4. ISBN 978-3-642-21610-7. "Historical background […] Logical operator NOR named Peirce arrow and also known as Webb-operation."  (xiii+1+123+7 pages) (NB. The back cover of this book erroneously states volume 4, whereas it actually is volume 101.)
  8. "Who is Donald L. Webb?". 2018-05-24. https://hsm.stackexchange.com/questions/5680/who-is-donald-l-webb. 
  9. Quine, W. V (1981). Mathematical Logic (Revised ed.). Cambridge, London, New York, New Rochelle, Melbourne and Sydney: Harvard University Press. p. 45. 
  10. Church, A. (1996). Introduction to Mathematical Logic. New Jersey: Princeton University Press. p. 37. 
  11. Bocheński, J. M. (1954) (in French). Précis de logique mathématique. Netherlands: F. G. Kroonder, Bussum, Pays-Bas. p. 11. 

External links

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