Logical NOR

Logical NOR
NOR
Definition
Truth table
Logic gate
Normal forms
Disjunctive
Conjunctive
Zhegalkin polynomial
Post's lattices
0-preserving	no
1-preserving	no
Monotone	no
Affine	no
Self-dual	no
	v; t; e;

In Boolean logic, logical NOR,^[1] non-disjunction, or joint denial^[1] 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 $\neg (p\lor q)$ and $\neg p\land \neg q$ , where the symbol $\neg$ signifies logical negation, $\lor$ signifies OR, and $\land$ signifies AND.

Non-disjunction is usually denoted as $\downarrow$ or ${\overline {\vee }}$ or $X$ (prefix) or $\operatorname {NOR}$ .

As with its dual, the NAND operator (also known as the Sheffer stroke—symbolized as either $\uparrow$ , $\mid$ or $/$ ), 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]

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 $A\downarrow B$ is as follows:

$A$	$B$	$A\downarrow B$
F	F	T
F	T	F
T	F	F
T	T	F

Logical equivalences

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

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

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 ${\overline {\curlywedge }}$ for non-conjunction and $\curlywedge$ for non-disjunction (in fact, what Peirce himself used is $\curlywedge$ and he didn't introduce ${\overline {\curlywedge }}$ while Peirce's editors made such disambiguated use).^[4] Peirce called $\curlywedge$ the 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 $\sim$ , the Stamm hook), and non-disjunction (using $*$ , the Stamm star), and showed their functional completeness.^[5]^[6] Note that most uses in logical notation of $\sim$ use this for negation.

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

In 1935, Webb described non-disjunction for $n$ -valued logic, and use $\mid$ for the operator. So some people call it Webb operator,^[7] Webb operation^[8] or Webb function.^[9]

In 1940, Quine also described non-disjunction and use $\downarrow$ for the operator.^[10] So some people call the operator Peirce arrow or Quine dagger.

In 1944, Church also described non-disjunction and use ${\overline {\vee }}$ for the operator.^[11]

In 1954, Bocheński used $X$ in $Xpq$ for non-disjunction in Polish notation.^[12]

Properties

NOR is commutative but not associative, which means that $P\downarrow Q\leftrightarrow Q\downarrow P$ but $(P\downarrow Q)\downarrow R\not \leftrightarrow P\downarrow (Q\downarrow R)$ .^[13]

Functional completeness

The logical NOR, taken by itself, is a functionally complete set of connectives.^[14] This can be proved by first showing, with a truth table, that $\neg A$ is truth-functionally equivalent to $A\downarrow A$ .^[15] Then, since $A\downarrow B$ is truth-functionally equivalent to $\neg (A\lor B)$ ,^[15] and $A\lor B$ is equivalent to $\neg (\neg A\land \neg B)$ ,^[15] the logical NOR suffices to define the set of connectives $\{\land ,\lor ,\neg \}$ ,^[15] which is shown to be truth-functionally complete by the Disjunctive Normal Form Theorem.^[15]

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 $\downarrow$ , the usual operators of propositional logic are:

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

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

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

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

References

^ ^a ^b Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. p. 43. ISBN 978-0-415-13342-5.
^ Hall, Eldon C. (1996). Journey to the Moon: The History of the Apollo Guidance Computer. Reston, Virginia, USA: American Institute of Aeronautics and Astronautics. p. 196. ISBN 1-56347-185-X.
^ Peirce, C. S. (1933) [1880]. "A Boolian Algebra with One Constant". In Hartshorne, C.; Weiss, P. (eds.). Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 13–18.
^ ^a ^b ^c Peirce, C. S. (1933) [1902]. "The Simplest Mathematics". In Hartshorne, C.; Weiss, P. (eds.). Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 189–262.
^ Stamm, Edward Bronisław [in Polish] (1911). "Beitrag zur Algebra der Logik". Monatshefte für Mathematik und Physik (in German). 22 (1): 137–149. doi:10.1007/BF01742795. S2CID 119816758.
^ Zach, R. (2023-02-18). "Sheffer stroke before Sheffer: Edward Stamm". Retrieved 2023-07-02.
^ Webb, Donald Loomis (May 1935). "Generation of any n-valued logic by one binary operation". Proceedings of the National Academy of Sciences. 21 (5). USA: National Academy of Sciences: 252–254. Bibcode:1935PNAS...21..252W. doi:10.1073/pnas.21.5.252. PMC 1076579. PMID 16577665.
^ Vasyukevich, Vadim O. (2011). "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). Vol. 101 (1st ed.). Berlin / Heidelberg, Germany: Springer-Verlag. p. 20. doi:10.1007/978-3-642-21611-4. ISBN 978-3-642-21610-7. ISSN 1876-1100. LCCN 2011929655. p. 20: 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.)
^ Freimann, Michael; Renfro, Dave L.; Webb, Norman (2018-05-24) [2017-02-10]. "Who is Donald L. Webb?". History of Science and Mathematics. Stack Exchange. Archived from the original on 2023-05-18. Retrieved 2023-05-18.
^ Quine, W. V (1981) [1940]. Mathematical Logic (Revised ed.). Cambridge, London, New York, New Rochelle, Melbourne and Sydney: Harvard University Press. p. 45.
^ Church, A. (1996) [1944]. Introduction to Mathematical Logic. New Jersey: Princeton University Press. p. 37.
^ Bocheński, J. M. (1954). Précis de logique mathématique (in French). Netherlands: F. G. Kroonder, Bussum, Pays-Bas. p. 11.
^ Rao, G. Shanker (2006). Mathematical Foundations of Computer Science. I. K. International Pvt Ltd. p. 22. ISBN 978-81-88237-49-4.
^ Smullyan, Raymond M. (1995). First-order logic. New York: Dover. pp. 5, 11, 14. ISBN 978-0-486-68370-6.
^ ^a ^b ^c ^d ^e Howson, Colin (1997). Logic with trees: an introduction to symbolic logic. London; New York: Routledge. pp. 41–43. ISBN 978-0-415-13342-5.