Template:Short description Template:Use dmy dates Template:Use list-defined references Template:Infobox logical connective Template:Logical connectives sidebar
In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called non-conjunction, alternative denial (since it says in effect that at least one of its operands is false), or NAND ("not and").<ref name=":13">Template:Cite book</ref> In digital electronics, it corresponds to the NAND gate. It is named after Henry Maurice Sheffer and written as <math>\mid</math> or as <math>\uparrow</math> or as <math>\overline{\wedge}</math> or as <math>Dpq</math> in Polish notation by Łukasiewicz (but not as ||, often used to represent disjunction).
Its dual is the NOR operator (also known as the Peirce arrow, Quine dagger or Webb operator). Like its dual, NAND can be used by itself, without any other logical operator, to constitute a logical formal system (making NAND functionally complete). This property makes the NAND gate crucial to modern digital electronics, including its use in computer processor design.
DefinitionEdit
The non-conjunction is a logical operation on two logical values. It produces a value of true, if — and only if — at least one of the propositions is false.
Truth tableEdit
The truth table of <math>A \uparrow B</math> is as follows.
Logical equivalencesEdit
The Sheffer stroke of <math>P</math> and <math>Q</math> is the negation of their conjunction
| <math>P \uparrow Q</math> | <math>\Leftrightarrow</math> | <math>\neg (P \land Q)</math> |
| File:Venn1110.svg | <math>\Leftrightarrow</math> | <math>\neg</math> File:Venn0001.svg |
By De Morgan's laws, this is also equivalent to the disjunction of the negations of <math>P</math> and <math>Q</math>
| <math>P \uparrow Q</math> | <math>\Leftrightarrow</math> | <math>\neg P</math> | <math>\lor</math> | <math>\neg Q</math> |
| File:Venn1110.svg | <math>\Leftrightarrow</math> | File:Venn1010.svg | <math>\lor</math> | File:Venn1100.svg |
Alternative notations and namesEdit
Peirce was the first to show the functional completeness of non-conjunction (representing this as <math>\overline{\curlywedge}</math>) but didn't publish his result.<ref name="peirce1880">Template:Cite encyclopedia</ref><ref name="peirce1902">Template:Cite encyclopedia</ref> Peirce's editor added <math>\overline{\curlywedge}</math>) for non-disjunction.<ref name="peirce1902"/>
In 1911, Template:Ill was the first to publish a proof of the completeness of non-conjunction, representing this with <math>\sim</math> (the Stamm hook)<ref name="zach2023">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and non-disjunction in print at the first time and showed their functional completeness.<ref name="Stamm_1911"/>
In 1913, Sheffer described non-disjunction using <math>\mid</math> and showed its functional completeness. Sheffer also used <math>\wedge</math> for non-disjunction.<ref name="zach2023" /> Many people, beginning with Nicod in 1917, and followed by Whitehead, Russell and many othersTemplate:Who, mistakenly thought Sheffer had described non-conjunction using <math>\mid</math>, naming this symbol the Sheffer stroke.Template:Citation needed
In 1928, Hilbert and Ackermann described non-conjunction with the operator <math>/</math>.<ref name="hilbert-ackermann1928">Template:Cite book</ref><ref name="hilbert-ackermann1950">Template:Cite book</ref>
In 1929, Łukasiewicz used <math>D</math> in <math>Dpq</math> for non-conjunction in his Polish notation.<ref name="lukasiewicz1929">Template:Cite book</ref>
An alternative notation for non-conjunction is <math>\uparrow</math>. It is not clear who first introduced this notation, although the corresponding <math>\downarrow</math> for non-disjunction was used by Quine in 1940.<ref name="quine1940">Template:Cite book</ref>
HistoryEdit
The stroke is named after Henry Maurice Sheffer, who in 1913 published a paper in the Transactions of the American Mathematical Society<ref name="Sheffer_1913"/> providing an axiomatization of Boolean algebras using the stroke, and proved its equivalence to a standard formulation thereof by Huntington employing the familiar operators of propositional logic (AND, OR, NOT). Because of self-duality of Boolean algebras, Sheffer's axioms are equally valid for either of the NAND or NOR operations in place of the stroke. Sheffer interpreted the stroke as a sign for nondisjunction (NOR) in his paper, mentioning non-conjunction only in a footnote and without a special sign for it. It was Jean Nicod who first used the stroke as a sign for non-conjunction (NAND) in a paper of 1917 and which has since become current practice.<ref name="Nicod_1917"/><ref name="Church_1956"/> Russell and Whitehead used the Sheffer stroke in the 1927 second edition of Principia Mathematica and suggested it as a replacement for the "OR" and "NOT" operations of the first edition.
Charles Sanders Peirce (1880) had discovered the functional completeness of NAND or NOR more than 30 years earlier, using the term ampheck (for 'cutting both ways'), but he never published his finding. Two years before Sheffer, Template:Ill also described the NAND and NOR operators and showed that the other Boolean operations could be expressed by it.<ref name="Stamm_1911"/>
PropertiesEdit
NAND is commutative but not associative, which means that <math>P \uparrow Q \leftrightarrow Q \uparrow P</math> but <math>(P \uparrow Q) \uparrow R \not\leftrightarrow P \uparrow (Q \uparrow R)</math>.<ref>Template:Cite book</ref>
Functional completenessEdit
The Sheffer stroke, taken by itself, is a functionally complete set of connectives.<ref name=":18">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=":2">Template:Citation</ref> This can be seen from the fact that NAND does not possess any of the following five properties, each of which is required to be absent from, and the absence of all of which is sufficient for, at least one member of a set of functionally complete operators: truth-preservation, falsity-preservation, linearity, monotonicity, self-duality. (An operator is truth-preserving if its value is truth whenever all of its arguments are truth, or falsity-preserving if its value is falsity whenever all of its arguments are falsity.)<ref>Template:Cite book</ref>
It can also be proved by first showing, with a truth table, that <math>\neg A</math> is truth-functionally equivalent to <math>A \uparrow A</math>.<ref name=":132">Template:Cite book</ref> Then, since <math>A \uparrow B</math> is truth-functionally equivalent to <math>\neg (A \land B)</math>,<ref name=":132" /> and <math>A \lor B</math> is equivalent to <math>\neg(\neg A \land \neg B)</math>,<ref name=":132" /> the Sheffer stroke suffices to define the set of connectives <math>\{\land, \lor, \neg\}</math>,<ref name=":132" /> which is shown to be truth-functionally complete by the Disjunctive Normal Form Theorem.<ref name=":132" />
Other Boolean operations in terms of the Sheffer strokeEdit
Expressed in terms of NAND <math>\uparrow</math>, the usual operators of propositional logic are:
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See alsoEdit
- Boolean domain
- CMOS
- Gate equivalent (GE)
- Logical graph
- Minimal axioms for Boolean algebra
- NAND flash memory
- NAND logic
- Peirce's law
- Peirce arrow = NOR
- Sole sufficient operator
ReferencesEdit
Further readingEdit
- Template:Cite book (NB. Edited and translated from the French and German editions: Précis de logique mathématique)
- Template:Cite book
External linksEdit
- Sheffer stroke article in the Internet Encyclopedia of Philosophy
- http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nand.html
- Implementations of 2- and 4-input NAND gates
- Proofs of some axioms by Stroke function by Yasuo Setô @ Project Euclid
Template:Logical connectives Template:Common logical symbols