| Sheffer stroke | |
| Other Titles: | NAND |
| Venn Diagram: | Venn1110.svg |
| Definition: | \overline{x ⋅ y} |
| Truth Table: | (1110) |
| Logic Gate: | NAND_ANSI.svg |
| Dnf: | \overline{x}+\overline{y} |
| Cnf: | \overline{x}+\overline{y} |
| Zhegalkin: | 1 ⊕ xy |
| 0-Preserving: | no |
| 1-Preserving: | no |
| Monotone: | no |
| Affine: | no |
| Self-Dual: | no |
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, or alternative denial[1] (since it says in effect that at least one of its operands is false), or NAND ("not and"). In digital electronics, it corresponds to the NAND gate. It is named after Henry Maurice Sheffer and written as
\mid
\uparrow
\overline{\wedge}
Dpq
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.
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.
The truth table of
A\uparrowB
The Sheffer stroke of
P
Q
P\uparrowQ | \Leftrightarrow | \neg(P\landQ) | |
| \Leftrightarrow | \neg |
By De Morgan's laws, this is also equivalent to the disjunction of the negations of
P
Q
P\uparrowQ | \Leftrightarrow | \negP | \lor | \negQ | |
| \Leftrightarrow | \lor |
Peirce was the first to show the functional completeness of non-conjunction (representing this as
\overline{\curlywedge}
\overline{\curlywedge}
In 1911, was the first to publish a proof of the completeness of non-conjunction, representing this with
\sim
In 1913, Sheffer described non-disjunction using
\mid
\wedge
\mid
In 1928, Hilbert and Ackermann described non-conjunction with the operator
/
In 1929, Łukasiewicz used
D
Dpq
An alternative notation for non-conjunction is
\uparrow
\downarrow
The stroke is named after Henry Maurice Sheffer, who in 1913 published a paper in the Transactions of the American Mathematical Society 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. 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, also described the NAND and NOR operators and showed that the other Boolean operations could be expressed by it.
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- (or falsity-)preserving if its value is truth (falsity) whenever all of its arguments are truth (falsity).) Therefore is a functionally complete set.
This can also be realized as follows: All three elements of the functionally complete set can be constructed using only NAND. Thus the set must be functionally complete as well.
Expressed in terms of NAND
\uparrow
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The Sheffer stroke, taken by itself, is a functionally complete set of connectives.[9] This can be proved by first showing, with a truth table, that
\negA
A\uparrowA
A\uparrowB
\neg(A\landB)
A\lorB
\neg(\negA\land\negB)
\{\land,\lor,\neg\}
de:Albert Heinrich Menne
. revised . Otto . Bird . . Dordrecht, South Holland, Netherlands. (NB. Edited and translated from the French and German editions: Précis de logique mathématique)