Sheffer stroke

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 nand ('not and') or the alternative denial, since it says in effect that at least one of its operands is false. In digital electronics, it corresponds to the NAND gate. It is named after Henry M. Sheffer and written as ↑ or as | (but not as ||, often used to represent disjunction). In Bocheński notation it can be written as Dpq. 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 nand ('not and') or the alternative denial, since it says in effect that at least one of its operands is false. In digital electronics, it corresponds to the NAND gate. It is named after Henry M. Sheffer and written as ↑ or as | (but not as ||, often used to represent disjunction). In Bocheński notation it can be written as Dpq. Its dual is the NOR operator (also known as the Peirce arrow or Quine dagger). 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 NAND operation 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 P ↑ Q {displaystyle Puparrow Q} (also written as P | ⁡ Q {displaystyle Pmathop {|} Q} , or Dpq) is as follows The Sheffer stroke of P {displaystyle P} and Q {displaystyle Q} is the negation of their conjunction By De Morgan's Laws, this is also equivalent to the disjunction of the negations of P {displaystyle P} and Q {displaystyle Q} The stroke is named after Henry M. Sheffer, who in 1913 published a paper in the Transactions of the American Mathematical Society (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. 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. 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- (falsity-) preserving if its value is truth (falsity) whenever all of its arguments are truth (falsity).) Therefore {NAND} is a functionally complete set.