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The stroke is named after Henry Maurice Sheffer, who in 1913 published a paper in the Transactions of the American Mathematical Society [10] 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).
definition: is defined as metalanguage:= means "from now on, ... Sheffer stroke, the sign for the NAND operator (negation of conjunction).
For example, an axiom with six NAND operations and three variables is equivalent to Boolean algebra: [1] (()) ((())) = where the vertical bar represents the NAND logical operation (also known as the Sheffer stroke).
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.
NAND or Sheffer stroke - true when it is not the case that all inputs are true ("not both") NOR or logical nor - true when none of the inputs are true ("neither") XNOR or logical equality - true when both inputs are the same ("equal") An example of a more complicated function is the majority function (of an odd number of inputs).
Similarly, it is sufficient to have only and as logical connectives, or to have only the Sheffer stroke (NAND) or the Peirce arrow (NOR) operator. It is possible to entirely avoid function symbols and constant symbols, rewriting them via predicate symbols in an appropriate way.
↑ or Sheffer stroke, the logical connective "not both" or NAND; ↑, the APL function 'take' "Increased" (and similar meanings), in medical notation; ↑, a chemical symbol for production of gas, which bubbles up.
As shown by Alexander V. Kuznetsov, either of the following connectives – the first one ternary, the second one quinary – is by itself functionally complete: either one can serve the role of a sole sufficient operator for intuitionistic propositional logic, thus forming an analog of the Sheffer stroke from classical propositional logic: [6]