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Sheffer interpreted the stroke as a sign for nondisjunction 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.
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.
Because Sheffer's stroke (also known as NAND operator) is functionally complete, it can be used to create an entire formulation of propositional calculus. NAND formulations use a rule of inference called Nicod's modus ponens: , (). Nicod's axiom system: [5]
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.
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).
A "Boolean algebra" may be defined from any of several different bases. Three bases for Boolean algebra are in common use, the lattice basis, the ring basis, and the Sheffer stroke or NAND basis. These bases impart respectively a logical, an arithmetical, and a parsimonious character to the subject.