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, Edward Stamm also described the NAND and NOR operators and showed that the other Boolean operations could be expressed by it. Properties 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. This can also be realized as follows: All three elements of the functionally complete set {AND, OR, NOT} can be constructed using only NAND. Thus the set {NAND} must be functionally complete as well. Other Boolean operations in terms of the Sheffer stroke Expressed in terms of NAND , the usual operators of propositional logic are: Formal system based on the Sheffer stroke The following is an example of a formal system based entirely on the Sheffer stroke, yet having the functional expressiveness of the propositional logic: Symbols pn for natural numbers n: ( | ) The Sheffer stroke commutes but does not associate (e.g., (T | T) | F = T, but T | (T | F) = F). Hence any formal system including the Sheffer stroke as an infix symbol must also include a means of indicating grouping (grouping is automatic if the stroke is used as a prefix, thus: || TTF = T and | T | TF = F). We shall employ '(' and ')' to this effect. We also write p, q, r, … instead of p0, p1, p2. Syntax Construction Rule I: For each natural number n, the symbol pn is a well-formed formula (wff), called an atom. Construction Rule II: If X and Y are wffs, then (X | Y) is a wff. Closure Rule: Any formulae which cannot be constructed by means of the first two Construction Rules are not wffs. The letters U, V, W, X, and Y are metavariables standing for wffs. A decision procedure for determining whether a formula is well-formed goes as follows: "deconstruct" the formula by applying the Construction Rules backwards, thereby breaking the formula into smaller