And indeed it has been known since before 1900 that both Nand and Nor on their own work—a fact I already used on pages 617 and 775 . … Nand and Nor are the only primitive functions that work on their own.

Nand and Nor yield the smallest number of theorems.

[No text on this page] A cellular automaton whose behavior seems neither highly regular nor completely random.

Nor is somewhat rare, though Dutch has noch and Old English ne . (Modern English has only the compound form neither ... nor .) … Most people seem to find it difficult to think in terms of Nand ( Nand is for example not associative, but then neither is Nor ).

Universal logical functions The fact that combinations of Nand or Nor are adequate to reproduce any logical function was noted by Charles Peirce around 1880, and became widely known after the work of Henry Sheffer in 1913. … Nand and Nor are the only 2-input functions universal in this sense. ( {Equal} can for example reproduce only functions {9, 10, 12, 15} , {Implies} only functions {10, 11, 12, 13, 14, 15} , and {Equal, Implies} only functions {8, 9, 10, 11, 12, 13, 14, 15} .) … Of these, 6 are straightforward generalizations of Nand and Nor .

And if one assumes consistency then it follows that there must be strings where neither the string nor its negation can be The effect of adding transformations to the rules for a multiway system. The first multiway system is incomplete, in the sense that for some strings, it generates neither the string nor its negation.

And the first axiom system for which Nand and Nor are the only operators allowed that involve 2 possible values is {((b ∘ b) ∘ a) ∘ (a ∘ b)  a} . … Each axiom system given applies equally well to Nor as well as Nand .

If less than half the strings of a given length are ever produced, this means that there must be some strings where neither the string nor its negation can be proved, indicating incompleteness.

and Nor . … If one looks at axiom systems of the form {…  a, a ∘ b  b ∘ a} the first one that one finds that allows only Nand and Nor with 2-value operators is {(a ∘ a) ∘ (a ∘ a)  a, a ∘ b  b ∘ a} .

And finally, in case (d), neither the digit sequences nor the sizes of numbers are anything but trivial.