In his Tractatus, Wittgenstein sets out what he calls his N-operator notation which can be used to calculate whether an expression is a tautology. In his Laws of Form, George Spencer Brown offers what he calls a “primary algebra” for such calculation. Both systems are perplexing. But comparing two blurry images can reduce noise, producing a focus. This paper reveals that Spencer Brown independently rediscovered the quantifier-free part of the N-operator calculus. The comparison sheds a flood light on each and (...) from the letters of correspondence we shall find that Russell, as one might have surmised, was a catalyst for both. (shrink)

We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices.