Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

History [of logic axioms]

(See page 1151.) (c) was found by Henry Sheffer in 1913; (e) by Carew Meredith in 1967. Until this book, very little work appears to have been done on finding short axioms for logic in terms of Nand. Around 1949 Meredith found the axiom system

{f[f[a, f[b, c]], f[a, f[b, c]]] == f[f[f[c, a], a], f[f[b, a], a]], f[f[a, a], f[b, a]] == a}

In 1967 George Spencer Brown found (see page 1173)

{f[f[a, a], f[f[b, b], b]] == a, f[a, f[b, c]] == f[f[f[f[c, c], a], f[f[b, b], a]], f[f[f[c, c], a], f[f[b, b], a]]]}

and in 1969 Meredith also gave the system

{f[a, f[b, f[a, c]]] == f[a, f[b, f[b, c]]], f[f[a, a], f[b, a]] == a, f[a, b] == f[b, a]}

From Stephen Wolfram: A New Kind of Science [citation]