 In 1880 Peirce described the procedure for reducing formulae of the sentential calculus to conjunctive and disjunctive normal form, and also in unpublished work demonstrated that the sentential calculus can be obtained from the single connective of joint denial (“neither p nor q”). His 1881, “On the Logic of Number”, examined the foundations of arithmetic, and analyzed the natural numbers in terms of discrete, linearly ordered sets without a maximum element. He gave informal recursive definitions of addition and multiplication, and proved that both operations were associative and commutative.