Peirce's axioms for propositional calculus

Abstract

In 1885 Peirce axiomatised the propositional calculus on the basis of five ‘icons’ or axioms, which with Cpq for ‘If p then q’ and o for a false proposition, and with the source in Peirce's [2] beside each, may be represented as follows: 1. Cpp (3.376); 2. CCpCqrCqCpr (3.377); 3. CCpqCCqrCpr (3.379); 4. Cop (3.381); 5. CCCpqpp (3.384). From these Peirce proves, among other things, 6. CpCCpqq (3.377), 7. CpCqp (3.378), and with Np (Not p) for Cpo, 8. CNNpp (3.384).By a well-known result of Wajsberg's II. 93, the entire two-valued propositional calculus is derivable by substitution and detachment from 3, 4, 5, and 7 (with Df. N). In Berry [1] this is taken to show the completeness of Peirce's basis; but this proof will not do as it stands. For Peirce's proof of 7 consists in first passing from 1 to CqCpp by the argument that ‘to say that (x ⥽ x) is generally true is to say that it is so in every state of things, say in that in which y is true’; and then from CqCpp to 7 by 2. This procedure clearly involves, formally, the use not only of substitution and detachment but of a rule to infer ⊦ Cqα. from ⊦α. Peirce's basis is sufficient with substitution and detachment nevertheless; for by a less well-known result of Wajsberg's II. 93, 3, 4, 5, and 6 are sufficient in this sense, and Peirce does prove 6 by substitution and detachment from 1 and 2.