Peano’s Logic

10.1. Implication and mathematics. In its birth Mathematical Logic was the theory of classes. The first person to have the opinion that the theory of statements (with or without variables) is more important was Hugh McColl (1837-1909) who in a group of papers on ‘The calculus of equivalent statements’ (1878 and later) put forward the belief that the only business of logic is with the theory of statements and that the chief statement connection is some sort of implication. The thought that the theory of statements and not that of classes is the root of Mathematical Logic, and the thought that implication of one sort or another is the chief relation to be given attention in logic, soon became the normal beliefs of those at the head of this field. For example, Frege and Peirce were interested in the logic of statement connections as a branch of logic separate from the algebra of class logic and implication was specially important in their systems. Before Peano (1858-1932), however, no one made use of the logic of statements for making clear the arguments of everyday mathematics, and so viewing logic as an instrument for getting clear and tight reasoning in such mathematics. (Peano was the first to give the new logic the name of ‘Mathematical Logic’, because of his view of it as an instrument for mathematics: for him, Mathematical Logic is the logic of mathematics.) And it had not been pointed out before Peano that implication is the chief relation in mathematics, all or almost all the statements that are true in any system of mathematics being implications. (It was because of this teaching of Peano’s about the theorems of mathematics being implications that Russell, in the opening of his Principles of Mathematics, gave as his definition of mathematics that this was the class of all statements of the form ‘if s₁ then s₂ s₁ and s₂ being limited in certain ways.) And the idea that it was possible, by the use of logic, for all the statements in mathematics, and not only in arithmetic, to be put in the form of a language of made-up signs and for the demonstrations of all its theorems to be done by changes and exchanges of these signs, starting from axioms and definitions as sign complexes, had not been acted on in detail before Peano. The other experts in Mathematical Logic at that time were interested in logic for itself or were interested, as Frege was, in turning a bit of mathematics into logic; unlike Peano they were not interested in the value of logic as an instrument for everyday mathematics as everyday mathematics (and not as logic or physics or philosophy).