The Principles of Mathematics (1903)

§ 15

Our calculus studies the relation of implication between propositions. This relation must be distinguished from the relation of formal implication, which holds between propositional functions when the one implies the other for all values of the variable. Formal implication is also involved in this calculus, but is not explicitly studied: we do not consider propositional functions in general, but only certain definite propositional functions which occur in the propositions of our calculus. How far formal implication is definable in terms of implication simply, or material implication as it may be called, is a difficult question, which will be discussed in Chapter III. What the difference is between the two, an illustration will explain. The fifth proposition of Euclid follows from the fourth: if the fourth is true, so is the fifth, while if the fifth is false, so is the fourth. This is a case of material implication, for both propositions are absolute constants, not dependent for their meaning upon the assigning of a value to a variable. But each of them states a formal implication. The fourth states that if x and y be triangles fulfilling certain other conditions, then x and y are triangles fulfilling ertain other conditions, and that this implication holds for all values of x and y; and the fifth states that if x is an isosceles triangle, x has the angles at the base equal. The formal implication involved in each of these two propositions is quite a different thing from the material implication holding between the propositions as wholes; both notions are required in the propositional calculus, but it is the study of material implication which specially distinguishes this subject, for formal implication occurs throughout the whole of mathematics.(§ 15 ¶ 1)

It has been customary, in treatises on logic, to confound the two kinds of implication, and often to be really considering the formal kind where the material kind only was apparently involved. For example, when it is said that Socrates is a man, therefore Socrates is a mortal, Socrates is felt as a variable: he is a type of humanity, and one feels that any other man would have done as well. If, instead of therefore, which implies the truth of hypothesis and consequent, we put Socrates is a man implies Socrates is mortal, it appears at once that we may substitute not only another man, but any other entity whatever, in the place of Socrates. Thus although what is explicitly stated, in such a case, is a material implication, what is meant is a formal implication; and some effort is needed to confine our imagination to material implication.(§ 15 ¶ 2)