The propositional calculus is characterized by the fact that all its propositions have as hypothesis and as consequent the assertion of a material implication. Usually the hypothesis is of the form

etc. which (§ 16) is equivalent to the assertion that the letters which occur in the consequent are propositions. Thus the consequents consist of propositional functions which are true of all propositions. It is important to observe that, though the letters employed are symbols for variables, and the consequents are true when the variables are given values which are propositions, these values must be genuine propositions, not propositional functions. The hypothesis `p` implies `p`,

is not satisfied if for `p` is a proposition`p` we put

but it is satisfied if we put `x` is a man,Socrates is a man

or if we put

Shortly, we may say that the propositions represented by single letters in this calculus are variables, but do not contain variables—in the case, that is to say, where the hypotheses of the propositions which the calculus asserts are satisfied.(§ 14 ¶ 1)`x` is a man implies `x` is a mortal for all values of `x`.

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.