must be noted that a proposition is true, or it is false. But a propositional function is always true, or sometimes true. This notion of being "always true" is quite different from being "true," for it means "true in all cases," whereas there are no cases of a proposition. It is this fact that gives rise to difficulty in the propositional interpretation of the class calculus. Let us write "4.2 always" for "42 is always true." Then, just as "-p" means "p is false," so "-0 always" means "02 is always false." Similarly, "-cg sometimes" means "4)2 is sometimes false." We can now define the logical product and the logical sum of sets of propositions: The logical product of a set of propositions is a propositional function that is always true, i.e., true for all values of the variable.
The logical sum of a set of propositions is a propositional func tion that is sometimes true, i.e., true for one or more values of the variable.
Russell employs the following convenient notation: "(x) • 43•V expresses "For all values of x, 4)x," i.e., the logical product. "(ax) • 4)2" expresses "For some value of x, 4x," i.e., the logical sum. Thus, if 4 represents "is mortal," then "(x) . would mean "Everything is mortal"; whilst "(ax) • .0" would mean "There is something which is mortal." Again, "Socrates" Here we have made explicit the distinction between the two premises. Now consider the syllogism, "all men are mortal, all mortals are fallible, therefore all men are fallible." Using the
same symbols as before and adding y to denote the class of "fal lible beings" the syllogism can be stated:
< •aD7 (2) Again we see that the two syllogisms are of different forms. Syl logism (I) and (2) could be written in the notation of proposi tional functions as follows: (x) • (/)x < tkx : dry : < & y
(x) • ckx
The earlier symbolic logicians made little, or no, attempt to deal with relations. Yet their im portance is obvious. Most propositions of everyday life express relations between two or more terms, e.g., "Brutus killed Caesar," "Peter gave Paul a sum of money for a reward," and so on. Still keeping to the extensional point of view, we can see that relations can be regarded as a class of ordered couples, a class of ordered trios, and so on. That is, just as a class is the set of values satis fying a propositional function involving one variable, so a relation is the set of values satisfying a propositional function involving two or more variables. Relations have, however, some properties which have no analogues for classes. These properties are of the utmost importance in the logistic development of mathematics, but they cannot be discussed here (vide MATHEMATICS).