The subject of Symbolic Logic consists of three parts, the
calculus of propositions, the calculus of classes, and the calculus of
relations. Between the first two, there is, within limits, a certain
parallelism, which arises as follows: In any symbolic expression, the letters
may be interpreted as classes or as propositions, and the relation of inclusion
in the one case may be replaced by that of formal implication in the other.
Thus, for example, in the principle of the syllogism, if `a`,
`b`, `c` be classes, and `a`
is contained in `b`, `b` in
`c`, then `a` is contained in
`c`; but if `a`, `b`,
`c` be propositions, and `a` implies
`b`, `b` implies `c`, then
`a` implies `c`. A great deal has been made of
this duality, and in the later editions of the Formulaire, Peano appears to have sacrificed logical precision to its
preservation^{[9]}. But, as a matter of fact, there are many ways in which
the calculus of propositions differs from that of classes. Consider, for
example, the following: If

This proposition is true; but its correlative is
false, namely: `p`, `q`,
`r` are propositions, and `p` implies
`q` or `r`, then `p` implies
`q` or `p` implies
`r`.If

For example, English people are all either men or
women, not are not all men nor yet all women. The fact is that the duality holds
for propositions asserting of a variable term that it belongs to a class, i.e. such propositions as
`a`, `b`,
`c` are classes, and `a` is contained in
`b` or `c`, then `a` is
contained in `b` or `a` is contained in
`c`.

provided that the implication
involved be formal, i.e. one
which holds for all values of `x` is a man,`x`. But

is itself not a proposition at all,
being neither true nor false; and it is not with such entities that we are
concerned in the propositional calculus, but with genuine propositions. To
continue the above illustration: It is true that for all values of
`x` is a man`x`,

either implies `x` is a man or a woman

or
`x` is a man

But it is false that
`x` is a woman.

either implies
`x` is a man or woman

or `x` is a man

for all values of `x` is a
woman`x`. Thus the implication
involved, which is always one of the two, is not formal, since it does not hold
for all values of `x`, being not always the same one of the
two. The symbolic affinity of the propositional and the class logic is, in fact,
something of a snare, and we have to decide which of the two we are to make
fundamental. Mr McColl, in an
important series of papers^{[10]} , has contended for the view
that implication and propositions are more fundamental than inclusion and
classes; and in this opinion I agree with him. But he does not appear to me to
realize adequately the distinction between genuine propositions and such as
contain a real variable: thus he is led to speak of propositions as sometimes
true and sometimes false, which of course is impossible with a genuine
proposition. As the distinction involved is of very great importance, I shall
dwell on it before proceeding further. A proposition, we may say, is anything
that is true or that is false. An expression such as

is therefore not a proposition, for it is neither true nor
false. If we give to `x`
is a man`x` any constant value whatever, the
expression becomes a proposition: it is thus as it were a schematic form
standing for any one of a whole class of propositions. And when we say

we are not asserting a single
implication, but a class of implications; we have now a genuine proposition, in
which, though the letter `x` is a man implies `x` is mortal for
all values of `x`,`x` appears, there is no real
variable: the variable is absorbed in the same kind of way as the
`x` under the integral sign in a definite integral, so that the
result is no longer a function of `x`. Peano distinguishes a
variable which appears in this way as `apparent`, since the
proposition does not depend upon the variable; whereas in

there are different propositions
for different values of the variable, and the variable is what Peano calls `x` is a man*real*^{[11]}. I shall speak of propositions exclusively where there is
no real variable: where there are one or more real variables, and for all values
of the variables the expression involved is a proposition, I shall call the
expression a *propositional function*. The study of genuine propositions
is, in my opinion, more fundamental than that of classes; but the study of
propositional functions appears to be strictly on a par with that of classes,
and indeed scarcely distinguishable therefrom. Peano, like McColl, at first
regarded propositions as more fundamental than classes, but he, even more
definitely, considered propositional functions rather than propositions. From
this criticism, Schröder is exempt: his second volume deals with genuine
propositions, and point out their formal differences from classs.(§ 13 ¶ 1)

§ 13 n. 1. On the points where the duality breaks down, cf. Schröder, op. cit., Vol. II, Lecture 21. ↩

§ 13 n. 2. Cf. The Calculus of Equivalent Statements, Proceedings of the London Mathematical Society, Vol. IX and subsequent volumes; Symbolic Reasoning, Mind, Jan. 1880, Oct. 1897, and Jan. 1900; La Logique Symbolique et ses Applications, Bibliothèque du Congrès International de Philosophie, Vol. III (Paris, 1901). I shall in future quote the proceedings of the above Congress by the title Congrès. ↩

§ 13 n. 3. F. 1901, p. 2. ↩

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