Most of the propositions of the class-calculus are easily deduced
from those of the propositional calculus. The logical product or common part of
two classes `a` nad `b` is the class of `x`’s such
that the logical product of

and
`x` is an `a`

is true. Similiarly we define the logical
sum of two classes (`x` is a `b``a` or `b`), and the negation of a class
(not-`a`). A new idea is introduced by the logical product and sum of a
class of classes. If `k` is a class of classes, its logical product is
the class of terms belonging to each of the classes of `k`, i.e. the class of terms
`x` such that

implies
`u` is a `k`

for all values of `x` is a `u``u`. The
logical sum isthe class which is contained in every class of the class
`k` is contained, i.e. the class of terms `x` such that, if

implies `u`
is a `k`

for all values of `u` is contained in `c``u`, then, for all values of `c`,
`x` is a `c`. And we say that a class `a` is
contained in class `b` when

implies `x` is an `a`

for all values of `x` is a `b``x`. In
like manner with the above we may define the product and sum of a class of
propositions. Another very important notion is what is called the
*existence* of a class—a word hwich must not be supposed to mean what
existence means in philosophy. A class is said to exist when it has at least one
term. A formal definition is as follows: `a` is an existent class when
and only when any proposition is true provided

always implies it whatevervalue we may give to `x` is an
`a``x`. It
must be understood that the proposition implied must be a genuine proposition,
not a propositional function of `x`. A class `a` exists when
the logical sum of all propositions of the form

is true,
i.e. when not all such
propositions are false.(§ 25 ¶ 1)`x` is an
`a`

It is important to understand clearly the manner in which
propositions in the class-calculus are obtained from those in the propositional
calculus. Consider, for example, the syllogism. We have

and `p` implies
`q`

imply
`q` implies `r`

Now put `p` implies `r`.`x` is an
`a`,`x` is a `b`,

for `x` is a
`c``p`, `q`, `r`, where
`x` must have some definite value, but it is not necessary to decide
what value. We then find that if, for the value of `x` in question,
`x` is an `a` implies `x` is a `b`, and
`x` is a `b` implies `x` is a `c`, then
`x` is an `a` implies `x` is a `c`. Since
the value of `x` is irrelevant, we may vary `x`, and thus we
find that if `a` is contained in `b`, and `b` in
`c`, then `a` is contained in `c`. This is the
class-syllogism. But in applying this process it is necessary to employ the
utmost caution if fallacies are to be successfully avoided. In this connection
it will be instructive to examine a point upon which a dispute has arisen between Schröder and Mr McColl^{[19]}.
Schröder asserts that if `p`, `q`, `r` are
propositions,

is equivalent
to the disjunction `p``q` implies `r`

Mr
McColl admits that the disjunction implies the other, but denies the converse
implication. The reason for the divergence is, that Schröder is thinking of
propositions and material implication, while Mr McColl is thinking of
propositional functions and formal implication. As regards propositions, the
truth of the principle may be easily made plain by the following considerations.
If `p` or `q` implies `r`.`p``q` implies `r`, then, if either `p`
or `q` be false, the one of them which is false implies `r`,
because false propositions imply all propositions. But if both be true,
`p``q` is true, and therefore `r` is true, and
therefore `p` implies `r` and `q` implies
`r`, because true propositions are implied by every proposition. Thus
in any case, one at least of the propositions `p` and `q` must
imply `r`. (This is not a proof, but an elucidation.) But Mr McColl
objects: Suppose `p` and `q` to be mutually contradictory, and
`r` to be the null proposition, then `p``q` implies
`r` but neither `p` nor `q` implies `r`.
Here we are dealing with propositional functions and formal implication. A
propositional function is said to be null when it is false for all values of
`x`; and the class of `x`’s satisfying the function is called
the null-class, being in fact a class of no terms. Either the function or the
class, following Peano, I shall denote by `Λ`. Now let our `r`
be replaced by `ϕ``x`, and our `q` by
not-`ϕ``x`, where `ϕ``x` is any
propositional function. Then `p``q` is false for all values of
`x`, and therefore implies `Λ`. Thus the above formula can
only be truly interpreted in the propositional calculus: in the class-calculus
it is false. This may be easily rendered obvious by the following
considerations: Let `ϕ``x`, `ψ``x`,
`χ``x` be three propositional functions. Then

implies, for all vlues of `ϕ``x` . `ψ``x` implies
`χ``x``x`, that either
`ϕ``x` implies `χ``x` or
`ψ``x` implies `χ``x`. But it does not imply
that either `ϕ``x` implies `χ``x` for all
values of `x`, or `ψ``x` implies
`χ``x` for all values of `x`. The disjunction is what
I shall call a *variable* disjunction, as opposed to a constant one: that
is, in some cases one alternative is true, in others the other, whereas in a
constant disjunction there is one of the alternatives (thought it is not stated
which) that is always true. Wherever disjunctions occur in regard to
propositional functions, they will only be transformable into statements in the
class-calculus in cases where the disjunction is constant. This is a point which
is both important in itself and instructive in its bearings. Another way of
stating the matter is this: In the proposition: If `ϕ``x` .
`ψ``x` implies `χ``x`, then either
`ϕ``x` implies `χ``x` or
`ψ``x` implies `χ``x`, the implication
indicated by *if* and `then` is formal, while the subordinate
implications are material; hence the subordinate implications do not lead to the
inclusion of one class in another, which results only from formal implication.(§ 25 ¶ 2)

The formal laws of addition, multiplication, tautology and
negation are the same as regards classes and propositions. The law of tautology
states that no change is made when a class or proposition is added to or
multiplied by itself. A new feature of the class-calculus is the null-class, or
class having no terms. This may be defined as the class of terms that belong to
every class, as the class which does not exist (in the sense defined above), as
the class which is contained in every class, as the class `Λ` which is
such that the propositional function

is
false for all values of `x` is a `Λ``x`, or as the class of `x`’s
satisfying any propositional function `ϕ``x` which is false
for all values of `x`. All those definitions are easily shown to be
equivalent.(§ 25 ¶ 3)

§ 25 n. 1. Schröder, Algebra der Logik, Vol. II, pp. 258-9; McColl, Calculus of Equivalent Statements, fifth paper, Proc. Lond. Math. Soc. Vol. XXVIII, p. 182. ↩

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