The ten axioms are the following. (1) If `p` implies `q`, then
`p` implies `q`^{[14]}: in other words, whatever `p` and
`q` may be,

is a proposition.
(2) If `p` implies `q``p` implies `q`, then `p` implies
`p`: in other words, whatever implies anything is also a proposition.
(3) If `p` implies `q`, then `q` implies
`q`; in other words, whatever is implied by anything is a proposition.
(4) A true hypothesis in an implication may be dropped, and the consequent
asserted. This is a principle incapable of formal symbolic statement, and
illustrating the essential limitations of formalism—a point to which I shall
return at a later stage. This definition is highly artificial, and illustrates the great distinction between mathematical and philosophical definitions. It is as
follows: If `p` implies `q`, then, if `q` implies
`q`, `p``q` (the logical product of `p` and
`q`) means that if `p` implies that `q` implies
`r`, then `r` is true. In other words, if `p` and
`q` are propositions, their joint assertion is equivalent to saying
that every proposition is true which is such that the first implies that the
second implies it. We cannot, with formal correctness, state our definition in
this shorter form, for the hypothesis

is already the logical product of `p` and `q` are
propositions

and `p` is a
proposition

We cannow state the
six main principles of inference, to each of which, owing to its importance, a
name is to be given; of these all except the last will be found in Peano’s
accounts of the subject. (5) If `q` is a proposition.`p` implies `p` and
`q` implies `q`, then `p``q` implies
`p`. this is called *simplification*, and asserts merely that
the joint assertion of two propositions implies the assertion of the first of
the two. (6) If `p` implies `q` and `q` implies
`r`, then `p` implies `r`. This will be called the
*syllogism*. (7) If `q` implies `q` and `r`
implies `r`, and if `p` implies that `q` implies
`r`, then `p``q` implies `r`. This is the
principle of *importation*. In the hypothesis, we have a product of three
propositions; but this can of course be defined by means of the product of two.
The principle states that if `p` implies that `q` implies
`r`, then `r` follows from the joint assertion of `p`
and `q`. For example: If I call on so-and-so, then if she is at home
I shall be admitted

implies If I call on so-and-so and she is at home, I
shall be admitted.

(8) If `p` implies `p` and `q`
implies `q`, then, if `p``q` implies `r`,
then `p` implies that `q` implies `r`. This is the converse of the preceding principle, and is
called *exportation*^{[15]}. The previous illustration reversed will
illustrate the principle. (9) If `p` implies `q` and
`p` implies `r`, then `p` implies
`q``r`: in other words, a proposition which implies each of
two propositions implies them both. This is called the principle of
*composition*. (10) If `p` implies `p` and `q`
implies `q`, then

implies

implies p`p` implies q`p`. This is called the principle of *reduction*; it has
less self-evidence than the previous principles, but is equivalent to many
propositions that are self-evident. I prefer it to these, because it is
explicitly concerned, like its predecessors, with implication, and has the same
kind of logical character as they have. If we remember that

is equivalent to `p`
implies `q`

we can easily convince ourselves that the above principle
is true; for `q` or
not-`p`,

is equivalent to

implies
`p` implies `q``p`

i.e. to `p` or the denial of
`q` or not-`p`,

i.e.
to `p` or

,`p` and not
`q``p`. But this way of persuading ourselves that the principle of
reduction is true involves many logical principles which have not yet been
demonstrated, and cannot be demonstrated except by reduction or some
equivalent.The principle is especially useful in connection with negation.
Without its help, by means of the first nine principles, we can prove the law of
contradiction; we can prove, if `p` and `q` be propositions,
that `p` implies not-not-`p`; that

is equivalent to `p` implies
not-`q`

and to not-`q` implies
not-`p``p``q`; that

implies `p`
implies `q`not-

; that `q` implies
not-`p``p` implies that not-`p` implies
`p`; that not-`p` is equivalent to

; and that `p` implies
not-`p`

is
equivalent to `p` implies not-`q`not-not-

But we
cannot prove without reduction or some equivalent (so far at least as I have
been able to discover) that `p` implies not-`q`.`p` or not-`p` must be true (the
law of the excluded middle); that every proposition is equivalent to the
negation of some other proposition; that not-not-`p` implies
`p`; that not-

implies
`q` implies not-`p`

; that `p` implies `q`not-

implies `p` implies
`p``p`, or that

implies `p` implies
`q`

Each of these
assumptions is equivalent to the principle of reduction, and may, if we choose,
be substituted for it. Some of them—especially excluded middle and double
negation—appear to have far more self-evidence. Bu when we have seen how to
define disjunction and engation in terms of implication, we shall see that the
supposed simplicity vanishes, and that, for formal purposes at any rate,
reduction is simpler than any of the possible alternatives. For this reason I
retain it among my premisses in preference to more usual and more superficially
obvious propositions.(§ 18 ¶ 1)`q` or not-`p`.

§ 18 n. 1. Note that the implications denoted by *if* and *then*, in these axioms, are formal, while those denoted by *implies* are material. ↩

§ 18 n. 2. (7) and (8) cannot (I think) be deduced from the definition of the logical product, because they are required for passing from If

to `p` is a proposition, then

implies etc.`q` is a propositionIf

↩`p` and `q` are propositions, then etc.

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