Logic | Page 4

far as
they are not laws of quantity, are tested by the principles of Logic, if
they at all admit of proof.
But it is plain that the process of proving cannot go on for ever;
something must be taken for granted; and this is usually considered to
be the case (1) with particular facts that can only be perceived and
observed, and (2) with those highest laws that are called 'axioms' or
'first principles,' of which we can only say that we know of no
exceptions to them, that we cannot help believing them, and that they
are indispensable to science and to consistent thought. Logic, then, may
be briefly defined as the science of proof with respect to qualitative
laws and propositions, except those that are axiomatic.
§ 2. Proof may be of different degrees or stages of completeness.
Absolute proof would require that a proposition should be shown to
agree with all experience and with the systematic explanation of
experience, to be a necessary part of an all-embracing and
self-consistent philosophy or theory of the universe; but as no one
hitherto has been able to frame such a philosophy, we must at present
put up with something less than absolute proof. Logic, assuming certain
principles to be true of experience, or at least to be conditions of
consistent discourse, distinguishes the kinds of propositions that can be
shown to agree with these principles, and explains by what means the
agreement can best be exhibited. Such principles are those of
Contradiction (chap. vi.), the Syllogism (chap. ix.), Causation (chap.
xiv.), and Probabilities (chap. xx.). To bring a proposition or an
argument under them, or to show that it agrees with them, is logical

proof.
The extent to which proof is requisite, again, depends upon the present
purpose: if our aim be general truth for its own sake, a systematic
investigation is necessary; but if our object be merely to remove some
occasional doubt that has occurred to ourselves or to others, it may be
enough to appeal to any evidence that is admitted or not questioned.
Thus, if a man doubts that some acids are compounds of oxygen, but
grants that some compounds of oxygen are acids, he may agree to the
former proposition when you point out that it has the same meaning as
the latter, differing from it only in the order of the words. This is called
proof by immediate inference.
Again, suppose that a man holds in his hand a piece of yellow metal,
which he asserts to be copper, and that we doubt this, perhaps
suggesting that it is really gold. Then he may propose to dip it in
vinegar; whilst we agree that, if it then turns green, it is copper and not
gold. On trying this experiment the metal does turn green; so that we
may put his argument in this way:--
Whatever yellow metal turns green in vinegar is copper; This yellow
metal turns green in vinegar; Therefore, this yellow metal is copper.
Such an argument is called proof by mediate inference; because one
cannot see directly that the yellow metal is copper; but it is admitted
that any yellow metal is copper that turns green in vinegar, and we are
shown that this yellow metal has that property.
Now, however, it may occur to us, that the liquid in which the metal
was dipped was not vinegar, or not pure vinegar, and that the greenness
was due to the impurity. Our friend must thereupon show by some
means that the vinegar was pure; and then his argument will be that,
since nothing but the vinegar came in contact with the metal, the
greenness was due to the vinegar; or, in other words, that contact with
that vinegar was the cause of the metal turning green.
Still, on second thoughts, we may suspect that we had formerly
conceded too much; we may reflect that, although it had often been

shown that copper turned green in vinegar, whilst gold did not, yet the
same might not always happen. May it not be, we might ask, that just at
this moment, and perhaps always for the future gold turns, and will turn
green in vinegar, whilst copper does not and never will again? He will
probably reply that this is to doubt the uniformity of causation: he may
hope that we are not serious: he may point out to us that in every action
of our life we take such uniformity for granted. But he will be obliged
to admit that, whatever he may say to induce us to assent to the
principle of Nature's uniformity, his arguments will not amount to
logical proof, because every argument in some way assumes that
principle. He has come, in fact, to the limits of Logic. Just