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 as Euclid does not try to prove that ’two magnitudes equal to the same third are equal to one another,’ so the Logician (as such) does not attempt to prove the uniformity of causation and the other principles of his science.