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.