The general principle of inference having been explained in the previous chapters, and a suitable system of symbols provided, we have now before us the comparatively easy task of tracing out the most common and important forms of deductive reasoning. The general problem of deduction is as follows:—From one or more propositions called premises to draw such other propositions as will necessarily be true when the premises are true. By deduction we investigate and unfold the information contained in the premises; and this we can do by one single rule—For any term occurring in any proposition substitute the term which is asserted in any premise to be identical with it. To obtain certain deductions, especially those involving negative conclusions, we shall require to bring into use the second and third Laws of Thought, and the process of reasoning will then be called Indirect Deduction. In the present chapter, however, I shall confine my attention to those results which can be obtained by the process of Direct Deduction, that is, by applying to the premises themselves the rule of substitution. It will be found that we can combine into one harmonious system, not only the various moods of the ancient syllogism but a great number of equally important forms of reasoning, which had no recognised place in the old logic. We can at the same time dispense entirely with the elaborate apparatus of logical rules and mnemonic lines, which were requisite so long as the vital principle of reasoning was not clearly expressed.
Immediate Inference.
Probably the simplest of all forms of inference is that which has been called Immediate Inference, because it can be performed upon a single proposition. It consists in joining an adjective, or other qualifying clause of the same nature, to both sides of an identity, and asserting the equivalence of the terms thus produced. For instance, since
it follows that
If we suppose that
it follows that
In general terms, from the identity
we can infer the identity
This is but a case of plain substitution; for by the first Law of Thought it must be admitted that
and if, in the second side of this identity, we substitute for A its equivalent B, we obtain
In like manner from the partial identity
we may obtain
by an exactly similar act of substitution; and in every other case the rule will be found capable of verification by the principle of inference. The process when performed as here described will be quite free from the