Venus, and Venus must differ from Mars. The Earth differs from Jupiter in density; therefore Jupiter must differ from the Earth. Speaking generally, if AB we shall also have BA, and these two forms may be considered expressions of the same difference. But the reader will notice that the relation of differing things is not wholly reciprocal. The density of Jupiter does not differ from that of the Earth in the same way that that of the Earth differs from that of Jupiter. The change of sensation which we experience in passing from Venus to Mars is not the same as what we experience in passing back to Venus, but just the opposite in nature. The colour of the sky is lighter than that of the ocean; therefore that of the ocean cannot be lighter than that of the sky, but darker. In these and all similar cases we gain a notion of direction or character of change, and results of immense importance may be shown to rest on this notion. For the present we shall be concerned with the mere fact of identity existing or not existing. Twofold Interpretation of Propositions. Terms, as we have seen (p. 31), may have a meaning either in extension or intension; and according as one or the other meaning is attributed to the terms of a proposition, so may a different interpretation be assigned to the proposition itself. When the terms are abstract we must read them in intension, and a proposition connecting such terms must denote the identity or nonidentity of the qualities respectively denoted by the Thus if we say terms. Equality Identity of magnitude, = the assertion means that the circumstance of being equal exactly corresponds with the circumstance of being identical in magnitude. Similarly in Opacity Incapability of transmitting light, = the quality of being incapable of transmitting light is declared to be the same as the intended meaning of the word opacity. When general names form the terms of a proposition we may apply a double interpretation. Thus means either that the qualities which belong to all exogens are the same as those which belong to all dicotyledons, or else that every individual falling under one name falls equally under the other. Hence it may be said that there are two distinct fields of logical thought. We may argue either by the qualitative meaning of names or by the quantitative, that is, the extensive meaning. Every argument involving concrete plural terms might be converted into one involving only abstract singular terms, and vice versa. But there are many reasons for believing that the intensive or qualitative form of reasoning is the primary and fundamental one. It is sufficient to point out that we may use abstract terms which contain no reference to an extensive meaning; and when there is a mode which we must sometimes and may always adopt, it is higher in importance than a mode which we never need adopt necessarily. CHAPTER IV. DEDUCTIVE REASONING. 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 or expression substitute the expression 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 in 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 distinct 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 Conductors of electricity Non-electrics, it follows that = Liquid conductors of electricity = Liquid non-electrics. If we suppose that Plants Bodies decomposing carbonic acid, it follows that = Microscopic plants Microscopic bodies decomposing carbonic acid. In general symbols, from the identity we can infer the identity A =B AC=BC. This is but a case of plain substitution; for by the first Law of Thought it must be admitted that AC=AC, and if in the second side of this identity we substitute for A its equivalent B, we obtain by an exactly similar form 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 found free from the liability to error which I have shown a to exist in Immediate Inference by added Determinants, as described by Dr. Thomson". Inference with Two Simple Identities. One of the most common forms of inference, and one to which I shall especially direct attention, is practised with two simple identities. From the two statements that 'London is the capital of England' and 'London is the most populous city in the world,' we instantaneously draw the conclusion that The capital of England is the most populous city in the world.' Similarly, from the identities Hydrogen Substance of least density = Hydrogen = Substance of least atomic weight, we infer Substance of least density = Substance of least atomic weight. The general form of the argument is exhibited in the symbols We may describe the result by saying that terms identical with the same term are identical with each other; and it is impossible to overlook the analogy to the first axiom of Euclid that things equal to the same thing are equal to each other.' It has been very commonly supposed that this was a fundamental principle of thought incapable of reduction to anything simpler. But I entertain no doubt that this form of reasoning is only one case a Elementary Lessons in Logic,' p. 86. |