§ 9. The preceding considerations it, is the entire theory of the ascerenable us to understand the true tainment of reasoned or inferred truth. nature of what is termed, by recent Formal Logic, therefore, which Sir writers, Formal Logic, and the rela- William Hamilton from his own point tion between it and Logic in the of view, and Archbishop Whately widest sense. Logic, as I conceive from his, have represented as the an abuse of language to say, that the proof tion is proved by the larger : while I conthat Socrates is mortal is that all men are tend that both assertions are proved tomortal. Turn it in what way we will, this gether by the same evidence, namely, the seems to me to be asserting that a thing is grounds of experience on which the genethe proof of itself. Whoever pronounces ral assertion was made, and by which it the words, All men are mortal, has affirmed must be justified. that Socrates is mortal, though he may never have heard of Socrates; for since Socrates, whether known to be so or not, really is a man, he is included in the words, All men, and in every assertion of which they are the subject. If the reviewer does not see that there is a difficulty here, I can only advise him to reconsider the subject until he does after which he will be a better judge of the success or failure of an attempt to remove the difficulty. That be had reflected very little on the point when he wrote his remarks is shown by his oversight respecting the dictum de omni et nullo. He acknowledges that this maxim as commonly expressed, "Whatever is true of a class is true of everything included in the class," is a mere identical proposition, since the class is nothing but the things included in it. But he thinks this defect would be cured by wording the maxim thus, "Whatever is true of a class is true of everything which can be shown to be a member of the class:" as if a thing could "be shown" to be a member of the class without being one. If a class means the sum of all the things included in the class, the things which can "be shown" to be included in it are part of the sum, and the dictum is as much an identical proposition with respect to them as to the rest. One would almost imagine that in the reviewer's opinion things are not members of a class until they are called up publicly to take their place in it-that so long, in fact, as Socrates is not known to be a man he is not a man, and any assertion which can be made concerning men does not at all regard him, nor is affected as to its truth or falsity by anything in which he is concerned. The reviewer says, that if the major premise included the conclusion, "we should be able to affirm the conclusion without the intervention of the minor premise; but every one sees that that is impossible." A similar argument is urged by Mr. De Morgan (Formal Logic, p. 259): "The whole objection tacitly assumes the superfluity of the minor; that is, tacitly assumes we know Socrates* to be a man as soon as we know him to be Socrates." The objection would be well grounded if the assertion that the major premise includes the conclusion, meant that it individually specifies all it includes. As, however, the only indication it gives is a description by marks, we have still to compare any new individual with the marks; and to show that this comparison has been made is the office of the minor. But since, by supposition, the new individual has the marks, whether we have ascertained him to have them or not; if we have affirmed the major premise, we have asserted him to be mortal. Now my position is that this assertion cannot be a necessary part of the argument. It cannot be a necessary condition of reasoning that we should begin by making an assertion which is afterwards to be employed in proving a part of itself. I can conceive only one way out of this difficulty, viz, that what really forms the proof is the other part of the assertion; the portion of it. the truth of which has been ascertained previously; and that the unproved part is bound up in one formula with the proved part in mere anticipation, and as a memorandum of the nature of the conclusions which we are prepared to prove. With respect to the minor premise in its formal shape, the minor as it stands in the syllogism, predicating of Socrates a definite class name, I readily admit that it is no more a necessary part of reasoning than the major. When there is a major, doing its work by means of a class name, minors are needed to interpret it: but reasoning can be carried on without either the one or the other. They are not the conditions of reasoning, but a precaution The difference between the reviewer's theory and mine may be thus stated. Both admit that when we say, All men are mortal, we make an assertion reaching beyond the sphere of our knowledge of individual cases; and that when a new individual, Socrates, is brought within the field of our knowledge by means of the minor premise, we learn that we have already made an assertion respecting Socrates without knowing it: our own general formula being, to that extent, for the first time interpreted to us. But according to the reviewer's theory, the smaller asser-exemplum, * Mr. De Morgan says "Plato," but to prevent confusion I have kept to my own whole of Logic properly so called, is really a very subordinate part of it, not being directly concerned with the process of Reasoning or Inference in the sense in which that process is a part of the Investigation of Truth. What, then, is Formal Logic? The name seems to be properly applied to all that portion of doctrine which relates to the equivalence of different modes of expression; the rules for determining when assertions in a given form imply or suppose the truth or falsity of other assertions. This includes the theory of the Import of Propositions, and of their Conversion, Equipollence, and Opposition of those falsely called Inductions (to be hereafter spoken of *), in which the apparent generalisation is a mere abridged statement of cases known individually; and finally, of the syllogism while the theory of Naming, and of (what is inseparably connected against erroneous reasoning. The only minor premise necessary to reasoning in the example under consideration is, Socrates is like A, B, C, and the other indithis is the only universal type of that step in the reasoning process which is represented by the minor. Experience, however, of the uncertainty of this loose mode of inference teaches the expediency of determining beforehand what kind of likeness to the cases observed is necessary to bring an unobserved case within the same predicate; and the answer to this question is the major. The minor then identifies the precise kind of likeness possessed by Socrates, as being the kind required by the formula. Thus the syllogistic major and the syllogistic minor start into existence together, and are called forth by the same exigency. When we conclude from personal experience without referring to any record-to any general theorems, either written, or traditional, or mentally registered by ourselves as conclusions of our own drawing-we do not use, in our thoughts, either a major or a minor, such as the syllogism puts into words. When, however, we revise this rough inference from particulars to particulars, and substitute a careful one, the revision consists in selecting two syllogistic premises. But this neither alters nor adds to the evidence we had before; it only puts us in a better position for judging whether our inference from particulars to particulars is well grounded. viduals who are known to have died. And * Infra, book iii. ch. ii, with it) Definition, though belonging still more to the other and larger kind of logic than to this, is a necessary preliminary to this. The end aimed at by Formal Logic, and attained by the observance of its precepts, is not truth, but consistency. It has been seen that this is the only direct purpose of the rules of the syllogism; the intention and effect of which is simply to keep our inferences or conclusions in complete consistency with our general formulæ or directions for drawing them. The Logic of Consistency is a necessary auxiliary to the Logic of Truth, not only because what is inconsistent with itself or with other truths cannot be true, but also because truth can only be successfully pursued by drawing inferences from experience, which, if warrantable at all, admit of being generalised, and, to test their warrantableness, require to be exhibited in a generalised form; after which the correctness of their application to particular cases is a question which specially concerns the Logic of Consistency. This Logic, not requiring any preliminary knowledge of the processes or conclusions of the various sciences, may be studied with benefit in a much earlier stage of education than the Logic of Truth: and the practice which has empirically obtained of teaching it apart, through elementary treatises which do not attempt to include anything else, though the reasons assigned for the practice are in general very far from philosophical, admits of philosophical justification. CHAPTER IV. OF TRAINS OF REASONING AND § 1. IN our analysis of the syllogism, it appeared that the minor premise always affirms a resemblance between a new case and some cases previously known; while the major premise asserts something which, having been found true of those | flame, which spot is soluble in hypoknown cases, we consider ourselves chloride of calcium, is arsenic; the warranted in holding true of any substance before me conforms to this other case resembling the former in certain given particulars. If all ratiocinations resembled, as to the minor premise, the examples which were exclusively employed in the preceding chapter; if the resemblance, which that premise asserts, were obvious to the senses, as in the proposition "Socrates is a man," or were at once ascertainable by direct observation; there would be no necessity for trains of reasoning, and Deductive or Ratiocinative Sciences would not exist. Trains of reasoning exist only for the sake of extending an induction founded, as all induc tions must be, on observed cases, to other cases in which we not only cannot directly observe the fact which is to be proved, but cannot directly observe even the mark which is to prove it. § 2. Suppose the syllogism to be, All cows ruminate; the animal which is before me is a cow; therefore it ruminates. The minor, if true at all, is obviously so: the only premise the establishment of which requires any anterior process of inquiry is the major; and provided the induction of which that premise is the expression was correctly performed, the conclusion respecting the animal now present will be instantly drawn ; because, as soon as she is compared with the formula, she will be identified as being included in it. But suppose the syllogism to be the following:-All arsenic is poisonous, the substance which is before me is arsenic, therefore it is poisonous. The truth of the minor may not here be obvious at first sight; it may not be intuitively evident, but may itself be known only by inference. It may be the conclusion of another argument, which, thrown into the syllogistic form, would stand thus :-Whatever when lighted produces a dark spot on a piece of white porcelain held in the condition; therefore it is arsenic. To establish, therefore, the ultimate conclusion, The substance before me is poisonous, requires a process which, in order to be syllogistically expressed, stands in need of two syllogisms; and we have a Train of Reasoning. When, however, we thus add syllogism to syllogism, we are really adding induction to induction. Two separate inductions must have taken place to render this chain of inference possible; inductions founded, probably, on different sets of individual instances, but which converge in their results, so that the instance which is the subject of inquiry comes within the range of them both. The record of these inductions is contained in the majors of the two syllogisms. First, we, or others for us, have examined various objects which yielded under the given circumstances a dark spot with the given property, and found that they possessed the properties connoted by the word arsenic; they were metallic, volatile, their vapour had a smell of garlic, and so forth. Next, we, or others for us, have examined various specimens which possessed this metallic and volatile character, whose vapour had this smell, &c., and have invariably found that they were poisonous. The first observation we judge that we may extend to all substances whatever which yield that particular kind of dark spot; the second, to all metallic and volatile substances resembling those we examined; and consequently, not to those only which are seen to be such, but to those which are concluded to be such by the prior induction. The substance before us is only seen to come within one of these inductions; but by means of this one it is brought within the other. We are still, as before, concluding from particulars to particulars; but we are now concluding from particulars observed, to other particulars which are not, as in the simple case, seen to resemble them in material points, but inferred to do so, because resembling them in something else, which we have been led by quite a different set of instances to consider as a mark of the former resemblance. This first example of a train of reasoning is still extremely simple, the series consisting of only two syllogisms. The following is somewhat more complicated:-No government, which earnestly seeks the good of its subjects is likely to be overthrown; some particular government earnestly seeks the good of its subjects, therefore it is not likely to be overthrown. The major premise in this argument we shall suppose not to be derived from considerations à priori, but to be a generalisation from history, which, whether correct or erroneous, must have been founded on observation of governments concerning whose desire of the good of their subjects there was no doubt. It has been found, or thought to be found, that these were not easily overthrown, and it has been deemed that those instances warranted an extension of the same predicate to any and every government which resembles them in the attribute of desiring earnestly the good of its subjects. But does the government in question thus resemble them? This may be debated pro and con by many arguments, and must, in any case, be proved by another induction; for we cannot directly observe the sentiments and desires of the persons who carry on the government. To prove the minor, therefore, we require an argument in this form: Every government which acts in a certain manner desires the good of its subjects: the supposed government acts in that particular manner, therefore it desires the good of its subjects. But is it true that the government acts in the manner supposed? This minor also may require proof; still another induction, as thus-What is asserted by intelligent and disinterested witnesses may be believed to be true; that the government acts in this manner is asserted by such witnesses, therefore it may be believed to be true. The argument hence consists of three steps. Having the evidence of our senses that the case of the government under consideration resembles a number of former cases, in the circumstance of having something asserted respecting it by intelligent and disinterested witnesses, we infer, first, that, as in those former instances, so in this instance, the assertion is true. Secondly, what was asserted of the government being that it acts in a particular manner, and other governments or persons having been observed to act in the same manner, the government in question is brought into known resemblance with those other governments or persons; and since they were known to desire the good of the people, it is thereupon, by a second induction, inferred that the particular government spoken of desires the good of the people. This brings that government into known resemblance with the other governments which were thought likely to escape revolution, and thence, by a third induction, it is concluded that this particular government is also likely to escape. This is still reasoning from particulars to particulars, but we now reason to the new instance from three distinct sets of former instances: to one only of those sets of instances do we directly perceive the new one to be similar; but from that similarity we inductively infer that it has the attribute by which it is assimilated to the next set, and brought within the corresponding induction; after which by a repetition of the same operation we infer it to be similar to the third set, and hence a third induction conducts us to the ultimate conclusion. §3. Notwithstanding the superior complication of these examples, compared with those by which in the preceding chapter we illustrated the general theory of reasoning, every doctrine which we then laid down sufficient for an induction, and we have marks to show whether any new case is one of those to which, if then known, the induction would have been deemed to extend. These marks we either recognise at once, or by the aid of other marks, which by another preIf vious induction we collected to be marks of the first. Even these marks of marks may only be recognised through a third set of marks; and we may have a train of reasoning, of any length, to bring a new case within the scope of an induction grounded on particulars its similarity to which is only ascertained in this indirect manner. holds equally true in these more intricate cases. The successive general propositions are not steps in the reasoning, are not intermediate links in the chain of inference between the particulars observed and those to which we apply the observation. we had sufficiently capacious memories, and a sufficient power of maintaining order among a huge mass of details, the reasoning could go on without any general propositions; they are mere formulæ for inferring particulars from particulars. The principle of general reasoning is (as before explained), that if, from observation of certain known particulars, what was seen to be true of them can be inferred to be true of any others, it may be inferred of all others which are of a certain description. And in order that we may never fail to draw this conclusion in a new case when it can be drawn correctly, and may avoid drawing it when it cannot, we determine once for all what are the distinguishing marks by which such cases may be recognised. The subsequent process is merely that of identifying an object, and ascertaining it to have those marks; whether we identify it by the very marks themselves, or by others which we have ascertained (through another and a similar process) to be marks of those marks. The real inference is always from particulars to particulars, from the observed instances to an unobserved one; but in drawing this inference, we conform to a formula which we have adopted for our guidance in such operations, and which is a record of the criteria by which we thought we had ascertained that we might distinguish when the inference could, and when it could not, be drawn. The real premises are the individual observations, even though they may have been forgotten, or, being the observations of others and not of ourselves, may, to us, never have been known but we have before us proof that we or others once thought them Thus, in the preceding example, the ultimate inductive inference was, that a certain government was not likely to be overthown; this inference was drawn according to a formula in which desire of the public good was set down as a mark of not being likely to be overthrown; a mark of this mark was acting in a particular manner; and a mark of acting in that manner was being asserted to do so by intelligent and disinterested witnesses: this mark the government under discussion was recognised by the senses as possessing. Hence that government fell within the last induction, and by it was brought within all the others. The perceived resemblance of the case to one set of observed particular cases brought it into known resemblance with another set, and that with a third. In the more complex branches of knowledge, the deductions seldom consist, as in the examples hitherto exhibited, of a single chain, a a mark of b, b of c, c of d, therefore a a mark of d. They consist (to carry on the same metaphor) of several chains united at the extremity, as thus: a a mark of d, b of e, c of f, d e f of n, therefore a b c a mark of n. Suppose, for example, the following combination of circumstances: 1st, rays of light impinging on a reflecting surface; 2nd, that surface parabolic ; 3rd, those rays parallel to each other |