colon (:) between the term and the copula when the quantity is universal, and a comma (,) when the quantity is particular. Thus we readily express the following affirmative propositions. C: M Some C's are some M's (I) and so on. Any affirmative proposition can be converted into the corresponding negative proposition by drawing a stroke through the line denoting the copula, as in the following C: C C, : M No C is any M : M Some C is not any M (0) M Some C is not some M (w) Any syllogism can be represented by placing M the middle term in the centre and connecting it on each side with the other terms. The copula representing the conclusion can then be placed below; Barbara is expressed as follows Sir W. Hamilton also proposed a new law or supreme canon of the syllogism by which the validity of all forms of the syllogism might be tested. This was stated in the following words: "What worse relation of subject and predicate subsists between either of two terms and a common third term, with which both are related, and one at least positively so-that relation subsists between these two terms themselves." By a worse relation, Sir William means that a negative relation is worse than an affirmative and a particular than a universal. This canon thus expresses the rules that if there be a negative premise the conclusion must be negative, and if there be a particular premise the conclusion must be particular. Special canons were also developed for each of the three figures, but in thus rendering the system complex the advantages of the quantified form of proposition seem to be lost. Prof. De Morgan also discovered the advantages of the quantified predicate, and invented a system differing greatly from that of Sir W. Hamilton. It is fully explained in his Formal Logic, The Syllabus of a new System of Logic, and various important memoirs on the Syllogism in the Transactions of the Cambridge Philosophical Society. In these works is also given a complete explanation of the "Numerically Definite Syllogism." Mr De Morgan pointed out that two particular premises may often give a valid conclusion provided that the actual quantities of the two terms are stated, and when added together exceed the quantity of the middle term. Thus if the majority of a public meeting vote for the first resolution, and a majority also vote for the second, it follows necessarily that some who voted for the first voted also for the second. The two majorities added together exceed the whole number of the meeting, so that they could not consist of entirely different people. They may indeed consist of exactly the same people; but all that we can deduce from the premises is that the excess of the two majorities added together over the number of the meeting must have voted in favour of each resolution. This kind of inference has by Sir W. Hamilton been said to depend on ultra-total distribution; and the name of Plurative Propositions has been proposed for all those which give a distinct idea of the fraction or number of the subject involved in the assertion. T. Spencer Baynes, Essay on the new Analytic of Prof. Bowen's Treatise on Logic or the Laws of Pure LESSON XXIII. BOOLE'S SYSTEM OF LOGIC. IT would not in the least be possible to give in an elementary work a notion of the system of indirect inference first discovered by the late Dr Boole, the Professor of Mathematics at the Queen's College, Cork. This system was founded as mentioned in the last lesson upon the Quantification of the Predicate, but Dr Boole regarded Logic as a branch of Mathematics, and believed that he could arrive at every possible inference by the principles of algebra. The process as actually employed by him is very obscure and difficult; and hardly any attempt to introduce it into elementary text-books of Logic has yet been made. I have been able to arrive at exactly the same results as Dr Boole without the use of any mathematics; and though the very simple process which I am going to describe can hardly be said to be strictly Dr Boole's logic, it is yet very similar to it and can prove everything that Dr Boole proved. This Method of Indirect Inference is founded upon the three primary Laws of Thought stated in Lesson XIV., and the reader who may have thought them mere useless truisms will perhaps be surprised to find how extensive and elegant a system of deduction may be derived from them. The law of excluded middle enables us to assert that anything must either have a given quality or must have it not. Thus if iron be the thing, and combustibility the quality, anyone must see that "Iron is either combustible or incombustible." This division of alternatives may be repeated as often as we like. Thus let Book be the class of things to be divided, and English and Scientific two qualities. Then any book must be either English or not English; again an English book must be either Scientific or not Scientific, and the same may be said of books which are not English. Thus we can at once divide books into four classes Books, English and Scientific. Books, English and not-Scientific. This is what we may call an exhaustive division of the class Books; for there is no possible book which does not fall into one division or other of these four, on account of the simple reason, that if it does not fall into any of the three first it must fall into the last. The process can be repeated without end, as long as any new circumstance can be suggested as the ground of division. Thus we might divide each class again according as the books are octavo or not octavo, bound or unbound, published in London or elsewhere, and so on. We shall call this process of twofold division, which is really the process of Dichotomy mentioned in p. 107, the development of a term, because it enables us always to develope the utmost number of alternatives which need be considered. As a general rule it is not likely that all the alternatives thus unfolded or developed can exist, and the next point is to ascertain how many do or may exist. The Law of Contradiction asserts that nothing can combine contradictory attributes or qualities, and if we meet with any term which is thus self-contradictory we are authorized at once to strike it out of the list. Now consider our old example of a syllogism : Iron is a metal; All metals are elements; Therefore iron is an element. We can readily prove this conclusion by the indirect method. For if we develope the term iron, we have four alternatives; thus Iron, metal, element. Iron, metal, not-element. Iron, not-metal, not-element. But if we compare each of these alternatives with the premises of the syllogism, it will be apparent that several of them are incapable of existing. Iron, we are informed, is a metal. Hence no class of things "iron, not-metal" can exist. Thus we are enabled by the first premise to strike out both of the last two alternatives which combine iron and not-metal. The second alternative, again, combines metal and not-element; but as the second premise informs us that "all metals are elements," it must be struck out. There remains, then, only one alternative |