It should be understood that this solution applies only to the terms of the example quoted above, and not to the general problem for which De Morgan intended it to serve as an illustration. As a second instance, let us take the following question :-The whole number of voters in a borough is a; the number against whom objections have been lodged by liberals is b; and the number against whom objections have been lodged by conservatives is c; required the number, if any, who have been objected to on both sides. Taking A = voter, B = objected to by liberals, C objected to by conservatives, = then we require the value of (ABC). Now the following equation is identically true (1) (ABC) = (AB) + (AC) + (Abc) — (A). For if we develop all the terms on the second side we obtain (ABC) + (ABC) + (ABC) + (AbC) + (Abc) (ABC) – (ABC) — (AbC) — (Abc); and striking out the corresponding positive and negative terms, we have left only (ABC) = (ABC). Since then (1) is necessarily true, we have only to insert the known values, and we have (ABC)=b+ca + (Abc). Hence the number who have received objections from both sides is equal to the excess, if any, of the whole number of objections over the number of voters together with the number of voters who have received no objection (Abc). The following problem illustrates the expression for the common part of any three classes:-The number of paupers who are blind males, is equal to the excess, if any, of the sum of the whole number of blind persons, added to the whole number of male persons, added to the number of those who being paupers are neither blind nor males, above the sum of the whole number of paupers added to the number of those who, not being paupers, are blind, and to the number of those who, not being paupers, are male. The reader is requested to prove the truth of the above statement, (1) by his own unaided common sense; (2) by the Aristotelian Logic; (3) by the method of numerical logic just expounded; and then to decide which method. is most satisfactory. Numerical meaning of Logical Conditions. In many cases classes of objects may exist under special logical conditions, and we must consider how these conditions can be interpreted numerically. Every logical proposition gives rise to a corresponding numerical equation. Sameness of qualities occasions sameness of numbers. Hence if A = B denotes the identity of the qualities of A and B, we may conclude that (A) = (B). It is evident that exactly those objects, and those objects only, which are comprehended under A must be comprehended under B. It follows that wherever we can draw an equation of qualities, we can draw a similar equation of numbers. Thus, from meaning that the numbers of A's and C's are equal to the number of B's, we can infer (A) = (C). But, curiously enough, this does not apply to negative propositions and inequalities. For if means that A is identical with B, which differs from D, it does not follow that (A) = (B) ~ (D). Two classes of objects may differ in qualities, and yet they may agree in number. This point strongly confirms me in the opinion which I have already expressed, that all inference really depends upon equations, not differences. The Logical Alphabet thus enables us to make a complete analysis of any numerical problem, and though the symbolical statement may sometimes seem prolix, I con ceive that it really represents the course which the mind must follow in solving the question. Although thought may outstrip the rapidity with which the symbols can be written down, yet the mind does not really follow a different course from that indicated by the symbols. For a fuller explanation of this natural system of Numerically Definite Reasoning, with more abundant illustrations and an analysis of De Morgan's Numerically Definite Syllogism, I must refer the reader to the paper1 in the Memoirs of the Manchester Literary and Philosophical Society, already mentioned, portions of which, however, have been embodied in the present section. The reader may be referred, also, to Boole's writings upon the subject in the Laws of Thought, chap. xix. p. 295, and in a paper on "Propositions Numerically Definite," communicated by De Morgan, in 1868, to the Cambridge Philosophical Society, and printed in their Transactions," vol. xi. part ii. 1 It has been pointed out to me by Mr. C. J. Monroe, that section 14 (p. 339) of this paper is erroneous, and ought to be cancelled. The problem concerning the number of paupers illustrates the answer which should have been obtained. Mr. A. J. Ellis, F.R.S., had previously observed that my solution in the paper of De Morgan's problem about men in the house" did not answer the conditions intended by De Morgan, and I therefore give in the text a more satisfactory solution. ། ་ CHAPTER IX. THE VARIETY OF NATURE, OR THE DOCTRINE OF NATURE may be said to be evolved from the monotony of non-existence by the creation of diversity. It is plausibly asserted that we are conscious only so far as we experience difference. Life is change, and perfectly uniform existence would be no better than non-existence. Certain it is that life demands incessant novelty, and that nature, though it probably never fails to obey the same fixed laws, yet presents to us an apparently unlimited series of varied combinations of events. It is the work of science to observe and record the kinds and comparative numbers of such combinations of phenomena, occurring spontaneously or produced by our interference. Patient and skilful examination of the records may then disclose the laws imposed on matter at its creation, and enable us more or less successfully to predict, or even to regulate, the future occurrence of any particular combination. The Laws of Thought are the first and most important of all the laws which govern the combinations of phenomena, and, though they be binding on the mind, they may also be regarded as verified in the external world. The Logical Alphabet develops the utmost variety of things and events which may occur, and it is evident that as each new quality is introduced, the number of combinations is doubled. Thus four qualities may occur in 16 combinations; five qualities in 32; six qualities in 64; and so on. In general language, if n be the number of qualities, 2" is the number of varieties of things which may be formed from them, if there be no conditions but those of logic. This number, it need hardly be said, increases after the first few terms, in an extraordinary manner, so that it would require 302 figures to express the number of combinations in which 1,000 qualities might conceivably present themselves. If all the combinations allowed by the Laws of Thought occurred indifferently in nature, then science would begin and end with those laws. To observe nature would give us no additional knowledge, because no two qualities would in the long run be oftener associated than any other two. We could never predict events with more certainty than we now predict the throws of dice, and experience would be without use. But the universe, as actually created, presents a far different and much more interesting problem. The most superficial observation shows that some things are constantly associated with other things. The more mature our examination, the more we become convinced that each event depends upon the prior occurrence of some other series of events. Action and reaction are gradually discovered to underlie the whole scene, and an independent or casual occurrence does not exist except in appearance. Even dice as they fall are surely determined in their course by prior conditions and fixed laws. Thus the combinations of events which can really occur are found to be comparatively restricted, and it is the work of science to detect these restricting conditions. In the English alphabet, for instance, we have twentysix letters. Were the combinations of such letters perfectly free, so that any letter could be indifferently sounded with any other, the number of words which could be formed without any repetition would be 226 — 1, or 67,108,863, equal in number to the combinations of the twenty-seventh column of the Logical Alphabet, excluding one for the case in which all the letters would be absent. But the formation of our vocal organs prevents us from using the far greater part of these conjunctions of letters. At least one vowel must be present in each word; more than two consonants cannot usually be brought together; and to produce words capable of smooth utterance a number of other rules must be |