required class or term as regards the terms involved in the premises. 2. For each term in these alternatives substitute its description as given in the premises. 3. Strike out every alternative which is then found to break the Law of Contradiction. 4. The remaining terms may be equated to the term in question as the desired description. Mr. Venn's Problem. The need of some logical method more powerful and comprehensive than the old logic of Aristotle is strikingly illustrated by Mr. Venn in his most interesting and able article on Boole's logic. An easy example, originally got, as he says, by the aid of my method as simply described in the Elementary Lessons in Logic, was proposed in examination and lecture-rooms to some hundred and fifty students as a problem in ordinary logic. It was answered by, at most, five or six of them. It was afterwards set, as an example on Boole's method, to a small class who had attended a few lectures on the nature of these symbolic methods. It was readily answered by half or more of their number. The problem was as follows:- "The members of a board were all of them either bondholders, or shareholders, but not both; and the bondholders as it happened, were all on the board. What conclusion can be drawn?" The conclusion wanted is, "No shareholders are bondholders." Now, as Mr. Venn says, nothing can look simpler than the following reasoning, when stated: "There can be no bondholders who are shareholders; for if there were they must be either on the board, or off it. But they are not on it, by the first of the given statements; nor off it, by the second." Yet from the want of any systematic mode of treating such a question only five or six of some hundred and fifty students could succeed in so simple a problem. 1 Mind; a Quarterly Review of Psychology and Philosophy; October, 1876, vol. i. p. 487. By symbolic statement the problem is instantly solved. Taking The class C or shareholders may in respect of A and B be developed into four alternatives, C ABC AbCaBC abC. = But substituting for A in the first and for B in the third alternative we get C = ABC | ABC | ABC + aABC + abC. The first, second, and fourth alternatives in the above are self-contradictory combinations, and only these; striking them out there remain C = AbCabC = bc, the required answer. This symbolic reasoning is, I believe, the exact equivalent of Mr. Venn's reasoning, and I do not believe that the result can be attained in a simpler manner. Mr. Venn adds that he could adduce other similar instances, that is, instances showing the necessity of a better logical method. Abbreviation of the Process. Before proceeding to further illustrations of the use of this method, I must point out how much its practical employment can be simplified, and how much more easy it is than would appear from the description. When we want to effect at all a thorough solution of a logical problem it is best to form, in the first place, a complete series of all the combinations of terms involved in it. If there be two terms A and B, the utmost variety of combinations in which they can appear are AB aB ab. The term A appears in the first and second; B in the first and third; a in the third and fourth; and b in the second and fourth. Now if we have any premise, say we must ascertain which of these combinations will be rendered self-contradictory by substitution; the second and third will have to be struck out, and there will remain only AB Hence we draw the following inferences ab, b = ab. A = AB, B = AB, a = Exactly the same method must be followed when a question involves a greater number of terms. Thus by the Law of Duality the three terms A, B, C, give rise to eight conceivable combinations, namely a BC (€) The development of the term A is formed by the first four of these; for B we must select (a), (3), (e), (); C consists of (a), (y), (e), (n) ; b of (7), (8), (7), (l'), and so on. Now if we want to investigate completely the meaning of the premises A = AB (I) (2) we examine each of the eight combinations as regards each premise; (y) and (8) are contradicted by (1), and (B) and (5) by (2), so that there remain only (0) To describe any term under the conditions of the premises (1) and (2), we have simply to draw out the proper combinations from this list; thus, A is represented only by ABC, that is to say similarly For B we have two A = ABC, c = abc. alternatives thus stated, and for b we have b=abc abc. When we have a problem involving four distinct terms. we need to double the number of combinations, and as we add each new term the combinations become twice as numerous. Thus А, В A, B, C, D A, B, C, D, E A, B, C, D, E, F and so on. I propose to call any such series of combinations the Logical Alphabet. It holds in logical science a position the importance of which cannot be exaggerated, and as we proceed from logical to mathematical considerations, it will become apparent that there is a close connection between these combinations and the fundamental theorems of mathematical science. For the convenience of the reader who may wish to employ the Alphabet in logical questions, I have had printed on the next page a complete series of the combinations up to those of six terms. At the very commencement, in the first column, is placed a single letter X, which might seem to be superfluous. This letter serves to denote that it is always some higher class which is divided up. Thus the combination AB really means ABX, or that part of some larger class, say X, which has the qualities of A and B present. The letter X is omitted in the greater part of the table merely for the sake of brevity and clearness. In a later chapter on Combinations it will become apparent that the introduction of this unit class is requisite in order to complete the analogy with the Arithmetical Triangle there described. The reader ought to bear in mind that though the Logical Alphabet seems to give mere lists of combinations, these combinations are intended in every case to constitute the development of a term of a proposition. Thus the four combinations AB, Ab, aB, ab really mean that any class X is described by the following proposition, X = XAB. XAb · XaB ·|· Xab. If we select the A's, we obtain the following proposition AX XAB XAb. = Thus whatever group of combinations we treat must be conceived as part of a higher class, summum genus or universe symbolised in the term X; but, bearing this in mind, it is needless to complicate our formulæ by always introducing the letter. All inference consists in passing from propositions to propositions, and combinations per se have no meaning. They are consequently to be regarded in all cases as forming parts of propositions. VII. ABCDEF ABCDE ABCDe F ABC Def A B C a E F ABCdEƒ ABC de F ABC def A Be D E F ABC DE ABC De F ABC Def A Bed EF A Bed Ef A Bcde F A B c d e f ABCDEF AbCDEƒ AbC DeF AbC Def Ab Cả EF Ab CdEƒ Ab Cde F Ab Cde f AbcD E F Abc DEƒ Abc De F Abc Def Abcd EP A b c d E Abcde F A b c d e f a BCDEF a B C D E f a BC De F a BC Def a BCdEF a BCdEf a BC de F a BC de a BC DE F A B C D Eƒ aBc De F a Bc Def a Bed EF a Bcd Eƒ aBc de F a B c d e f ab CDE F ab C D E ƒ ab C De F ab C Def ab CdE F ab Cd Ef ab Cde F a b c d e f a b c DE F a b c D E f abc De F a b c Def abcd EF abcd Eƒ abcde F a b c d e f |