variations of which the law is capable. There might be also added, as a sixteenth case, that case where no special logical condition exists, so that all the eight combinations remain.

There are sixty-three series of combinations derived from self-contradictory premises, which with 192, the sum of the numbers of distinct logical variations stated in the third column of the table, and with the one case where there are no conditions or laws at all, make up the whole conceivable number of 256 series.

We learn from this table, for instance, that two propositions of the form A = AB, B = BC, which are such as constitute the premises of the old syllogism Barbara, exclude as impossible four of the eight combinations in which three terms may be united, and that these propositions are capable of taking twenty-four variations by transpositions of the terms or the introduction of negatives. This table then presents the results of a complete analysis of all the possible logical relations arising in the case of three terms, and the old syllogism forms hut one out of fifteen typical forms. Generally speaking, every form can be converted into apparently different propositions; thus the fourth type A = B, B = BC may appear in the form A = ABC, a = ab, or again in the form of three propositions A AB, B = BC, aB = aBc; but all these sets of premises yield identically the same series of combinations,

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and are therefore of equivalent logical meaning. The fifth type, or Barbara, can also be thrown into the equivalent forms A = ABC, aB = aBC and A = AC, B A | aBC. In other cases I have obtained the very same logical conditions in four modes of statements. As regards mere appearance and form of statement, the number of possible premises would be very great, and difficult to exhibit exhaustively.

The most remarkable of all the types of logical condition is the fourteenth, namely, A = BC bc. It is that which expresses the division of a genus into two doubly marked species, and might be illustrated by the example—" Component of the physical universe = matter, gravitating, or not-matter (ether), not-gravitating." It is capable of only two distinct logical variations, namely, ABC be and A BebC. By transposition or negative change of the letters we can indeed obtain six different expressions of each of these propositions; but when their meanings are analysed, by working out the combinations, they are found to be logically equivalent to one or other of the above two. Thus the proposition ABC be can be written in any of the following five other modes,

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I do not think it needful to publish at present the complete table of 193 series of combinations and the premises corresponding to each. Such a table enables us by mere inspection to learn the laws obeyed by any set of combinations of three things, and is to logic what a table of factors and prime numbers is to the theory of numbers, or a table of integrals to the higher mathematics. The table already given (p. 140) would enable a person with but little labour to discover the law of any combinations. If there be seven combinations (one contradicted) the law must be of the eighth type, and the proper variety will be apparent. If there be six combinations (two contradicted), either the second, eleventh, or twelfth type applies, and a certain number of trials will disclose the proper type and variety. If there be but two combinations the law must be of the third type, and so on.

The above investigations are complete as regards the possible logical relations of two or three terms. But

when we attempt to apply the same kind of method to the relations of four or more terms, the labour becomes impracticably great. Four terms give sixteen combinations compatible with the laws of thought, and the number of possible selections of combinations is no less than 216 or 65,536. The following table shows the extraordinary manner in which the number of possible logical relations increases with the number of terms involved.

Some years of continuous labour would be required to ascertain the types of laws which may govern the combinations of only four things, and but a small part of such laws would be exemplified or capable of practical application in science. The purely logical inverse problem, whereby we pass from combinations to their laws, is solved in the preceding pages, as far as it is likely to be for a long time to come; and it is almost impossible that it should ever be carried more than a single step further.

In the first edition, vol i. p. 158, I stated that I had not been able to discover any mode of calculating the number of cases in which inconsistency would be implied in the selection of combinations from the Logical Alphabet. The logical complexity of the problem appeared to be so great that the ordinary modes of calculating numbers of combinations failed, in my opinion, to give any aid, and exhaustive examination of the combinations in detail seemed to be the only method applicable. This opinion, however, was mistaken, for both Mr. R. B. Hayward, of Harrow, and Mr. W. H. Brewer have calculated the numbers of inconsistent cases both for three and for four terms, without much difficulty. In the case of four terms they find that there are 1761 inconsistent selections and 63,774 consistent, which with one case where no

condition exists, make up the total of 65,536 possible selections.

The inconsistent cases are distributed in the manner shown in the following table:

When more than eight combinations of the Logical Alphabet (p. 94, column V.) remain unexcluded, there cannot be inconsistency. The whole numbers of ways of selecting 0, 1, 2, &c., combinations out of 16 are given in the 17th line of the Arithmetical Triangle given further on in the Chapter on Combinations and Permutations, the sum of the numbers in that line being 65,536.

Professor Clifford on the Types of Compound Statement involving Four Classes.

In the first edition (vol. i. p. 163), I asserted that some years of labour would be required to ascertain even the precise number of types of law governing the combinations of four classes of things. Though I still believe that some years' labour would be required to work out the types themselves, it is clearly a mistake to suppose that the numbers of such types cannot be calculated with a reasonable amount of labour, Professor W. K. Clifford having actually accomplished the task. His solution of the numerical problem involves the use of a complete new system of nomenclature and is far too intricate to be fully described here. I can only give a brief abstract of the results, and refer readers, who wish to follow out the reasoning, to the Proceedings of the Literary and Philosophical Society of Manchester, for the 9th January, 1877, vol. xvi., p. 88, where Professor Clifford's paper is printed in full.

By a simple statement Professor Clifford means the denial of the existence of any single combination or cross

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division, of the classes, as in ABCD O, or AbCd: = 0. The denial of two or more such combinations is called a compound statement, and is further said to be twofold, threefold, &c., according to the number denied. Thus ABC o is a twofold compound statement in regard to four classes, because it involves both ABCD

= 0 and ABCd = O. When two compound statements can be converted into one another by interchange of the classes, A, B, C, D, with each other or with their complementary classes, a, b, c, d, they are called similar, and all similar statements are said to belong to the same type.

Two statements are called complementary when they deny between them all the sixteen combinations without both denying any one; or, which is the same thing, when each denies just those combinations which the other permits to exist. It is obvious that when two statements are similar, the complementary statements will also be similar, and consequently for every type of n-fold statement, there is a complementary type of (16—n)-fold statement. It follows that we need only enumerate the types as far as the eighth order; for the types of more-than-eight-fold statement will already have been given as complementary to types of lower orders.

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One combination, ABCD, may be converted into another AbCd by interchanging one or more of the classes with the complementary classes. The number of such changes is called the distance, which in the above case is 2. two similar compound statements the distances of the combinations denied must be the same; but it does not follow that when all the distances are the same, the statements are similar. There is, however, only one example of two dissimilar statements having the same distances. When the distance is 4, the two combinations are said to be obverse to one another, and the statements denying them are called obverse statements, as in ABCD = 0 and abcd = 0 or again Abcd o and a BcD = 0. When any one combination is given, called the origin, all the others may be grouped in respect of their relations to it as follows:-Four are at distance one from it, and may be called proximates; six are at distance two, and may be called mediates; four are at distance three, and may be called ultimates; finally the obverse is at distance four.

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