CHAPTER VII. INDUCTION.

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 but 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, a B =  a B c ; but all these sets of premises yield identically the same series of combinations, and are therefore of equivalent logical meaning. The fifth type, or Barbara, can also be thrown into the equivalent forms A = ABC, a B =  a BC and A = AC, B = A ꖌ  a BC. 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 ꖌ 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, A = BC ꖌ bc and A = Bc ꖌ bC. 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 A = BC ꖌ bc can be written in any of the following five other modes,

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.