of possible laws then cannot exceed the number of selections which we can make from these four combinations. Since each combination may be present or absent, the number of cases to be considered is 2 x 2 x 2 x 2, or sixteen; and these cases are all shown in the following table, in which the sign o indicates absence or non-existence of the combination shown at the left-hand column in the same line, and the mark 1 its presence : AB 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Thus in column sixteen we find that all the conceivable combinations are present, which means that there are no special laws in existence in such a case, and that the combinations are governed only by the universal Laws of Identity and Difference. The example of metals and conductors of electricity would be represented by the twelfth column; and every other mode in which two things or qualities might present themselves is shown in one or other of the columns. More than half the cases may indeed be at once rejected, because they involve the entire absence of a term or its negative. It has been shown to be a logical principle that every term must have its negative (p. 111), and when this is not the case, inconsistency between the conditions of combination must exist. Thus if we laid down the two following propositions, "Graphite conducts electricity," and "Graphite does not conduct electricity," it would amount to asserting the impossibility of graphite existing at all; or in general terms, A is B and A is not B result in destroying altogether the combinations containing A, a case shown in the fourth column of the above table. We therefore restrict our attention to those cases which may be represented in natural phenomena when at least two combinations are present, and which correspond to those columns of the table in which each of A, a, B, b appears. These cases Four cases exhibiting three combinations, = It has already been pointed out that a proposition of the form A AB destroys one combination, Ab, so that this is the form of law applying to the twelfth column. But by changing one or more of the terms in A = AB into its negative, or by interchanging A and B, a and b, we obtain no less than eight different varieties of the one form; thus— = B = The reader of the preceding sections will see that each proposition in the lower line is logically equivalent to, and is in fact the contrapositive of, that above it (p. 83). Thus the propositions A = Ab and B aB both give the same combinations, shown in the eighth column of the table, and trial shows that the twelfth, eighth, fifteenth and fourteenth columns are thus accounted for. We come to this conclusion then-The general form of proposition A AB admits of four logically distinct varieties, each capable of expression in two modes. = In two columns of the tenth, we observe that Now a simple identity A = = = table, namely the seventh and two combinations are missing. B renders impossible both Ab and aB, accounting for the tenth case; and if we change B into b the identity A b accounts for the seventh case. There may indeed be two other varieties of the simple identity, namely a b and a = B; but it has already been shown repeatedly that these are equivalent respectively to AB and Ab (p. 115). As the sixteenth column has already been accounted for as governed by no special conditions, we come to the following general conclusion:-The laws governing the combinations of two terms must be capable of expression either in a partial identity or a simple identity; the partial identity is capable of only four logically distinct varieties, and the simple identity of two. Every logical relation between two terms must be expressed in one of these six forms of law, or must be logically equivalent to one of them. In short, we may conclude that in treating of partial and complete identity, we have exhaustively treated the modes in which two terms or classes of objects can be related. Of any two classes it can be said that one must either be included in the other, or must be identical with it, or a like relation must exist between one class and the negative of the other. We have thus completely solved the inverse logical problem concerning two terms.1 The Inverse Logical Problem involving Three Classes. No sooner do we introduce into the problem a third term C, than the investigation assumes a far more complex character, so that some readers may prefer to pass over this section. Three terms and their negatives may be combined, as we have frequently seen, in eight different combinations, and the effect of laws or logical conditions is to destroy any one or more of these combinations. Now we may make selections from eight things in 28 or 256 ways; so that we have no less than 256 different cases to treat, and the complete solution is at least fifty times as troublesome as with two terms. Many series of combinations, indeed, are contradictory, as in the simpler problem, and may be passed over, the test of consistency being that each of the letters A, B, C, a, b, c, shall appear somewhere in the series of combinations. My mode of solving the problem was as follows:Having written out the whole of the 256 series of combinations, I examined them separately and struck out such as did not fulfil the test of consistency. I then chose some form of proposition involving two or three terms, and varied it in every possible manner, both by the circular interchange of letters (A, B, C into B, C, A and then into C, A, B), and by the substitution for any one or more of the terms of the corresponding negative terms. The contents of this and the following section nearly correspond with those of a paper read before the Manchester Literary and Philosophical Society on December 26th, 1871. See Proceedings of the Society, vol. xi. pp. 65-68, and Memoirs, Third Series, vol. v. PP. 119-130. so on. == = = ABc, ab = = For instance, the proposition AB = ABC can be first varied by circular interchange so as to give BC = BCA and then CA = CAB. Each of these three can then be thrown into eight varieties by negative change. Thus AB = ABC gives aB = aBC, Ab = АБС, АВ abC, and Thus there may possibly exist no less than twentyfour varieties of the law having the general form AB = ABC, meaning that whatever has the properties of A and B has those also of C. It by no means follows that some of the varieties may not be equivalent to others; and trial shows, in fact, that AB ABC is exactly the same in meaning as Ac Abc or Bc Bca. Thus the law in question has but eight varieties of distinct logical meaning. I now ascertain by actual deductive reasoning which of the 256 series of combinations result from each of these distinct laws, and mark them off as soon as found. I then proceed to some other form of law, for instance A ABC, meaning that whatever has the qualities of A has those also of B and C. I find that it admits of twenty-four variations, all of which are found to be logically distinct; the combinations being worked out, I am able to mark off twenty-four more of the list of 256 series. I proceed in this way to work out the results of every form of law which I can find or invent. If in the course of this work I obtain any series of combinations which had been previously marked off, I learn at once that the law giving these combinations is logically equivalent to some law previously treated. It may be safely inferred that every variety of the apparently new law will coincide in meaning with some variety of the former expression of the same law. I have sufficiently verified this assumption in some cases, and have never found it lead to error. Thus as AB ABC is equivalent to Ac ab = = = find that Abc, so we abC is equivalent to ac = acB. Among the laws treated were the two A AB and A B which involve only two terms, because it may of course happen that among three things two only are in special logical relation, and the third independent; and the series of combinations representing such cases of relation are sure to occur in the complete enumeration. All single propositions which I could invent having been treated, pairs of propositions were next investigated. Thus we have the relations, "All A's are B's and all B's are C's," of which the old logical syllogism is the development. We may also have "all A's are all B's, and all B's are C's," or even "all A's are all B's, and all B's are all C's." All such premises admit of variations, greater or less in number, the logical distinctness of which can only be determined by trial in detail. Disjunctive propositions either singly or in pairs were also treated, but were often found to be equivalent to other propositions of a simpler form; thus AABC Abc is exactly the same in meaning as AB = AC. This mode of exhaustive trial bears some analogy to that ancient mathematical process called the Sieve of Eratosthenes. Having taken a long series of the natural numbers, Eratosthenes is said to have calculated out in succession all the multiples of every number, and to have marked them off, so that at last the prime numbers alone remained, and the factors of every number were exhaustively discovered. My problem of 256 series of combinations is the logical analogue, the chief points of difference being that there is a limit to the number of cases, and that prime numbers have no analogue in logic, since every series of combinations corresponds to a law or group of conditions. But the analogy is perfect in the point that they are both inverse processes. There is no mode of ascertaining that a number is prime but by showing that it is not the product of any assignable factors. So there is no mode of ascertaining what laws are embodied in any series of combinations but trying exhaustively the laws which would give them. Just as the results of Eratosthenes' method have been worked out to a great extent and registered in tables for the convenience of other mathematicians, I have endeavoured to work out the inverse logical problem to the utmost extent which is at present practicable or useful. I have thus found that there are altogether fifteen conditions or series of conditions which may govern the combinations of three terms, forming the premises of fifteen essentially different kinds of arguments. The following table contains a statement of these conditions, together with the numbers of combinations which are contradicted or destroyed by each, and the numbers of logically distinct |