The third answer, given by Mr. R. B. Hayward, M.A., is in the simplest terms. The propositions inserted as the fifth answer are those from which I formed the combinations deductively. The student may prove that any one of these answers is deducible from any other without descending explicitly to the combinations; thus C AC is the contrapositive of a = ac; B C D is equivalent to b cd, and so forth. = CHAPTER XXVI ELEMENTS OF NUMERICAL LOGIC 1. LET a logical term, when enclosed in brackets, acquire a quantitative meaning, so as to denote the number of individual objects which possess the qualities connoted by the logical term. Then (A) = number of objects possessing qualities of A, or say, for the sake of brevity, the number of As. Every logical equation now 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 A=B = C, we infer A = C; and similarly from (A) = (B) = (C), meaning the number of As and Cs are equal to the number of Bs, we can infer (A)=(C). But, curiously enough, this does not apply to negative propositions and inequalities. For if A = B~D 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. 2. The sign being used to stand for the disjunctive conjunction or, but in an unexclusive sense, it follows that I is not identical in meaning with +. It does not follow from the statement that A is either B or C, that the number o As is equal to the number of Bs added to the number of Cs; some objects, or possibly all, may have been counted twice in this addition. Thus, if we say An elector is either an elector for a borough, or for a county, or for a university, it does not follow that the total number of electors is equal to the number of borough, county, and university electors added together; for some men will be found in two or three of the classes. This difficulty, however, is avoided with great ease; for we need only develop each alternative into all its possible subclasses and strike out any subclass which appears more than once, and then convert into numbers, connected by the sign of addition. Thus, from ABC we get A = BC | Bc. BC .|. ¿C; but striking out one of the terms BC as being superfluous, we have ABC Bcl- bC. The alternatives are now strictly exclusive, or devoid of any common part, so that we may draw the numerical equation Thus, if A = elector, (A) = (BC) + (Bc) + (bC). B = borough elector, C = county elector, D= university elector, we may from the proposition A B C D draw the numerical equation (A) = (BCD) + (BCď) + (BcD) + (Bcd) + (bCD) + (bCd) + (bcD). 3. The data of any problem in Numerical Logic will be of two kinds : (1) The logical conditions governing the combinations of certain qualities or classes of things, expressed in propositions. (2) The numbers of individuals in certain logical classes existing under those conditions. The quaesita of the problem will consist in determining the numbers of individuals in certain other logical classes existing under the same logical conditions, so far as such numbers are rendered determinable by the data. The usefulness of the method will, indeed, often consist in showing whether or not the magnitude of a class is determined or not, or in indicating what further hypotheses or data are required. It will appear, too, that where an exact result is not determinable we may yet assign limits within which an unknown quantity must lie. 4. In a certain statistical investigation, among 100 As there are found 45 Bs and 53 Cs; that is to say, in 45 out of 100 cases where A occurs B also occurs, and in 53 cases C occurs. Suppose it to be also known that wherever B is, C also necessarily exists. It is required to determine (1) The number of cases (all being As) where C exists without B. (2) The number of cases (all being As) where neither B nor C exists. The logical equation asserts that the class B is identical with the class BC, which is the true mode of asserting that all Bs are Cs. Two distinct results follow from this, namely: 1st, that the number of the class BC is identical with the number of the class B; and 2nd, that there are no such things as Bs which are not Cs. The logical equation is thus equivalent to two additional numerical equations, namely, We have now means of solving the problem; for, by the To obtain the number of Abcs, we have Hence (Abc)= 47. |