It was first discovered by De Morgan that many arguments are valid which combine logical and numerical reasoning, although they cannot be included in the ancient logical formulas. He developed the doctrine of the “Numerically Definite Syllogism,” fully explained in his *Formal Logic* (pp. 141–170). Boole also devoted considerable attention to the determination of what he called “Statistical Conditions,” meaning the numerical conditions of logical classes. In a paper published among the Memoirs of the Manchester Literary and Philosophical Society, Third Series, vol. IV. p. 330 (Session 1869–70), I have pointed out that we can apply arithmetical calculation to the Logical Alphabet. Having given certain logical conditions and the numbers of objects in certain classes, we can either determine the numbers of objects in other classes governed by those conditions, or can show what further data are required to determine them. As an example of the kind of questions treated in numerical logic, and the mode of treatment, I give the following problem suggested by De Morgan, with my mode of representing its solution. “For every man in the house there is a person who is aged; some of the men are not aged. It follows that some persons in the house are not men.”[92] [92] *Syllabus of a Proposed System of Logic*, p. 29. Now let A = person in house, B = male, C = aged. By enclosing a logical symbol in brackets, let us denote the number of objects belonging to the class indicated by the symbol. Thus let (A) = number of persons in house, (AB) = number of male persons in house, (ABC) = number of aged male persons in house, and so on. Now if we use *w* and *w*′ to denote unknown numbers, the conditions of the problem may be thus stated according to my interpretation of the words-- (AB) = (AC) - *w*, (1) that is to say, the number of persons in the house who are aged is at least equal to, and may exceed, the number of male persons in the house; (AB*c*) = *w*′, (2) that is to say, the number of male persons in the house who are not aged is some unknown positive quantity. If we develop the terms in (1) by the Law of Duality (pp. 74, 81, 89), we obtain (ABC) + (AB*c*) = (ABC) + (A*b*C) - *w*. Subtracting the common term (ABC) from each side and substituting for (AB*c*) its value as given in (2), we get at once (A*b*C) = *w* + *w*′, and adding (A*bc*) to each side, we have (A*b*) = (A*bc*) + *w* + *w*′. The meaning of this result is that the number of persons in the house who are not men is at least equal to *w* + *w*′, and exceeds it by the number of persons in the house who are neither men nor aged (A*bc*). It should be understood that this solution applies only to the terms of the example quoted above, and not to the general problem for which De Morgan intended it to serve as an illustration.