Boole treats this problem in the fourth page of his Memoir On Propositions Numerically Definite (Cambridge Philosophical Transactions, vol. xi. part ii.). Taking 1 to represent the company which is the universe of the proposition, the class possessing coats, y the class possessing waistcoats, and using the letter N, according to Boole's notation, as equivalent to the words 'number of,'

p=Nx, q=Ny, r=N1,

he finds, as we have found in a preceding page (p. 264, No. 7),

Nxy=p+q-r+ N1 − x 1-y.

N1 − x 1-y=r-p− q + Nxy.

He proceeds, 'Again, let us form the equation

=

x 1 - y)

Nx 1-y-2Ny 1-x-N 1 − x 1 -y.

From which we have

Nx 1-y=2p - q − r + 2Ny 1 − x + N 1−x I –y.

Hence we might deduce that the number who had coats but not waistcoats would exceed the number 2p -q-r by twice the number who had waistcoats without coats together with the number who had neither coats nor waistcoats. This is not, indeed, the simplest result with reference to the class in question, but it is a correct one.'

The student is requested to verify this result.

On going over this paper of Boole's again, it becomes apparent to my mind that his method is identical with that developed in this chapter and in my previous paper

T

on the same subject (Memoirs of the Manchester Literary and Philosophical Society, Third Series, vol. iv. p. 330, Session 1869-70), written with a knowledge, as stated on p. 331, of Boole's publication on the subject.

16. Can we represent a syllogism in the extensive form by means of numerical symbols?

In a very interesting and remarkable paper read to the Belfast Philosophical Society in 1875, Mr. Joseph John Murphy has given a kind of numerical notation for the syllogism. He has since printed a more condensed and matured account of his views in Mind, January, 1877.

Taking the syllogism-Chlorine is one of the class of imperfect gases; imperfect gases are part of the class of substances freely soluble in water; therefore, chlorine is one of the class of substances freely soluble in water'—he assumes the symbols

x= Chlorine,

y= Imperfect gases,

%= substances freely soluble in water.

He expresses the first premise in the form

y = x+p,

p being a positive numerical quantity indicating that there are other things besides chlorine in the class of imperfect gases. The second premise takes the form

z = y + 9,

similarly indicating that besides imperfect gases there are q things in the class of substances freely soluble in water. Substitution gives z = x + p + q,

which would seem to

prove that besides chlorine (x) there

are pq things in the class of substances freely soluble in

water.

The student who wishes to master the difficulties of the modern logical views should read these papers with great care. Space does not admit of my arguing the matter out at full length, and I can therefore only briefly express my objections to Mr. Murphy's views as follows:-His equations are equations in extension, and, with his use of + and they can only hold true when his terms are numerical quantities. Under this assumption his equations show with perfect correctness the numbers of certain classes; but they are not therefore equivalent to syllogisms. Because z=x+p+q, we learn that the number z exceeds x by p + q, but it does not therefore follow that chlorine belongs to the class of substances represented by z. In short, as I have pointed out at the beginning of this chapter (p. 259), from logical equations arithmetical ones follow, but not vice versâ. (See also Principles of Science, p. 171; first edit. vol. i. p. 193.) I hold, therefore, that Mr. Murphy's forms are not really representations of syllogisms; but at the same time I am quite willing to admit that this is a question never yet settled and demanding further investigation. It is very remarkable that Hallam inserted in his History of Literature (ed. 1839, vol. iii. pp. 287-8) a long note containing a theory of the syllogism somewhat similar to that of Mr. Murphy, but which has hitherto remained unknown to Mr. Murphy and apparently to all other logical writers.

CHAPTER XXVII

PROBLEMS IN NUMERICAL LOGIC

1. IF from the number of members of Parliament we subtract the number of them who are not military men, we get the same result as if from the whole number of military men we subtract the number of them who are not members of Parliament. Prove this.

2. In a company of x individuals it is discovered that y are Cambridge men, and z are lawyers. Find an expression for the number of Cambridge men in the company who are lawyers, and assign its greatest and least possible values.

[BOOLE.]

3. Prove that in any population the difference between the number of females and the number of minors is equal to the difference between the number of females who are not minors, and of minors who are not females.

4. Show that if to the number of metals which are red, we add the number which are brittle, the sum is equal to that of the whole number of metals after addition of the number of metals which are both red and brittle, and after subtraction of the number of metals which are neither red nor brittle.

5. What is the value of the following expression

6. Prove that the number of quadrupeds in the world added to the number of beings not quadrupeds which possess stomachs is equal to the whole number of things having stomachs together with the number of things not having stomachs which are quadrupeds.

7. If x and y be respectively the numbers of things which are X and Y, while m is the whole number which are both X and Y, and n the number which are either X alone or Y alone, what is the relation between m + n and x+y?

8. Let u be the whole number of things under consideration, the number which are A, and y the number which are B; then if m be the number of things which are both A and B, show that m+u-x-y is the number which are neither A nor B.

9. Taking each logical term to represent the number of things included in its class, verify the following equa

ABC AD + ABD + ACD - ABCD = Abcd.

10. What is the product of the logical multiplication of the four factors

(A-AB) (AAC) (A - AD) (A - AE)?

Give another expression for its value.

11. Show that the following equation is necessarily true : B+ AC + bС + Abc = A + C + aBc.

12. What happens in Problem 8 if it be discovered that the class B does not exist at all?

13. Find an expression for the difference between (A) and (B) + (C).