These shall be discussed separately.

[pg088]§ 2.
Solution by Method of Separate Syllogisms.

The Rules, for doing this, are as follows:—

(1) Name the ‘Universe of Discourse’.
(2) Construct a Dictionary, making a, b, c, &c. represent the Terms.
(3) Put the Proposed Premisses into subscript form.
(4) Select two which, containing between them a pair of codivisional Classes, can be used as the Premisses of a Syllogism.
(5) Find their Conclusion by Formula.
(6) Find a third Premiss which, along with this Conclusion, can be used as the Premisses of a second Syllogism.
(7) Find a second Conclusion by Formula.
(8) Proceed thus, until all the proposed Premisses have been used.
(9) Put the last Conclusion, which is the Complete Conclusion of the Sorites, into concrete form.

[As an example of this process, let us take, as the proposed Set of Premisses,

(1) “All the policemen on this beat sup with our cook;
(2) No man with long hair can fail to be a poet;
(3) Amos Judd has never been in prison;
(4) Our cook’s ‘cousins’ all love cold mutton;
(5) None but policemen on this beat are poets;
(6) None but her ‘cousins’ ever sup with our cook;
(7) Men with short hair have all been in prison.”

Univ. “men”; a = Amos Judd; b = cousins of our cook; c = having been in prison; d = long-haired; e = loving cold mutton; h = poets; k = policemen on this beat; l = supping with our cook

[pg089]We now have to put the proposed Premisses into subscript form. Let us begin by putting them into abstract form. The result is

(1) ”All k are l;
(2) No d are h′;
(3) All a are c′;
(4) All b are e;
(5) No k′ are h;
(6) No b′ are l;
(7) All d′ are c.”

And it is now easy to put them into subscript form, as follows:—

(1) k1l′0
(2) dh′0
(3) a1c0
(4) b1e′0
(5) k′h0
(6) b′l0
(7) d′1c′0

We now have to find a pair of Premisses which will yield a Conclusion. Let us begin with No. (1), and look down the list, till we come to one which we can take along with it, so as to form Premisses belonging to Fig. I. We find that No. (5) will do, since we can take k as our Eliminand. So our first syllogism is

(1) k1l′0
(5) k′h0
∴ l′h0 … (8)

We must now begin again with l′h0 and find a Premiss to go along with it. We find that No. (2) will do, h being our Eliminand. So our next Syllogism is

(8) l′h0
(2) dh′0
∴ l′d0 … (9)

We have now used up Nos. (1), (5), and (2), and must search among the others for a partner for l′d0. We find that No. (6) will do. So we write

(9) l′d0
(6) b′l0
∴ db′0 … (10)

Now what can we take along with db′0? No. (4) will do.

(10) db′0
(4) b1e′0
∴ de′0 … (11)

[pg090]Along with this we may take No. (7).

(11) de′0
(7) d′1c′0
∴ c′e′0 … (12)

And along with this we may take No. (3).

(12) c′e′0
(3) a1c0
∴ a1e′0

This Complete Conclusion, translated into abstract form, is

“All a are e”;

and this, translated into concrete form, is

“Amos Judd loves cold mutton.”

In actually working this Problem, the above explanations would, of course, be omitted, and all, that would appear on paper, would be as follows:—

(1) k1l′0
(2) dh′0
(3) a1c0
(4) b1e′0
(5) k′h0
(6) b′l0
(7) d′1c′0

(1) k1l′0
(5) k′h0
∴ l′h0 … (8)

(8) l′h0
(2) dh′0
∴ l′d0 … (9)

(9) l′d0
(6) b′l0
∴ db′0 … (10)

(10) db′0
(4) b1e′0
∴ de′0 … (11)

(11) de′0
(7) d′1c′0
∴ c′e′0 … (12)

(12) c′e′0
(3) a1c0
∴ a1e′0

Note that, in working a Sorites by this Process, we may begin with any Premiss we choose.]

[pg091]§ 3.
Solution by Method of Underscoring.

Consider the Pair of Premisses

xm0 † ym′0