[Note that the question, whether the Conclusion is or is not consequent from the Premisses, is not affected by the actual truth or falsity of any one of the Propositions which make up the Sorites, by depends entirely on their relationship to one another.
[pg086]As a specimen-Sorites, let us take the following Set of 5 Propositions:—
(1) ”No a are b′;
(2) All b are c;
(3) All c are d;
(4) No e′ are a′;
(5) All h are e′”.Here the first and second, taken together, yield “No a are c′”.
This, taken along with the third, yields “No a are d′”.
This, taken along with the fourth, yields “No d′ are e′”.
And this, taken along with the fifth, yields “All h are d”.
Hence, if the original Set were true, this would also be true.
Hence the original Set, with this tacked on, is a Sorites; the original Set is its Premisses; the Proposition “All h are d” is its Conclusion; the Terms a, b, c, e are its Eliminands; and the Terms d and h are its Retinends.
Hence we may write the whole Sorites thus:—
”No a are b′;
All b are c;
All c are d;
No e′ are a′;
All h are e′.
∴ All h are d”.In the above Sorites, the 3 Partial Conclusions are the Positions “No a are e′”, “No a are d′”, “No d′ are e′”; but, if the Premisses were arranged in other ways, other Partial Conclusions might be obtained. Thus, the order 41523 yields the Partial Conclusions “No c′ are b′”, “All h are b”, “All h are c”. There are altogether nine Partial Conclusions to this Sorites, which the Reader will find it an interesting task to make out for himself.]
[pg087]CHAPTER II.
PROBLEMS IN SORITESES.
§ 1.
Introductory.
The Problems we shall have to solve are of the following form:—
“Given three or more Propositions of Relation, which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them.”
We will limit ourselves, at present, to Problems which can be worked by the Formulæ of Fig. I. (See [p. 75].) Those, that require other Formulæ, are rather too hard for beginners.
Such Problems may be solved by either of two Methods, viz.
(1) The Method of Separate Syllogisms;
(2) The Method of Underscoring.