pg053CHAPTER IV.
INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS OR DIGITS.
The problem before us is, given a marked Triliteral Diagram, to ascertain what Propositions of Relation, in terms of x and y, are represented on it.
The best plan, for a beginner, is to draw a Biliteral Diagram alongside of it, and to transfer, from the one to the other, all the information he can. He can then read off, from the Biliteral Diagram, the required Propositions. After a little practice, he will be able to dispense with the Biliteral Diagram, and to read off the result from the Triliteral Diagram itself.
To transfer the information, observe the following Rules:—
(1) Examine the N.W. Quarter of the Triliteral Diagram.
(2) If it contains a “I”, in either Cell, it is certainly occupied, and you may mark the N.W. Quarter of the Biliteral Diagram with a “I”.
(3) If it contains two “O”s, one in each Cell, it is certainly empty, and you may mark the N.W. Quarter of the Biliteral Diagram with a “O”.
pg054(4) Deal in the same way with the N.E., the S.W., and the S.E. Quarter.
[Let us take, as examples, the results of the four Examples worked in the previous Chapters.
(1) In the N.W. Quarter, only one of the two Cells is marked as empty: so we do not know whether the N.W. Quarter of the Biliteral Diagram is occupied or empty: so we cannot mark it.
In the N.E. Quarter, we find two “O”s: so this Quarter is certainly empty; and we mark it so on the Biliteral Diagram.
In the S.W. Quarter, we have no information at all.
In the S.E. Quarter, we have not enough to use.
We may read off the result as “No x are y′”, or “No y′ are x,” whichever we prefer.
(2) In the N.W. Quarter, we have not enough information to use.
In the N.E. Quarter, we find a “I”. This shows us that it is occupied: so we may mark the N.E. Quarter on the Biliteral Diagram with a “I”.
In the S.W. Quarter, we have not enough information to use.
In the S.E. Quarter, we have none at all.
We may read off the result as “Some x are y′”, or “Some y′ are x”, whichever we prefer.
In the N.W. Quarter, we have no information. (The “I”, sitting on the fence, is of no use to us until we know on which side he means to jump down!)
In the N.E. Quarter, we have not enough information to use.
Neither have we in the S.W. Quarter.
The S.E. Quarter is the only one that yields enough information to use. It is certainly empty: so we mark it as such on the Biliteral Diagram.
We may read off the results as “No x′ are y′”, or “No y′ are x′”, whichever we prefer.
(4)
The N.W. Quarter is occupied, in spite of the “O” in the Outer Cell. So we mark it with a “I” on the Biliteral Diagram.
The N.E. Quarter yields no information.
The S.W. Quarter is certainly empty. So we mark it as such on the Biliteral Diagram.
The S.E. Quarter does not yield enough information to use.
We read off the result as “All y are x.”]
pg056BOOK V.
SYLLOGISMS.
CHAPTER I.
INTRODUCTORY
When a Trio of Biliteral Propositions of Relation is such that
(1) all their six Terms are Species of the same Genus,
(2) every two of them contain between them a Pair of codivisional Classes,
(3) the three Propositions are so related that, if the first two were true, the third would be true,
the Trio is called a ‘Syllogism’; the Genus, of which each of the six Terms is a Species, is called its ‘Universe of Discourse’, or, more briefly, its ‘Univ.’; the first two Propositions are called its ‘Premisses’, and the third its ‘Conclusion’; also the Pair of codivisional Terms in the Premisses are called its ‘Eliminands’, and the other two its ‘Retinends’.
The Conclusion of a Syllogism is said to be ‘consequent’ from its Premisses: hence it is usual to prefix to it the word “Therefore” (or the Symbol “∴”).
pg057[Note that the ‘Eliminands’ are so called because they are eliminated, and do not appear in the Conclusion; and that the ‘Retinends’ are so called because they are retained, and do appear in the Conclusion.
Note also 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 of the Trio, but depends entirely on their relationship to each other.
As a specimen-Syllogism, let us take the Trio
“No x-Things are m-Things;
No y-Things are m′-Things.
No x-Things are y-Things.”which we may write, as explained at p. 26, thus:—
“No x are m;
No y are m′.
No x are y”.Here the first and second contain the Pair of codivisional Classes m and m′; the first and third contain the Pair x and x; and the second and third contain the Pair y and y.
Also the three Propositions are (as we shall see hereafter) so related that, if the first two were true, the third would also be true.
Hence the Trio is a Syllogism; the two Propositions, “No x are m” and “No y are m′”, are its Premisses; the Proposition “No x are y” is its Conclusion; the Terms m and m′ are its Eliminands; and the Terms x and y are its Retinends.
Hence we may write it thus:—
“No x are m;
No y are m′.
∴ No x are y”.As a second specimen, let us take the Trio
“All cats understand French;
Some chickens are cats.
Some chickens understand French”.These, put into normal form, are
“All cats are creatures understanding French;
Some chickens are cats.
Some chickens are creatures understanding French”.Here all the six Terms are Species of the Genus “creatures.”
Also the first and second Propositions contain the Pair of codivisional Classes “cats” and “cats”; the first and third contain the Pair “creatures understanding French” and “creatures understanding French”; and the second and third contain the Pair “chickens” and “chickens”.
pg058Also the three Propositions are (as we shall see at p. 64) so related that, if the first two were true, the third would be true. (The first two are, as it happens, not strictly true in our planet. But there is nothing to hinder them from being true in some other planet, say Mars or Jupiter—in which case the third would also be true in that planet, and its inhabitants would probably engage chickens as nursery-governesses. They would thus secure a singular contingent privilege, unknown in England, namely, that they would be able, at any time when provisions ran short, to utilise the nursery-governess for the nursery-dinner!)
Hence the Trio is a Syllogism; the Genus “creatures” is its ‘Univ.’; the two Propositions, “All cats understand French“ and ”Some chickens are cats”, are its Premisses, the Proposition “Some chickens understand French” is its Conclusion; the Terms “cats” and “cats” are its Eliminands; and the Terms, “creatures understanding French” and “chickens”, are its Retinends.
Hence we may write it thus:—
“All cats understand French;
Some chickens are cats;
∴ Some chickens understand French”.]