Complete Works of Lewis Carroll
Page 103
Similarly for the fifteen similar Propositions, in terms of x and m, or of y and m.
These thirty-two Propositions of Relation are the only ones that we shall have to represent on this Diagram.
The Reader should now get his genial friend to question him on the following four Tables.
The Victim should have nothing before him but a blank Triliteral Diagram, a Red Counter, and 2 Grey ones, with which he is to represent the various Propositions named by the Inquisitor, e.g. “No y′ are m”, “Some xm′ exist”, &c., &c.
TABLE V.
Some xm exist
= Some x are m
= Some m are x
No xm exist
= No x are m
= No m are x
Some xm′ exist
= Some x are m′
= Some m′ are x
No xm′ exist
= No x are m′
= No m′ are x
Some x′m exist
= Some x′ are m
= Some m are x′
No x′m exist
= No x′ are m
= No m are x′
Some x′m′ exist
= Some x′ are m′
= Some m′ are x′
No x′m′ exist
= No x′ are m′
= No m′ are x′
TABLE VI.
Some ym exist
= Some y are m
= Some m are y
No ym exist
= No y are m
= No m are y
Some ym′ exist
= Some y are m′
= Some m′ are y
No ym′ exist
= No y are m′
= No m′ are y
Some y′m exist
= Some y′ are m
= Some m are y′
No y′m exist
= No y′ are m
= No m are y′
Some y′m′ exist
= Some y′ are m′
= Some m′ are y′
No y′m′ exist
= No y′ are m′
= No m′ are y′
TABLE VII.
All x are m
All x are m′
All x′ are m
All x′ are m′
All m are x
All m are x′
All m′ are x
All m′ are x′
TABLE VIII.
All y are m
All y are m′
All y′ are m
All y′ are m′
All m are y
All m are y′
All m′ are y
All m′ are y′
CHAPTER III.
REPRESENTATION OF TWO PROPOSITIONS OF RELATION, ONE IN TERMS OF x AND m, AND THE OTHER IN TERMS OF y AND m, ON THE SAME DIAGRAM.
The Reader had better now begin to draw little Diagrams for himself, and to mark them with the Digits “I” and “O”, instead of using the Board and Counters: he may put a “I” to represent a Red Counter (this may be interpreted to mean “There is at least one Thing here”), and a “O” to represent a Grey Counter (this may be interpreted to mean “There is nothing here”).
The Pair of Propositions, that we shall have to represent, will always be, one in terms of x and m, and the other in terms of y and m.
When we have to represent a Proposition beginning with “All”, we break it up into the two Propositions to which it is equivalent.
When we have to represent, on the same Diagram, Propositions, of which some begin with “Some” and others with “No”, we represent the negative ones first. This will sometimes save us from having to put a “I” “on a fence” and afterwards having to shift it into a Cell.
[Let us work a few examples.
(1)
“No x are m′;
No y′ are m”.
Let us first represent “No x are m′”. This gives us Diagram a.
Then, representing “No y′ are m” on the same Diagram, we get Diagram b.
a
b
(2)
“Some m are x;
No m are y”.
If, neglecting the Rule, we were begin with “Some m are x”, we should get Diagram a.
And if we were then to take “No m are y”, which tells us that the Inner N.W. Cell is empty, we should be obliged to take the “I” off the fence (as it no longer has the choice of two Cells), and to put it into the Inner N.E. Cell, as in Diagram c.
This trouble may be saved by beginning with “No m are y”, as in Diagram b.
And now, when we take “Some m are x”, there is no fence to sit on! The “I” has to go, at once, into the N.E. Cell, as in Diagram c.
a
b
c
(3)
“No x′ are m′;
All m are y”.
Here we begin by breaking up the Second into the two Propositions to which it is equivalent. Thus we have three Propositions to represent, viz.—
(1) “No x′ are m′;
(2) Some m are y;
(3) No m are y′”.
These we will take in the order 1, 3, 2.
First we take No. (1), viz. “No x′ are m′”. This gives us Diagram a.
Adding to this, No. (3), viz. “No m are y′”, we get Diagram b.
This time the “I”, representing No. (2), viz. “Some m are y,” has to sit on the fence, as there is no “O” to order it off! This gives us Diagram c.
a
b
c
(4)
“All m are x;
All y are m”.
Here we break up both Propositions, and thus get four to represent, viz.—
(1) “Some m are x;
(2) No m are x′;
(3) Some y are m;
(4) No y are m′”.
These we will take in the order 2, 4, 1, 3.
First we take No. (2), viz. “No m are x′”. This gives us Diagram a.
To this we add No. (4), viz. “No y are m′”, and thus get Diagram b.
If we were to add to this No. (1), viz. “Some m are x”, we should have to put the “I” on a fence: so let us try No. (3) instead, viz. “Some y are m”. This gives us Diagram c.
And now there is no need to trouble about No. (1), as it would not add anything to our information to put a “I” on the fence. The Diagram already tells us that “Some m are x”.]
a
b
c
[Work Examples § 1, 9–12 (p. 97); § 2, 1–20 (p. 98).]
CHAPTER 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”.
(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.
(3)
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.”]
[Review Tables V, VI (pp. 46, 47). Work Examples § 1, 13–16 (p. 97); § 2, 21–32 (p. 98); § 3, 1–20 (p. 99).]
BOOK 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 “∴”).
[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”.
Also 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”.]
CHAPTER II.
PROBLEMS IN SYLLOGISMS.
§ 1.
Introductory.
When the Terms of a Proposition are represented by words, it is said to be ‘concrete’; when by letters, ‘abstract.’
To translate a Proposition from concrete into abstract form, we fix on a Univ., and regard each Term as a Species of it, and we choose a letter to represent its Differentia.
[For example, suppose we wish to translate “Some soldiers are brave” into abstract form. We may take “men” as Univ., and regard “soldiers” and “brave men” as Species of the Genus “men”; and we may choose x to represent the peculiar Attribute (say “military”) of “soldiers,” and y to represent “brave.” Then the Proposition may be written “Some military men are brave men”; i.e. “Some x-men are y-men”; i.e. (omitting “men,” as explained at p. 26) “Some x are y.”
In practice, we should merely say “Let Univ. be “men”, x = soldiers, y = brave”, and at once translate “Some soldiers are brave” into “Some x are y.”]
The Problems we shall have to solve are of two kinds, viz.
(1) “Given a Pair of Propositions of Relation, which contain between them a pair of codivisional Classes, and which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them.”
(2) “Given a Trio of Propositions of Relation, of which every two contain a pair of codivisional Classes, and which are proposed as a Syllogism: to ascertain whether the proposed Conclusion is consequent from the proposed Premisses, and, if so, whether it is complete.”
These Problems we will discuss separately.
§ 2.
Given a Pair of Propositions of Relation, which contain between them a pair of codivisional Classes, and which are proposed as Premisses:
to ascertain what Conclusion, if any, is consequent from them.
The Rules, for doing this, are as follows:—
(1) Determine the ‘Universe of Discourse’.
(2) Construct a Dictionary, making m and m (or m and m′) represent the pair of codivisional Classes, and x (or x′) and y (or y′) the other two.
(3) Translate the proposed Premisses into abstract form.
(4) Represent them, together, on a Triliteral Diagram.
(5) Ascertain what Proposition, if any, in terms of x and y, is also represented on it.
(6) Translate this into concrete form.
It is evident that, if the proposed Premisses were true, this other Proposition would also be true. Hence it is a Conclusion consequent from the proposed Premisses.
[Let us work some examples.
(1)
“No son of mine is dishonest;
People always treat an honest man with respect”.
Taking “men” as Univ., we may write these as follows:—