Complete Works of Lewis Carroll
Page 105
“No gluttons are healthy;
No unhealthy men are strong”.
proposed as Premisses. Taking “men” as our ‘Universe’, and making m = healthy; x = gluttons; y = strong; we might translate the Pair into abstract form, thus:—
“No x are m;
No m′ are y”.
These, in subscript form, would be
xm0 † m′y0
which are identical with those in our Formula. Hence we at once know the Conclusion to be
xy0
that is, in abstract form,
“No x are y”;
that is, in concrete form,
“No gluttons are strong”.]
I shall now take three different forms of Pairs of Premisses, and work out their Conclusions, once for all, by Diagrams; and thus obtain some useful Formulæ. I shall call them “Fig. I”, “Fig. II”, and “Fig. III”.
Fig. I.
This includes any Pair of Premisses which are both of them Nullities, and which contain Unlike Eliminands.
The simplest case is
xm0 † ym′0
∴ xy0
In this case we see that the Conclusion is a Nullity, and that the Retinends have kept their Signs.
And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as
m1x0 † ym′0 (which ¶ xy0)
xm′0 † m1y0 (which ¶ xy0)
x′m0 † ym′0 (which ¶ x′y0)
m′1x′0 † m1y′0 (which ¶ x′y′0).]
If either Retinend is asserted in the Premisses to exist, of course it may be so asserted in the Conclusion.
Hence we get two Variants of Fig. I, viz.
(α) where one Retinend is so asserted;
(β) where both are so asserted.
[The Reader had better work out, on Diagrams, examples of these two Variants, such as
m1x0 † y1m′0 (which proves y1x0)
x1m′0 † m1y0 (which proves x1y0)
x′1m0 † y1m′0 (which proves x′1y0 † y1x′0).]
The Formula, to be remembered, is
xm0 † ym′0 ¶ xy0
with the following two Rules:—
(1) Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.
(2) A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.
[Note that Rule (1) is merely the Formula expressed in words.]
Fig. II.
This includes any Pair of Premisses, of which one is a Nullity and the other an Entity, and which contain Like Eliminands.
The simplest case is
xm0 † ym1
∴ x′y1
In this case we see that the Conclusion is an Entity, and that the Nullity-Retinend has changed its Sign.
And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as
x′m0 † ym1 (which ¶ xy1)
x1m′0 † y′m′1 (which ¶ x′y′1)
m1x0 † y′m1 (which ¶ x′y′1).]
The Formula, to be remembered, is,
xm0 † ym1 ¶ x′y1
with the following Rule:—
A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.
[Note that this Rule is merely the Formula expressed in words.]
Fig. III.
This includes any Pair of Premisses which are both of them Nullities, and which contain Like Eliminands asserted to exist.
The simplest case is
xm0 † ym0 † m1
[Note that “m1” is here stated separately, because it does not matter in which of the two Premisses it occurs: so that this includes the three forms “m1x0 † ym0”, “xm0 † m1y0”, and “m1x0 † m1y0”.]
∴ x′y′1
In this case we see that the Conclusion is an Entity, and that both Retinends have changed their Signs.
And we should find this Rule to hold good with any Pair of Premisses which fulfil the given conditions.
[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as
x′m0 † m1y0 (which ¶ xy′1)
m′1x0 † m′y′0 (which ¶ x′y1)
m1x′0 † m1y′0 (which ¶ xy1).]
The Formula, to be remembered, is
xm0 † ym0 † m1 ¶ x′y′1
with the following Rule (which is merely the Formula expressed in words):—
Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.
In order to help the Reader to remember the peculiarities and Formulæ of these three Figures, I will put them all together in one Table.
TABLE IX.
Fig. I.
xm0 † ym′0 ¶ xy0
Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.
A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.
Fig. II.
xm0 † ym1 ¶ x′y1
A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.
Fig. III.
xm0 † ym0 † m1 ¶ x′y′1
Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.
I will now work out, by these Formulæ, as models for the Reader to imitate, some Problems in Syllogisms which have been already worked, by Diagrams, in Book V., Chap. II.
(1)
“No son of mine is dishonest;
People always treat an honest man with respect.”
Univ. “men”; m = honest; x = my sons; y = treated with respect.
xm′0 † m1y′0 ¶ xy′0 [Fig. I.
i.e. “No son of mine ever fails to be treated with respect.”
(2)
“All cats understand French;
Some chickens are cats.”
Univ. “creatures”; m = cats; x = understanding French; y = chickens.
m1x′0 † ym1 ¶ xy1 [Fig. II.
i.e. “Some chickens understand French.”
(3)
“All diligent students are successful;
All ignorant students are unsuccessful.”
Univ. “students”; m = successful; x = diligent; y = ignorant.
x1m′0 † y1m0 ¶ x1y0 † y1x0 [Fig. I (β).
i.e. “All diligent students are learned; and all ignorant students are idle.”
(4)
“All soldiers are strong;
All soldiers are brave.
Some strong men are brave.”
Univ. “men”; m = soldiers; x = strong; y = brave.
m1x′0 † m1y′0 ¶ xy1 [Fig. III.
Hence proposed Conclusion is right.
(5)
“I admire these pictures;
When I admire anything, I wish to examine it thoroughly.
I wish to examine some of these pictures thoroughly.”
Univ. “things”; m = admired by me; x = these; y = things which I wish to examine thoroughly.
x1m′0 † m1y′0 ¶ x1y′0 [Fig. I (α).
Hence proposed Conclusion, xy1, is incomplete, the complete one being “I wish to examine all these pictures thoroughly.”
(6)
“None but the brave deserve the fair;
Some braggarts are cowards.
Some braggarts do not deserve the fair.”
Univ. “persons”; m = brave; x = deserving of the fair; y = braggarts.
m′x0 † ym′1 ¶ x′y1 [Fig. II.
Hence proposed Conclusion is right.
(7)
”No one, who means to go by the train
and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.
This party of tourists need not run.”
Univ. “persons meaning to go by the train, and unable to get a conveyance”; m = having enough time to walk to the station; x = needing to run; y = these tourists.
m′x′0 † y1m′0 do not come under any of the three Figures. Hence it is necessary to return to the Method of Diagrams, as shown at p. 69.
Hence there is no Conclusion.
[Work Examples § 4, 12–20 (p. 100); § 5, 13–24 (pp. 101, 102); § 6, 1–6 (p. 106); § 7, 1–3 (pp. 107, 108). Also read Note (A), at p. 164.]
§ 3.
Fallacies.
Any argument which deceives us, by seeming to prove what it does not really prove, may be called a ‘Fallacy’ (derived from the Latin verb fallo “I deceive”): but the particular kind, to be now discussed, consists of a Pair of Propositions, which are proposed as the Premisses of a Syllogism, but yield no Conclusion.
When each of the proposed Premisses is a Proposition in I, or E, or A, (the only kinds with which we are now concerned,) the Fallacy may be detected by the ‘Method of Diagrams,’ by simply setting them out on a Triliteral Diagram, and observing that they yield no information which can be transferred to the Biliteral Diagram.
But suppose we were working by the ‘Method of Subscripts,’ and had to deal with a Pair of proposed Premisses, which happened to be a ‘Fallacy,’ how could we be certain that they would not yield any Conclusion?
Our best plan is, I think, to deal with Fallacies in the same was as we have already dealt with Syllogisms: that is, to take certain forms of Pairs of Propositions, and to work them out, once for all, on the Triliteral Diagram, and ascertain that they yield no Conclusion; and then to record them, for future use, as Formulæ for Fallacies, just as we have already recorded our three Formulæ for Syllogisms.
Now, if we were to record the two Sets of Formulæ in the same shape, viz. by the Method of Subscripts, there would be considerable risk of confusing the two kinds. Hence, in order to keep them distinct, I propose to record the Formulæ for Fallacies in words, and to call them “Forms” instead of “Formulæ.”
Let us now proceed to find, by the Method of Diagrams, three “Forms of Fallacies,” which we will then put on record for future use. They are as follows:—
(1) Fallacy of Like Eliminands not asserted to exist.
(2) Fallacy of Unlike Eliminands with an Entity-Premiss.
(3) Fallacy of two Entity-Premisses.
These shall be discussed separately, and it will be seen that each fails to yield a Conclusion.
(1) Fallacy of Like Eliminands not asserted to exist.
It is evident that neither of the given Propositions can be an Entity, since that kind asserts the existence of both of its Terms (see p. 20). Hence they must both be Nullities.
Hence the given Pair may be represented by (xm0 † ym0), with or without x1, y1.
These, set out on Triliteral Diagrams, are
xm0 † ym0
x1m0 † ym0
xm0 † y1m0
x1m0 † y1m0
(2) Fallacy of Unlike Eliminands with an Entity-Premiss.
Here the given Pair may be represented by (xm0 † ym′1) with or without x1 or m1.
These, set out on Triliteral Diagrams, are
xm0 † ym′1
x1m0 † ym′1
m1x0 † ym′1
(3) Fallacy of two Entity-Premisses.
Here the given Pair may be represented by either (xm1 † ym1) or (xm1 † ym′1).
These, set out on Triliteral Diagrams, are
xm1 † ym1
xm1 † ym′1
§ 4.
Method of proceeding with a given Pair of Propositions.
Let us suppose that we have before us a Pair of Propositions of Relation, which contain between them a Pair of codivisional Classes, and that we wish to ascertain what Conclusion, if any, is consequent from them. We translate them, if necessary, into subscript-form, and then proceed as follows:—
(1) We examine their Subscripts, in order to see whether they are
(a) a Pair of Nullities;
or (b) a Nullity and an Entity;
or (c) a Pair of Entities.
(2) If they are a Pair of Nullities, we examine their Eliminands, in order to see whether they are Unlike or Like.
If their Eliminands are Unlike, it is a case of Fig. I. We then examine their Retinends, to see whether one or both of them are asserted to exist. If one Retinend is so asserted, it is a case of Fig. I (α); if both, it is a case of Fig. I (β).
If their Eliminands are Like, we examine them, in order to see whether either of them is asserted to exist. If so, it is a case of Fig. III.; if not, it is a case of “Fallacy of Like Eliminands not asserted to exist.”
(3) If they are a Nullity and an Entity, we examine their Eliminands, in order to see whether they are Like or Unlike.
If their Eliminands are Like, it is a case of Fig. II.; if Unlike, it is a case of “Fallacy of Unlike Eliminands with an Entity-Premiss.”
(4) If they are a Pair of Entities, it is a case of “Fallacy of two Entity-Premisses.”
[Work Examples § 4, 1–11 (p. 100); § 5, 1–12 (p. 101); § 6, 7–12 (p. 106); § 7, 7–12 (p. 108).]
BOOK VII.
SORITESES.
CHAPTER I.
INTRODUCTORY.
When a Set of three or more Biliteral Propositions are such that all their Terms are Species of the same Genus, and are also so related that two of them, taken together, yield a Conclusion, which, taken with another of them, yields another Conclusion, and so on, until all have been taken, it is evident that, if the original Set were true, the last Conclusion would also be true.
Such a Set, with the last Conclusion tacked on, is called a ‘Sorites’; the original Set of Propositions is called its ‘Premisses’; each of the intermediate Conclusions is called a ‘Partial Conclusion’ of the Sorites; the last Conclusion is called its ‘Complete Conclusion,’ or, more briefly, its ‘Conclusion’; the Genus, of which all the Terms are Species, is called its ‘Universe of Discourse’, or, more briefly, its ‘Univ.’; the Terms, used as Eliminands in the Syllogisms, are called its ‘Eliminands’; and the two Terms, which are retained, and therefore appear in the Conclusion, are called its ‘Retinends’.
[Note that each Partial Conclusion contains one or two Eliminands; but that the Complete Conclusion contains Retinends only.]
The Conclusion is said to be ‘consequent’ from the Premisses; for which reason it is usual to prefix to it the word “Therefore” (or the symbol “∴”).
[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.
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′;
&
nbsp; 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.]
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. 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.
These shall be discussed separately.
§ 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.