Complete Works of Lewis Carroll (154 page)

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:—

“No sons of mine are dishonest men;

  All honest men are men treated with respect”.

We can now construct our Dictionary, viz.
m
 = honest;
x
 = sons of mine;
y
 = treated with respect.

(Note that the expression “
x
 = sons of mine” is an abbreviated form of “
x
 = the Differentia of ‘sons of mine’, when regarded as a Species of ‘men’”.)

The next thing is to translate the proposed Premisses into abstract form, as follows:—

“No
x
are
m
′
;

  All
m
are
y
”.

Next, by the process described at p.
50, we represent these on a Triliteral Diagram, thus:—

Next, by the process described at p.
53, we transfer to a Biliteral Diagram all the information we can.

The result we read as “No
x
are
y
′
” or as “No
y
′
are
x
,” whichever we prefer.
So we refer to our Dictionary, to see which will look best; and we choose

“No
x
are
y
′
”,

which, translated into concrete form, is

“No son of mine fails to be treated with respect”.

(2)

“All cats understand French;

  Some chickens are cats”.

Taking “creatures” as Univ., we write these as follows:—

“All cats are creatures understanding French;

  Some chickens are cats”.

We can now construct our Dictionary, viz.
m
 = cats;
x
 = understanding French;
y
 = chickens.

The proposed Premisses, translated into abstract form, are

“All
m
are
x
;

  Some
y
are
m
”.

In order to represent these on a Triliteral Diagram, we break up the first into the two Propositions to which it is equivalent, and thus get the
three
Propositions

(1) “Some
m
are
x
;

(2)   No
m
are
x
′
;

(3)   Some
y
are
m
”.

The Rule, given at p.
50, would make us take these in the order 2, 1, 3.

This, however, would produce the result

So it would be better to take them in the order 2, 3, 1.
Nos.
(2) and (3) give us the result here shown; and now we need not trouble about No.
(1), as the Proposition “Some
m
are
x
” is
already
represented on the Diagram.

Transferring our information to a Biliteral Diagram, we get

This result we can read either as “Some
x
are
y
” or “Some
y
are
x
”.

After consulting our Dictionary, we choose

“Some
y
are
x
”,

which, translated into concrete form, is

“Some chickens understand French.”

(3)

“All diligent students are successful;