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".
pg058
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".]
pg059
CHAPTER II.
_PROBLEMS IN SYLLOGISMS._
Sec. 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.
pg060
Sec. 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".
pg061
Next, by the process described at p. 50, we represent these on a
Triliteral Diagram, thus:--
.---------------.
|(O) | (O)|
| .---|---. |
| | |(O)| |
|---|(I)|---|---|
| | |(O)| |
| .---|---. |
| | |
.---------------.
Next, by the process described at p. 53, we transfer to a
Biliteral Diagram all the information we can.
.-------.
| |(O)|
|---|---|
| | |
.-------.
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
.-----------------.
| | |
| .----|----. |
| |(I)(I) | |
| |----|----| |
| |(O) | (O)| |
| .----|----. |
| | |
.-----------------.
pg062
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.
.---------------.
| | |
| .---|---. |
| |(I)| | |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
| | |
.---------------.
Transferring our information to a Biliteral Diagram, we get
.-------.
|(I)| |
|---|---|
| | |
.-------.
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;
All ignorant students are unsuccessful".
Let Univ. be "students"; m = successful; x = diligent;
y = ignorant.
These Premisses, in abstract form, are
"All x are m;
All y are m'".
These, broken up, give us the four Propositions
(1) "Some x are m;
(2) No x are m';
(3) Some y are m';
(4) No y are m".
which we will take in the order 2, 4, 1, 3.
Representing these on a Triliteral Diagram, we get
.---------------.
|(O) | (O)|
| .---|---. |
| |(O)|(I)| |
|---|---|---|---|
| |(O)| | |
| .---|---. |
|(I) | |
.---------------.
And this information, transferred to a Biliteral Diagram, is
.-------.
|(O)|(I)|
|---|---|
|(I)| |
.-------.
Here we get _two_ Conclusions, viz.
"All x are y';
All y are x'."
pg063
And these, translated into concrete form, are
"All diligent students are (not-ignorant, i.e.) learned;
All ignorant students are (not-diligent, i.e.) idle".
(See p. 4.)
(4)
"Of the prisoners who were put on their trial at the last
Assizes, all, against whom the verdict 'guilty' was
returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also
sentenced to hard labour".
Let Univ. be "the prisoners who were put on their trial at the
last Assizes"; m = who were sentenced to imprisonment;
x = against whom the verdict 'guilty' was returned; y = who were
sentenced to hard labour.
The Premisses, translated into abstract form, are
"All x are m;
Some m are y".
Breaking up the first, we get the three
(1) "Some x are m;
(2) No x are m';
(3) Some m are y".
Representing these, in the order 2, 1, 3, on a Triliteral
Diagram, we get
.---------------.
|(O) | (O)|
| .---|---. |
| | (I) | |
|---|(I)|---|---|
| | | | |
| .---|---. |
| | |
.---------------.
Here we get no Conclusion at all.
You would very likely have guessed, if you had seen _only_ the
Premisses, that the Conclusion would be
"Some, against whom the verdict 'guilty' was returned,
were sentenced to hard labour".
But this Conclusion is not even _true_, with regard to the
Assizes I have here invented.
"Not _true!_" you exclaim. "Then who _were_ they, who were
sentenced to imprisonment and were also sentenced to hard
labour? They _must_ have had the verdict 'guilty' returned
against them, or how could they be sentenced?"
Well, it happened like _this_, you see. They were three
ruffians, who had committed highway-robbery. When they were put
on their trial, they _pleaded_ 'guilty'. So no _verdict_ was
returned at all; and they were sentenced at once.]
I will now work out, in their briefest form, as models for the Reader to
imitate in working examples, the above four concrete Problems.
pg064
(1) [see p. 60]
"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.
.---------------. .-------.
"No x are m'; |(O) | (O)| | |(O)|
All m are y." | .---|---. | |---|---|
| | |(O)| | | | |
|---|(I)|---|---| .-------.
| | |(O)| |
| .---|---. | .'. "No x are y'."
| | |
.---------------.
i.e. "No son of mine ever fails to be treated with respect."
(2) [see p. 61]
"All cats understand French;
Some chickens are cats".
Univ. "creatures"; m = cats; x = understanding French; y = chickens.
.---------------. .-------.
"All m are x; | | | |(I)| |
Some y are m." | .---|---. | |---|---|
| |(I)| | | | | |
|---|---|---|---| .-------.
| |(O)|(O)| |
| .---|---. | .'. "Some y are x."
| | |
.---------------.
i.e. "Some chickens understand French."
(3) [see p. 62]
"All diligent students are successful;
All ignorant students are unsuccessful".
Univ. "students"; m = successful; x = diligent; y = ignorant.
.---------------. .-------.
"All x are m; |(O) | (O)| |(O)|(I)|
All y are m'." | .---|---. | |---|---|
| |(O)|(I)| | |(I)| |
|---|---|---|---| .-------.
| |(O)| | |
| .---|---. | .'. "All x are y';
|(I) | | All y are x'."
.---------------.
i.e. "All diligent students are learned; and all ignorant students are
idle".
pg065
(4) [see p. 63]
"Of the prisoners who were put on their trial at the last
Assizes, all, against whom the verdict 'guilty' was
returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also
sentenced to hard labour".
Univ. "prisoners who were put on their trial at the last Assizes",
m = sentenced to imprisonment; x = against whom the verdict 'guilty' was
returned; y = sentenced to hard labour.
.---------------.
"All x are m; |(O) | (O)|
Some m are y." | .---|---. |
| | (I) | |
|---|(I)|---|---|
| | | | |
| .---|---. | There is no
| | | Conclusion.
.---------------.
[Review Tables VII, VIII (pp. 48, 49). Work Examples Sec. =1=,
17-21 (p. 97); Sec. =4=, 1-6 (p. 100); Sec. =5=, 1-6 (p. 101).]
pg066
Sec. 3.
_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._
The Rules, for doing this, are as follows:--
(1) Take the proposed Premisses, and ascertain, by the process
described at p. 60, what Conclusion, if any, is consequent
from them.
(2) If there be _no_ Conclusion, say so.
(3) If there _be_ a Conclusion, compare it with the proposed
Conclusion, and pronounce accordingly.
I will now work out, in their briefest form, as models for the Reader to
imitate in working examples, six Problems.
(1)
"All soldiers are strong;
All soldiers are brave.
Some strong men are brave."
Univ. "men"; m = soldiers; x = strong; y = brave.
pg067
.---------------. .-------.
"All m are x; | | | |(I)| |
All m are y. | .---|---. | |---|---|
Some x are y." | |(I)|(O)| | | | |
|---|---|---|---| .-------.
| |(O)|(O)| |
| .---|---. | .'. "Some x are y."
| | |
.---------------.
Hence proposed Conclusion is right.
(2)
"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 pictures; y = things which
I wish to examine thoroughly.
.---------------. .-------.
"All x are m; |(O) | (O)| |(I)|(O)|
All m are y. | .---|---. | |---|---|
Some x are y." | |(I)|(O)| | | | |
|---|---|---|---| .-------.
| | |(O)| |
| .---|---. | .'. "All x are y."
| | |
.---------------.
Hence proposed Conclusion is _incomplete_, the _complete_ one being "I
wish to examine _all_ these pictures thoroughly".
(3)
"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.
.---------------. .-------.
"No m' are x; |(O) | (O)| | | |
Some y are m'. | .---|---. | |---|---|
Some y are x'." | | | | | |(I)| |
|---|---|---|---| .-------.
| | | | |
| .---|---. | .'. "Some y are x'."
|(I) | |
.---------------.
Hence proposed Conclusion is right.
pg068
(4)
"All soldiers can march;
Some babies are not soldiers.
Some babies cannot march".
Univ. "persons"; m = soldiers; x = able to march; y = babies.
.---------------.
"All m are x; | | |
Some y are m'. | .---|---. |
Some y are x'." | | (I) | |
|(I)|---|---|---|
| |(O)|(O)| |
| .---|---. | There is no
| | | Conclusion.
.---------------.
(5)
"All selfish men are unpopular;
All obliging men are popular.
All obliging men are unselfish".
Univ. "men"; m = popular; x = selfish; y = obliging.
.---------------. .-------.
"All x are m'; |(O) | (I)| |(O)|(I)|
All y are m. | .---|---. | |---|---|
All y are x'." | |(O)|(O)| | |(I)| |
|---|---|---|---| .-------.
| |(I)| | |
| .---|---. | .'. "All x are y';
|(O) | | All y are x'."
.---------------.
Hence proposed Conclusion is _incomplete_, the _complete_ one
containing, in addition, "All selfish men are disobliging".
(6)
"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.
pg069
.---------------.
"No m' are x'; |(O) | |
All y are m. | .---|---. |
All y are x'." | | | | |
|---|(I)|---|---|
| | | | |
| .---|---. | There is no
|(O) | (O)| Conclusion.
.---------------.
[Here is _another_ opportunity, gentle Reader, for playing a
trick on your innocent friend. Put the proposed Syllogism before
him, and ask him what he thinks of the Conclusion.
He will reply "Why, it's perfectly correct, of course! And if
your precious Logic-book tells you it _isn't_, don't believe it!
You don't mean to tell me those tourists _need_ to run? If _I_
were one of them, and knew the _Premisses_ to be true, I should
be _quite_ clear that I _needn't_ run--and I _should walk!_"
And _you_ will reply "But suppose there was a mad bull behind
you?"
And then your innocent friend will say "Hum! Ha! I must think
that over a bit!"
You may then explain to him, as a convenient _test_ of the
soundness of a Syllogism, that, if circumstances can be invented
which, without interfering with the truth of the _Premisses_,
would make the _Conclusion_ false, the Syllogism _must_ be
unsound.]
[Review Tables V-VIII (pp. 46-49). Work Examples Sec. =4=, 7-12 (p.
100); Sec. =5=, 7-12 (p. 101); Sec. =6=, 1-10 (p. 106); Sec. =7=, 1-6
(pp. 107, 108).]
pg070
BOOK VI.
THE METHOD OF SUBSCRIPTS.
CHAPTER I.
_INTRODUCTORY._
Let us agree that "x_{1}" shall mean "Some existing Things have the
Attribute x", i.e. (more briefly) "Some x exist"; also that "xy_{1}"
shall mean "Some xy exist", and so on. Such a Proposition may be called
an '=Entity=.'
[Note that, when there are _two_ letters in the expression, it
does not in the least matter which stands _first_: "xy_{1}" and
"yx_{1}" mean exactly the same.]
Also that "x_{0}" shall mean "No existing Things have the Attribute x",
i.e. (more briefly) "No x exist"; also that "xy_{0}" shall mean "No xy
exist", and so on. Such a Proposition may be called a '=Nullity='.
Also that "+" shall mean "and".
[Thus "ab_{1} + cd_{0}" means "Some ab exist and no cd exist".]
Also that ">" shall mean "would, if true, prove".
[Thus, "x_{0} > xy_{0}" means "The Proposition 'No x exist'
would, if true, prove the Proposition 'No xy exist'".]
When two Letters are both of them accented, or both _not_ accented, they
are said to have '=Like Signs=', or to be '=Like=': when one is
accented, and the other not, they are said to have '=Unlike Signs=', or
to be '=Unlike='.
pg071
CHAPTER II.
_REPRESENTATION OF PROPOSITIONS OF RELATION._
Let us take, first, the Proposition "Some x are y".
This, we know, is equivalent to the Proposition of Existence "Some xy
exist". (See p. 31.) Hence it may be represented by the expression
"xy_{1}".
The Converse Proposition "Some y are x" may of course be represented by
the _same_ expression, viz. "xy_{1}".
Similarly we may represent the three similar Pairs of Converse
Propositions, viz.--
"Some x are y'" = "Some y' are x",
"Some x' are y" = "Some y are x'",
"Some x' are y'" = "Some y' are x'".
Let us take, next, the Proposition "No x are y".
This, we know, is equivalent to the Proposition of Existence "No xy
exist". (See p. 33.) Hence it may be represented by the expression
"xy_{0}".
The Converse Proposition "No y are x" may of course be represented by
the _same_ expression, viz. "xy_{0}".
Similarly we may represent the three similar Pairs of Converse
Propositions, viz.--
"No x are y'" = "No y' are x",
"No x' are y" = "No y are x'",
"No x' are y'" = "No y' are x'".
pg072
Let us take, next, the Proposition "All x are y".
Now it is evident that the Double Proposition of Existence "Some x exist
and no xy' exist" tells us that _some_ x-Things exist, but that _none_
of them have the Attribute y': that is, it tells us that _all_ of them
have the Attribute y: that is, it tells us that "All x are y".
Also it is evident that the expression "x_{1} + xy'_{0}" represents this
Double Proposition.
Hence it also represents the Proposition "All x are y".
[The Reader will perhaps be puzzled by the statement that the
Proposition "All x are y" is equivalent to the Double
Proposition "Some x exist and no xy' exist," remembering that it
was stated, at p. 33, to be equivalent to the Double Proposition
"Some x are y and no x are y'" (i.e. "Some xy exist and no xy'
exist"). The explanation is that the Proposition "Some xy exist"
contains _superfluous information_. "Some x exist" is enough for
our purpose.]
This expression may be written in a shorter form, viz. "x_{1}y'_{0}",
since _each_ Subscript takes effect back to the _beginning_ of the
expression.
Similarly we may represent the seven similar Propositions "All x are
y'", "All x' are y", "All x' are y'", "All y are x", "All y are x'",
"All y' are x", and "All y' are x'".
[The Reader should make out all these for himself.]
It will be convenient to remember that, in translating a Proposition,
beginning with "All", from abstract form into subscript form, or _vice
versa_, the Predicate _changes sign_ (that is, changes from positive to
negative, or else from negative to positive).
[Thus, the Proposition "All y are x'" becomes "y_{1}x_{0}",
where the Predicate changes from x' to x.
Again, the expression "x'_{1}y'_{0}" becomes "All x' are y",
where the Predicate changes for y' to y.]
pg073
CHAPTER III.
_SYLLOGISMS._
Sec. 1.
_Representation of Syllogisms._
We already know how to represent each of the three Propositions of a
Syllogism in subscript form. When that is done, all we need, besides, is
to write the three expressions in a row, with "+" between the Premisses,
and ">" before the Conclusion.
[Thus the Syllogism
"No x are m';
All m are y.
.'. No x are y'."
may be represented thus:--
xm'_{0} + m_{1}y'_{0} > xy'_{0}
When a Proposition has to be translated from concrete form into
subscript form, the Reader will find it convenient, just at
first, to translate it into _abstract_ form, and _thence_ into
subscript form. But, after a little practice, he will find it
quite easy to go straight from concrete form to subscript form.]
pg074
Sec. 2.
_Formulae for solving Problems in Syllogisms._
When once we have found, by Diagrams, the Conclusion to a given Pair of
Premisses, and have represented the Syllogism in subscript form, we have
a _Formula_, by which we can at once find, without having to use
Diagrams again, the Conclusion to any _other_ Pair of Premisses having
the _same_ subscript forms.
[Thus, the expression
xm_{0} + ym'_{0} > xy_{0}
is a Formula, by which we can find the Conclusion to any Pair of
Premisses whose subscript forms are
xm_{0} + ym'_{0}
For example, suppose we had the Pair of Propositions
"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
xm_{0} + m'y_{0}
which are identical with those in our _Formula_. Hence we at
once know the Conclusion to be
xy_{0}
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 Formulae. I shall call them "Fig. I", "Fig. II", and "Fig. III".
pg075
Fig. I.
This includes any Pair of Premisses which are both of them Nullities,
and which contain Unlike Eliminands.
The simplest case is
.---------------. .-------.
xm_{0} + ym'_{0} |(O) | | |(O)| |
| .---|---. | |---|---|
| |(O)|(O)| | | | |
|---|---|---|---| .-------.
| | | | |
| .---|---. | .'. xy_{0}
|(O) | |
.---------------.
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
m_{1}x_{0} + ym'_{0} (which > xy_{0})
xm'_{0} + m_{1}y_{0} (which > xy_{0})
x'm_{0} + ym'_{0} (which > x'y_{0})
m'_{1}x'_{0} + m_{1}y'_{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.
(a) where _one_ Retinend is so asserted;
(b) where _both_ are so asserted.
[The Reader had better work out, on Diagrams, examples of these
two Variants, such as
m_{1}x_{0} + y_{1}m'_{0} (which proves y_{1}x_{0})
x_{1}m'_{0} + m_{1}y_{0} (which proves x_{1}y_{0})
x'_{1}m_{0} + y_{1}m'_{0} (which proves x'_{1}y_{0} + y_{1}x'_{0}).]
The Formula, to be remembered, is
xm_{0} + ym'_{0} > xy_{0}
with the following two Rules:--
(1) _Two Nullities, with Unlike Eliminands, yield a Nullity,
in which both Retinends keep their Signs._
pg076
(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
xm_{0} + ym_{1}
.---------------. .-------.
| | | | | |
| .---|---. | |---|---|
| |(O)|(O)| | |(I)| |
|---|---|---|---| .-------.
| |(I)| | |
| .---|---. | .'. x'y_{1}
| | |
.---------------.
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'm_{0} + ym_{1} (which > xy_{1})
x_{1}m'_{0} + y'm'_{1} (which > x'y'_{1})
m_{1}x_{0} + y'm_{1} (which > x'y'_{1}).]
The Formula, to be remembered, is,
xm_{0} + ym_{1} > x'y_{1}
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.]
pg077
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
xm_{0} + ym_{0} + m_{1}
[Note that "m_{1}" is here stated _separately_, because it does
not matter in which of the two Premisses it occurs: so that this
includes the _three_ forms "m_{1}x_{0} + ym_{0}", "xm_{0} +
m_{1}y_{0}", and "m_{1}x_{0} + m_{1}y_{0}".]
.---------------. .-------.
| | | | | |
| .---|---. | |---|---|
| |(O)|(O)| | | |(I)|
|---|---|---|---| .-------.
| |(O)|(I)| |
| .---|---. | .'. 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'm_{0} + m_{1}y_{0} (which > xy'_{1})
m'_{1}x_{0} + m'y'_{0} (which > x'y_{1})
m_{1}x'_{0} + m_{1}y'_{0} (which > xy_{1}).]
The Formula, to be remembered, is
xm_{0} + ym_{0} + m_{1} > 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 Formulae of
these three Figures, I will put them all together in one Table.
pg078
TABLE IX.
_______________________________________________________
| |
| Fig. I. |
| |
| xm_{0} + ym'_{0} > xy_{0} |
| |
| 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. |
| |
| xm_{0} + ym_{1} > x'y_{1} |
| |
| A Nullity and an Entity, with Like Eliminands, |
| yield an Entity, in which the Nullity-Retinend |
| changes its Sign. |
|_______________________________________________________|
| |
| Fig. III. |
| |
| xm_{0} + ym_{0} + m_{1} > 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 Formulae, as models for the Reader to
imitate, some Problems in Syllogisms which have been already worked, by
Diagrams, in Book V., Chap. II.
(1) [see p. 64]
"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} + m_{1}y'_{0} > xy'_{0} [Fig. I.
_i.e._ "No son of mine ever fails to be treated with respect."
pg079
(2) [see p. 64]
"All cats understand French;
Some chickens are cats."
Univ. "creatures"; m = cats; x = understanding French; y = chickens.
m_{1}x'_{0} + ym_{1} > xy_{1} [Fig. II.
_i.e._ "Some chickens understand French."
(3) [see p. 64]
"All diligent students are successful;
All ignorant students are unsuccessful."
Univ. "students"; m = successful; x = diligent; y = ignorant.
x_{1}m'_{0} + y_{1}m_{0} > x_{1}y_{0} + y_{1}x_{0} [Fig. I (b).
_i.e._ "All diligent students are learned; and all ignorant students are
idle."
(4) [see p. 66]
"All soldiers are strong;
All soldiers are brave.
Some strong men are brave."
Univ. "men"; m = soldiers; x = strong; y = brave.
m_{1}x'_{0} + m_{1}y'_{0} > xy_{1} [Fig. III.
Hence proposed Conclusion is right.
(5) [see p. 67]
"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.
x_{1}m'_{0} + m_{1}y'_{0} > x_{1}y'_{0} [Fig. I (a).
Hence proposed Conclusion, xy_{1}, is _incomplete_, the _complete_ one
being "I wish to examine _all_ these pictures thoroughly."
pg080
(6) [see p. 67]
"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'x_{0} + ym'_{1} > x'y_{1} [Fig. II.
Hence proposed Conclusion is right.
(7) [see p. 69]
"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} + y_{1}m'_{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 Sec. =4=, 12-20 (p. 100); Sec. =5=, 13-24 (pp. 101,
102); Sec. =6=, 1-6 (p. 106); Sec. =7=, 1-3 (pp. 107, 108). Also read
Note (A), at p. 164.]
pg081
Sec. 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 _Formulae for Fallacies_, just as
we have already recorded our three _Formulae for Syllogisms_.
pg082
Now, if we were to record the two Sets of Formulae 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 Formulae for _Fallacies_ in _words_, and to call
them "Forms" instead of "Formulae."
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 (xm_{0} + ym_{0}), with or
without x_{1}, y_{1}.
These, set out on Triliteral Diagrams, are
xm_{0} + ym_{0} x_{1}m_{0} + ym_{0}
.---------------. .---------------.
| | | | (I) |
| .---|---. | | .---|---. |
| |(O)|(O)| | | |(O)|(O)| |
|---|---|---|---| |---|---|---|---|
| |(O)| | | | |(O)| | |
| .---|---. | | .---|---. |
| | | | | |
.---------------. .---------------.
xm_{0} + y_{1}m_{0} x_{1}m_{0} + y_{1}m_{0}
.---------------. .---------------.
| | | | (I) |
| .---|---. | | .---|---. |
| |(O)|(O)| | | |(O)|(O)| |
|(I)|---|---|---| |(I)|---|---|---|
| |(O)| | | | |(O)| | |
| .---|---. | | .---|---. |
| | | | | |
.---------------. .---------------.
pg083
(2) _Fallacy of Unlike Eliminands with an Entity-Premiss._
Here the given Pair may be represented by (xm_{0} + ym'_{1}) with or
without x_{1} or m_{1}.
These, set out on Triliteral Diagrams, are
xm_{0} + ym'_{1} x_{1}m_{0} + ym'_{1} m_{1}x_{0} + ym'_{1}
.---------------. .---------------. .---------------.
| | | | (I) | | | |
| .---|---. | | .---|---. | | .---|---. |
| |(O)|(O)| | | |(O)|(O)| | | |(O)|(O)| |
|(I)|---|---|---| |(I)|---|---|---| |(I)|---|---|---|
| | | | | | | | | | | | (I) | |
| .---|---. | | .---|---. | | .---|---. |
| | | | | | | | |
.---------------. .---------------. .---------------.
(3) _Fallacy of two Entity-Premisses._
Here the given Pair may be represented by either (xm_{1} + ym_{1}) or
(xm_{1} + ym'_{1}).
These, set out on Triliteral Diagrams, are
xm_{1} + ym_{1} xm_{1} + ym'_{1}
.---------------. .---------------.
| | | | | |
| .---|---. | | .---|---. |
| | (I) | | | | (I) | |
|---|(I)|---|---| |(I)|---|---|---|
| | | | | | | | | |
| .---|---. | | .---|---. |
| | | | | |
.---------------. .---------------.
pg084
Sec. 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 (a);
if both, it is a case of Fig. I (b).
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 Sec. =4=, 1-11 (p. 100); Sec. =5=, 1-12 (p. 101);
Sec. =6=, 7-12 (p. 106); Sec. =7=, 7-12 (p. 108).]
pg085
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_.
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._
Sec. 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 Formulae of Fig. I. (See p. 75.) Those, that require _other_ Formulae,
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.
pg088
Sec. 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.
[As an example of this process, let us take, as the proposed Set
of Premisses,
(1) "All the policemen on this beat sup with our cook;
(2) No man with long hair can fail to be a poet;
(3) Amos Judd has never been in prison;
(4) Our cook's 'cousins' all love cold mutton;
(5) None but policemen on this beat are poets;
(6) None but her 'cousins' ever sup with our cook;
(7) Men with short hair have all been in prison."
Univ. "men"; a = Amos Judd; b = cousins of our cook; c = having
been in prison; d = long-haired; e = loving cold mutton;
h = poets; k = policemen on this beat; l = supping with our cook
pg089
We now have to put the proposed Premisses into _subscript_ form.
Let us begin by putting them into _abstract_ form. The result is
(1) "All k are l;
(2) No d are h';
(3) All a are c';
(4) All b are e;
(5) No k' are h;
(6) No b' are l;
(7) All d' are c."
And it is now easy to put them into _subscript_ form, as
follows:--
(1) k_{1}l'_{0}
(2) dh'_{0}
(3) a_{1}c_{0}
(4) b_{1}e'_{0}
(5) k'h_{0}
(6) b'l_{0}
(7) d'_{1}c'_{0}
We now have to find a pair of Premisses which will yield a
Conclusion. Let us begin with No. (1), and look down the list,
till we come to one which we can take along with it, so as to
form Premisses belonging to Fig. I. We find that No. (5) will
do, since we can take k as our Eliminand. So our first syllogism
is
(1) k_{1}l'_{0}
(5) k'h_{0}
.'. l'h_{0} ... (8)
We must now begin again with l'h_{0} and find a Premiss to go
along with it. We find that No. (2) will do, h being our
Eliminand. So our next Syllogism is
(8) l'h_{0}
(2) dh'_{0}
.'. l'd_{0} ... (9)
We have now used up Nos. (1), (5), and (2), and must search
among the others for a partner for l'd_{0}. We find that No. (6)
will do. So we write
(9) l'd_{0}
(6) b'l_{0}
.'. db'_{0} ... (10)
Now what can we take along with db'_{0}? No. (4) will do.
(10) db'_{0}
(4) b_{1}e'_{0}
.'. de'_{0} ... (11)
pg090
Along with this we may take No. (7).
(11) de'_{0}
(7) d'_{1}c'_{0}
.'. c'e'_{0} ... (12)
And along with this we may take No. (3).
(12) c'e'_{0}
(3) a_{1}c_{0}
.'. a_{1}e'_{0}
This Complete Conclusion, translated into _abstract_ form, is
"All a are e";
and this, translated into _concrete_ form, is
"Amos Judd loves cold mutton."
In actually _working_ this Problem, the above explanations
would, of course, be omitted, and all, that would appear on
paper, would be as follows:--
(1) k_{1}l'_{0}
(2) dh'_{0}
(3) a_{1}c_{0}
(4) b_{1}e'_{0}
(5) k'h_{0}
(6) b'l_{0}
(7) d'_{1}c'_{0}
(1) k_{1}l'_{0}
(5) k'h_{0}
.'. l'h_{0} ... (8)
(8) l'h_{0}
(2) dh'_{0}
.'. l'd_{0} ... (9)
(9) l'd_{0}
(6) b'l_{0}
.'. db'_{0} ... (10)
(10) db'_{0}
(4) b_{1}e'_{0}
.'. de'_{0} ... (11)
(11) de'_{0}
(7) d'_{1}c'_{0}
.'. c'e'_{0} ... (12)
(12) c'e'_{0}
(3) a_{1}c_{0}
.'. a_{1}e'_{0}
Note that, in working a Sorites by this Process, we may begin
with _any_ Premiss we choose.]
pg091
Sec. 3.
_Solution by Method of Underscoring._
Consider the Pair of Premisses
xm_{0} + ym'_{0}
which yield the Conclusion xy_{0}
We see that, in order to get this Conclusion, we must eliminate m and
m', and write x and y together in one expression.
Now, if we agree to _mark_ m and m' as eliminated, and to read the two
expressions together, as if they were written in one, the two Premisses
will then exactly represent the _Conclusion_, and we need not write it
out separately.
Let us agree to mark the eliminated letters by _underscoring_ them,
putting a _single_ score under the _first_, and a _double_ one under the
_second_.
The two Premisses now become
xm_{0} + ym'_{0}
- =
which we read as "xy_{0}".
In copying out the Premisses for underscoring, it will be convenient to
_omit all subscripts_. As to the "0's" we may always _suppose_ them
written, and, as to the "1's", we are not concerned to know _which_
Terms are asserted to _exist_, except those which appear in the
_Complete_ Conclusion; and for _them_ it will be easy enough to refer to
the original list.
pg092
[I will now go through the process of solving, by this method,
the example worked in Sec. 2.
The Data are
1 2 3 4
k_{1}l'_{0} + dh'_{0} + a_{1}c_{0} + b_{1}e'_{0} +
5 6 7
k'h_{0} + b'l_{0} + d'_{1}c'_{0}
The Reader should take a piece of paper, and write out this
solution for himself. The first line will consist of the above
Data; the second must be composed, bit by bit, according to the
following directions.
We begin by writing down the first Premiss, with its numeral
over it, but omitting the subscripts.
We have now to find a Premiss which can be combined with this,
_i.e._, a Premiss containing either k' or l. The first we find
is No. 5; and this we tack on, with a +.
To get the _Conclusion_ from these, k and k' must be eliminated,
and what remains must be taken as one expression. So we
_underscore_ them, putting a _single_ score under k, and a
_double_ one under k'. The result we read as l'h.
We must now find a Premiss containing either l or h'. Looking
along the row, we fix on No. 2, and tack it on.
Now these 3 Nullities are really equivalent to (l'h + dh'), in
which h and h' must be eliminated, and what remains taken as one
expression. So we _underscore_ them. The result reads as l'd.
We now want a Premiss containing l or d'. No. 6 will do.
These 4 Nullities are really equivalent to (l'd + b'l). So we
underscore l' and l. The result reads as db'.
We now want a Premiss containing d' or b. No. 4 will do.
Here we underscore b' and b. The result reads as de'.
We now want a Premiss containing d' or e. No. 7 will do.
Here we underscore d and d'. The result reads as c'e'.
We now want a Premiss containing c or e. No. 3 will do--in fact
_must_ do, as it is the only one left.
Here we underscore c' and c; and, as the whole thing now reads
as e'a, we tack on e'a_{0} as the _Conclusion_, with a >.
We now look along the row of Data, to see whether e' or a has
been given as _existent_. We find that a has been so given in
No. 3. So we add this fact to the Conclusion, which now stands
as > e'a_{0} + a_{1}, _i.e._ > a_{1}e'_{0}; i.e. "All a are e."
If the Reader has faithfully obeyed the above directions, his
written solution will now stand as follows:--
1 2 3 4
k_{1}l'_{0} + dh'_{0} + a_{1}c_{0} + b_{1}e'_{0} +
5 6 7
k'h_{0} + b'l_{0} + d'_{1}c'_{0}
1 5 2 6 4 7 3
kl' + k'h + dh' + b'l + be' + d'c' + ac
-- = - -= - = = = - =
> e'a_{0} + a_{1}
_i.e._ > a_{1}e'_{0};
_i.e._ "All a are e."
pg093
The Reader should now take a second piece of paper, and copy the
Data only, and try to work out the solution for himself,
beginning with some other Premiss.
If he fails to bring out the Conclusion a_{1}e'_{0}, I would
advise him to take a third piece of paper, and _begin again_!]
I will now work out, in its briefest form, a Sorites of 5 Premisses, to
serve as a model for the Reader to imitate in working examples.
(1) "I greatly value everything that John gives me;
(2) Nothing but this bone will satisfy my dog;
(3) I take particular care of everything that I greatly
value;
(4) This bone was a present from John;
(5) The things, of which I take particular care, are
things I do _not_ give to my dog".
Univ. "things"; a = given by John to me; b = given by me to my dog;
c = greatly valued by me; d = satisfactory to my dog; e = taken
particular care of by me; h = this bone.
1 2 3 4 5
a_{1}c'_{0} + h'd_{0} + c_{1}e'_{0} + h_{1}a'_{0} + e_{1}b_{0}
1 3 4 2 5
ac' + ce' + ha' + h'd + eb > db_{0}
-- =- -= = =
i.e. "Nothing, that I give my dog, satisfies him," or, "My dog is not
satisfied with _anything_ that I give him!"
[Note that, in working a Sorites by this process, we may begin
with _any_ Premiss we choose. For instance, we might begin with
No. 5, and the result would then be
5 3 1 4 2
eb + ce' + ac' + ha' + h'd > bd_{0}]
- -= -= -= =
[Work Examples Sec. =4=, 25-30 (p. 100); Sec. =5=, 25-30 (p. 102);
Sec. =6=, 13-15 (p. 106); Sec. =7=, 13-15 (p. 108); Sec. =8=, 1-4, 13,
14, 19, 24 (pp. 110, 111); Sec. =9=, 1-4, 26, 27, 40, 48 (pp. 112,
116, 119, 121).]
pg094
The Reader, who has successfully grappled with all the Examples hitherto
set, and who thirsts, like Alexander the Great, for "more worlds to
conquer," may employ his spare energies on the following 17
Examination-Papers. He is recommended not to attempt more than _one_
Paper on any one day. The answers to the questions about words and
phrases may be found by referring to the Index at p. 197.
I. Sec. =4=, 31 (p. 100); Sec. =5=, 31-34 (p. 102); Sec. =6=, 16, 17 (p.
106); Sec. =7=, 16 (p. 108); Sec. =8=, 5, 6 (p. 110); Sec. =9=, 5, 22,
42 (pp. 112, 115, 119). What is 'Classification'? And what is
a 'Class'?
II. Sec. =4=, 32 (p. 100); Sec. =5=, 35-38 (pp. 102, 103); Sec. =6=, 18
(p. 107); Sec. =7=, 17, 18 (p. 108); Sec. =8=, 7, 8 (p. 110);
Sec. =9=, 6, 23, 43 (pp. 112, 115, 119). What are 'Genus',
'Species', and 'Differentia'?
III. Sec. =4=, 33 (p. 100); Sec. =5=, 39-42 (p. 103); Sec. =6=, 19, 20 (p.
107); Sec. =7=, 19 (p. 109); Sec. =8=, 9, 10 (p. 111); Sec. =9=, 7,
24, 44 (pp. 113, 116, 120). What are 'Real' and 'Imaginary'
Classes?
IV. Sec. =4=, 34 (p. 100); Sec. =5=, 43-46 (p. 103); Sec. =6=, 21 (p.
107); Sec. =7=, 20, 21 (p. 109); Sec. =8=, 11, 12 (p. 111); Sec. =9=,
8, 25, 45 (pp. 113, 116, 120). What is 'Division'? When are
Classes said to be 'Codivisional'?
V. Sec. =4=, 35 (p. 100); Sec. =5=, 47-50 (p. 103); Sec. =6=, 22, 23 (p.
107); Sec. =7=, 22 (p. 109); Sec. =8=, 15, 16 (p. 111); Sec. =9=, 9,
28, 46 (pp. 113, 116, 120). What is 'Dichotomy'? What
arbitrary rule does it sometimes require?
pg095
VI. Sec. =4=, 36 (p. 100); Sec. =5=, 51-54 (p. 103); Sec. =6=, 24 (p.
107); Sec. =7=, 23, 24 (p. 109); Sec. =8=, 17 (p. 111); Sec. =9=, 10,
29, 47 (pp. 113, 117, 120). What is a 'Definition'?
VII. Sec. =4=, 37 (p. 100); Sec. =5=, 55-58 (pp. 103, 104); Sec. =6=, 25,
26 (p. 107); Sec. =7=, 25 (p. 109); Sec. =8=, 18 (p. 111); Sec. =9=,
11, 30, 49 (pp. 113, 117, 121). What are the 'Subject' and
the 'Predicate' of a Proposition? What is its 'Normal' form?
VIII. Sec. =4=, 38 (p. 100); Sec. =5=, 59-62 (p. 104); Sec. =6=, 27 (p.
107); Sec. =7=, 26, 27 (p. 109); Sec. =8=, 20 (p. 111); Sec. =9=, 12,
31, 50 (pp. 113, 117, 121). What is a Proposition 'in _I_'?
'In _E_'? And 'in _A_'?
IX. Sec. =4=, 39 (p. 100); Sec. =5=, 63-66 (p. 104); Sec. =6=, 28, 29 (p.
107); Sec. =7=, 28 (p. 109); Sec. =8=, 21 (p. 111); Sec. =9=, 13, 32,
51 (pp. 114, 117, 121). What is the 'Normal' form of a
Proposition of Existence?
X. Sec. =4=, 40 (p. 100); Sec. =5=, 67-70 (p. 104); Sec. =6=, 30 (p.
107); Sec. =7=, 29, 30 (p. 109); Sec. =8=, 22 (p. 111); Sec. =9=, 14,
33, 52 (pp. 114, 117, 122). What is the 'Universe of
Discourse'?
XI. Sec. =4=, 41 (p. 100); Sec. =5=, 71-74 (p. 104); Sec. =6=, 31, 32 (p.
107); Sec. =7=, 31 (p. 109); Sec. =8=, 23 (p. 111); Sec. =9=, 15, 34,
53 (pp. 114, 118, 122). What is implied, in a Proposition of
Relation, as to the Reality of its Terms?
XII. Sec. =4=, 42 (p. 100); Sec. =5=, 75-78 (p. 105); Sec. =6=, 33 (p.
107); Sec. =7=, 32, 33 (pp. 109, 110); Sec. =8=, 25 (p. 111);
Sec. =9=, 16, 35, 54 (pp. 114, 118, 122). Explain the phrase
"sitting on the fence".
XIII. Sec. =5=, 79-83 (p. 105); Sec. =6=, 34, 35 (p. 107); Sec. =7=, 34 (p.
110); Sec. =8=, 26 (p. 111); Sec. =9=, 17, 36, 55 (pp. 114, 118,
122). What are 'Converse' Propositions?
XIV. Sec. =5=, 84-88 (p. 105); Sec. =6=, 36 (p. 107); Sec. =7=, 35, 36 (p.
110); Sec. =8=, 27 (p. 111); Sec. =9=, 18, 37, 56 (pp. 114, 118,
123). What are 'Concrete' and 'Abstract' Propositions?
pg096
XV. Sec. =5=, 89-93 (p. 105); Sec. =6=, 37, 38 (p. 107); Sec. =7=, 37 (p.
110); Sec. =8=, 28 (p. 111); Sec. =9=, 19, 38, 57 (pp. 115, 118,
123). What is a 'Syllogism'? And what are its 'Premisses' and
its 'Conclusion'?
XVI. Sec. =5=, 94-97 (p. 106); Sec. =6=, 39 (p. 107); Sec. =7=, 38, 39 (p.
110); Sec. =8=, 29 (p. 111); Sec. =9=, 20, 39, 58 (pp. 115, 119,
123). What is a 'Sorites'? And what are its 'Premisses',
its 'Partial Conclusions', and its 'Complete Conclusion'?
XVII. Sec. =5=, 98-101 (p. 106); Sec. =6=, 40 (p. 107); Sec. =7=, 40 (p.
110); Sec. =8=, 30 (p. 111); Sec. =9=, 21, 41, 59, 60 (pp. 115,
119, 124). What are the 'Universe of Discourse', the
'Eliminands', and the 'Retinends', of a Syllogism? And of a
Sorites?
pg097
BOOK VIII.
EXAMPLES, ANSWERS, AND SOLUTIONS.
[N.B. Reference tags for Examples, Answers & Solutions will be
found in the right margin.]
CHAPTER I.
_EXAMPLES._
Sec. 1. EX1
_Propositions of Relation, to be reduced to normal form._
1. I have been out for a walk.
2. I am feeling better.
3. No one has read the letter but John.
4. Neither you nor I are old.
5. No fat creatures run well.
6. None but the brave deserve the fair.
7. No one looks poetical unless he is pale.
8. Some judges lose their tempers.
9. I never neglect important business.
10. What is difficult needs attention.
11. What is unwholesome should be avoided.
12. All the laws passed last week relate to excise.
13. Logic puzzles me.
14. There are no Jews in the house.
15. Some dishes are unwholesome if not well-cooked.
16. Unexciting books make one drowsy.
17. When a man knows what he's about, he can detect a sharper.
18. You and I know what we're about.
19. Some bald people wear wigs.
20. Those who are fully occupied never talk about their grievances.
21. No riddles interest me if they can be solved.
pg098
Sec. 2. EX2
_Pairs of Abstract Propositions, one in terms of x and m, and the other
in terms of y and m, to be represented on the same Triliteral Diagram._
1. No x are m;
No m' are y.
2. No x' are m';
All m' are y.
3. Some x' are m;
No m are y.
4. All m are x;
All m' are y'.
5. All m' are x;
All m' are y'.
6. All x' are m';
No y' are m.
7. All x are m;
All y' are m'.
8. Some m' are x';
No m are y.
9. All m are x';
No m are y.
10. No m are x';
No y are m'.
11. No x' are m';
No m are y.
12. Some x are m;
All y' are m.
13. All x' are m;
No m are y.
14. Some x are m';
All m are y.
15. No m' are x';
All y are m.
16. All x are m';
No y are m.
17. Some m' are x;
No m' are y'.
18. All x are m';
Some m' are y'.
19. All m are x;
Some m are y'.
20. No x' are m;
Some y are m.
21. Some x' are m';
All y' are m.
22. No m are x;
Some m are y.
23. No m' are x;
All y are m'.
24. All m are x;
No y' are m'.
25. Some m are x;
No y' are m.
26. All m' are x';
Some y are m'.
27. Some m are x';
No y' are m'.
28. No x are m';
All m are y'.
29. No x' are m;
No m are y'.
30. No x are m;
Some y' are m'.
31. Some m' are x;
All y' are m;
32. All x are m';
All y are m.
pg099
Sec. 3. EX3
_Marked Triliteral Diagrams, to be interpreted in terms of x and y._
1
.---------------.
|(O) | |
| .---|---. |
| |(I)|(O)| |
|---|---|---|---|
| | |(O)| |
| .---|---. |
|(O) | |
.---------------.
2
.---------------.
|(O) | |
| .---|---. |
| | |(O)| |
|---|---|---|(I)|
| | |(O)| |
| .---|---. |
|(O) | |
.---------------.
3
.---------------.
| | (O)|
| .---|---. |
| |(O)|(O)| |
|---|---|---|---|
| | |(I)| |
| .---|---. |
| | (O)|
.---------------.
4
.---------------.
|(O) | (O)|
| .---|---. |
| |(O)| | |
|---|---|(I)|---|
| |(O)| | |
| .---|---. |
| | |
.---------------.
5
.---------------.
| | (I)|
| .---|---. |
| | |(O)| |
|---|---|---|---|
| | |(O)| |
| .---|---. |
|(O) | (O)|
.---------------.
6
.---------------.
|(O) | (O)|
| .---|---. |
| | | | |
|---|---|---|---|
| | |(O)| |
| .---|---. |
|(I) | (O)|
.---------------.
7
.---------------.
|(O) | (O)|
| .---|---. |
| |(I)|(O)| |
|---|---|---|---|
| | |(O)| |
| .---|---. |
| | |
.---------------.
8
.---------------.
|(I) | |
| .---|---. |
| |(O)| | |
|---|---|---|---|
| |(O)|(I)| |
| .---|---. |
|(O) | |
.---------------.
9
.---------------.
| | |
| .---|---. |
| |(O)| | |
|---|---|---|---|
| |(O)|(I)| |
| .---|---. |
|(O) | (O)|
.---------------.
10
.---------------.
|(O) | (O)|
| .---|---. |
| |(O)|(I)| |
|---|---|---|---|
| |(O)| | |
| .---|---. |
| | |
.---------------.
11
.---------------.
|(O) | (O)|
| .---|---. |
| | (I) | |
|---|---|---|---|
| |(O)| | |
| .---|---. |
| | (O)|
.---------------.
12
.---------------.
|(O) | |
| .---|---. |
| | | | |
|---|(I)|---|---|
| | | | |
| .---|---. |
|(O) | (I)|
.---------------.
13
.---------------.
|(O) | (I)|
| .---|---. |
| | |(O)| |
|---|---|---|---|
| | |(O)| |
| .---|---. |
|(O) | |
.---------------.
14
.---------------.
|(O) | (O)|
| .---|---. |
| | |(O)| |
|---|---|---|---|
| | |(O)| |
| .---|---. |
| (I) |
.---------------.
15
.---------------.
| | |
| .---|---. |
| |(I)| | |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
| | |
.---------------.
16
.---------------.
|(I) | |
| .---|---. |
| | | | |
|---|---|---|---|
| |(O)| | |
| .---|---. |
|(O) | (O)|
.---------------.
17
.---------------.
| | (O)|
| .---|---. |
| | |(I)| |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
|(I) | (O)|
.---------------.
18
.---------------.
|(O) | (O)|
| .---|---. |
| |(O)|(I)| |
|---|---|---|---|
| |(O)| | |
| .---|---. |
|(I) | |
.---------------.
19
.---------------.
|(O) | (O)|
| .---|---. |
| |(I)|(O)| |
|---|---|---|---|
| | |(O)| |
| .---|---. |
| | (I)|
.---------------.
20
.---------------.
|(O) | |
| .---|---. |
| |(O)|(O)| |
|---|---|---|---|
| |(I)| | |
| .---|---. |
|(O) | |
.---------------.
pg100
Sec. 4. EX4
_Pairs of Abstract Propositions, proposed as Premisses: Conclusions to
be found._
1. No m are x';
All m' are y.
2. No m' are x;
Some m' are y'.
3. All m' are x;
All m' are y'.
4. No x' are m';
All y' are m.
5. Some m are x';
No y are m.
6. No x' are m;
No m are y.
7. No m are x';
Some y' are m.
8. All m' are x';
No m' are y.
9. Some x' are m';
No m are y'.
10. All x are m;
All y' are m'.
11. No m are x;
All y' are m'.
12. No x are m;
All y are m.
13. All m' are x;
No y are m.
14. All m are x;
All m' are y.
15. No x are m;
No m' are y.
16. All x are m';
All y are m.
17. No x are m;
All m' are y.
18. No x are m';
No m are y.
19. All m are x;
All m are y'.
20. No m are x;
All m' are y.
21. All x are m;
Some m' are y.
22. Some x are m;
All y are m.
23. All m are x;
Some y are m.
24. No x are m;
All y are m.
25. Some m are x';
All y' are m'.
26. No m are x';
All y are m.
27. All x are m';
All y' are m.
28. All m are x';
Some m are y.
29. No m are x;
All y are m'.
30. All x are m';
Some y are m.
31. All x are m;
All y are m.
32. No x are m';
All m are y.
33. No m are x;
No m are y.
34. No m are x';
Some y are m.
35. No m are x;
All y are m.
36. All m are x';
Some y are m.
37. All m are x;
No y are m.
38. No m are x;
No m' are y.
39. Some m are x';
No m are y.
40. No x' are m;
All y' are m.
41. All x are m';
No y are m'.
42. No m' are x;
No y are m.
pg101
Sec. 5. EX5
_Pairs of Concrete Propositions, proposed as Premisses: Conclusions to
be found._
1. I have been out for a walk;
I am feeling better.
2. No one has read the letter but John;
No one, who has _not_ read it, knows what it is about.
3. Those who are not old like walking;
You and I are young.
4. Your course is always honest;
Your course is always the best policy.
5. No fat creatures run well;
Some greyhounds run well.
6. Some, who deserve the fair, get their deserts;
None but the brave deserve the fair.
7. Some Jews are rich;
All Esquimaux are Gentiles.
8. Sugar-plums are sweet;
Some sweet things are liked by children.
9. John is in the house;
Everybody in the house is ill.
10. Umbrellas are useful on a journey;
What is useless on a journey should be left behind.
11. Audible music causes vibration in the air;
Inaudible music is not worth paying for.
12. Some holidays are rainy;
Rainy days are tiresome.
13. No Frenchmen like plumpudding;
All Englishmen like plumpudding.
14. No portrait of a lady, that makes her simper or scowl, is
satisfactory;
No photograph of a lady ever fails to make her simper or scowl.
15. All pale people are phlegmatic;
No one looks poetical unless he is pale.
16. No old misers are cheerful;
Some old misers are thin.
17. No one, who exercises self-control, fails to keep his temper;
Some judges lose their tempers.
pg102
18. All pigs are fat;
Nothing that is fed on barley-water is fat.
19. All rabbits, that are not greedy, are black;
No old rabbits are free from greediness.
20. Some pictures are not first attempts;
No first attempts are really good.
21. I never neglect important business;
Your business is unimportant.
22. Some lessons are difficult;
What is difficult needs attention.
23. All clever people are popular;
All obliging people are popular.
24. Thoughtless people do mischief;
No thoughtful person forgets a promise.
25. Pigs cannot fly;
Pigs are greedy.
26. All soldiers march well;
Some babies are not soldiers.
27. No bride-cakes are wholesome;
What is unwholesome should be avoided.
28. John is industrious;
No industrious people are unhappy.
29. No philosophers are conceited;
Some conceited persons are not gamblers.
30. Some excise laws are unjust;
All the laws passed last week relate to excise.
31. No military men write poetry;
None of my lodgers are civilians.
32. No medicine is nice;
Senna is a medicine.
33. Some circulars are not read with pleasure;
No begging-letters are read with pleasure.
34. All Britons are brave;
No sailors are cowards.
35. Nothing intelligible ever puzzles _me_;
Logic puzzles me.
36. Some pigs are wild;
All pigs are fat.
pg103
37. All wasps are unfriendly;
All unfriendly creatures are unwelcome.
38. No old rabbits are greedy;
All black rabbits are greedy.
39. Some eggs are hard-boiled;
No eggs are uncrackable.
40. No antelope is ungraceful;
Graceful creatures delight the eye.
41. All well-fed canaries sing loud;
No canary is melancholy if it sings loud.
42. Some poetry is original;
No original work is producible at will.
43. No country, that has been explored, is infested by dragons;
Unexplored countries are fascinating.
44. No coals are white;
No niggers are white.
45. No bridges are made of sugar;
Some bridges are picturesque.
46. No children are patient;
No impatient person can sit still.
47. No quadrupeds can whistle;
Some cats are quadrupeds.
48. Bores are terrible;
You are a bore.
49. Some oysters are silent;
No silent creatures are amusing.
50. There are no Jews in the house;
No Gentiles have beards a yard long.
51. Canaries, that do not sing loud, are unhappy;
No well-fed canaries fail to sing loud.
52. All my sisters have colds;
No one can sing who has a cold.
53. All that is made of gold is precious;
Some caskets are precious.
54. Some buns are rich;
All buns are nice.
55. All my cousins are unjust;
All judges are just.
pg104
56. Pain is wearisome;
No pain is eagerly wished for.
57. All medicine is nasty;
Senna is a medicine.
58. Some unkind remarks are annoying;
No critical remarks are kind.
59. No tall men have woolly hair;
Niggers have woolly hair.
60. All philosophers are logical;
An illogical man is always obstinate.
61. John is industrious;
All industrious people are happy.
62. These dishes are all well-cooked;
Some dishes are unwholesome if not well-cooked.
63. No exciting books suit feverish patients;
Unexciting books make one drowsy.
64. No pigs can fly;
All pigs are greedy.
65. When a man knows what he's about, he can detect a sharper;
You and I know what we're about.
66. Some dreams are terrible;
No lambs are terrible.
67. No bald creature needs a hairbrush;
No lizards have hair.
68. All battles are noisy;
What makes no noise may escape notice.
69. All my cousins are unjust;
No judges are unjust.
70. All eggs can be cracked;
Some eggs are hard-boiled.
71. Prejudiced persons are untrustworthy;
Some unprejudiced persons are disliked.
72. No dictatorial person is popular;
She is dictatorial.
73. Some bald people wear wigs;
All your children have hair.
74. No lobsters are unreasonable;
No reasonable creatures expect impossibilities.
pg105
75. No nightmare is pleasant;
Unpleasant experiences are not eagerly desired.
76. No plumcakes are wholesome;
Some wholesome things are nice.
77. Nothing that is nice need be shunned;
Some kinds of jam are nice.
78. All ducks waddle;
Nothing that waddles is graceful.
79. Sandwiches are satisfying;
Nothing in this dish is unsatisfying.
80. No rich man begs in the street;
Those who are not rich should keep accounts.
81. Spiders spin webs;
Some creatures, that do not spin webs, are savage.
82. Some of these shops are not crowded;
No crowded shops are comfortable.
83. Prudent travelers carry plenty of small change;
Imprudent travelers lose their luggage.
84. Some geraniums are red;
All these flowers are red.
85. None of my cousins are just;
All judges are just.
86. No Jews are mad;
All my lodgers are Jews.
87. Busy folk are not always talking about their grievances;
Discontented folk are always talking about their grievances.
88. None of my cousins are just;
No judges are unjust.
89. All teetotalers like sugar;
No nightingale drinks wine.
90. No riddles interest me if they can be solved;
All these riddles are insoluble.
91. All clear explanations are satisfactory;
Some excuses are unsatisfactory.
92. All elderly ladies are talkative;
All good-tempered ladies are talkative.
93. No kind deed is unlawful;
What is lawful may be done without scruple.
pg106
94. No babies are studious;
No babies are good violinists.
95. All shillings are round;
All these coins are round.
96. No honest men cheat;
No dishonest men are trustworthy.
97. None of my boys are clever;
None of my girls are greedy.
98. All jokes are meant to amuse;
No Act of Parliament is a joke.
99. No eventful tour is ever forgotten;
Uneventful tours are not worth writing a book about.
100. All my boys are disobedient;
All my girls are discontented.
101. No unexpected pleasure annoys me;
Your visit is an unexpected pleasure.
Sec. 6. EX6
_Trios of Abstract Propositions, proposed as Syllogisms: to be
examined._
1. Some x are m; No m are y'. Some x are y.
2. All x are m; No y are m'. No y are x'.
3. Some x are m'; All y' are m. Some x are y.
4. All x are m; No y are m. All x are y'.
5. Some m' are x'; No m' are y. Some x' are y'.
6. No x' are m; All y are m'. All y are x'.
7. Some m' are x'; All y' are m'. Some x' are y'.
8. No m' are x'; All y' are m'. All y' are x.
9. Some m are x'; No m are y. Some x' are y'.
10. All m' are x'; All m' are y. Some y are x'.
11. All x are m'; Some y are m. Some y are x'.
12. No x are m; No m' are y'. No x are y'.
13. No x are m; All y' are m. All y' are x'.
14. All m' are x'; All m' are y. Some y are x'.
15. Some m are x'; All y are m'. Some x' are y'.
16. No x' are m; All y' are m'. Some y' are x.
17. No m' are x; All m' are y'. Some x' are y'.
pg107
18. No x' are m; Some m are y. Some x are y.
19. Some m are x; All m are y. Some y are x'.
20. No x' are m'; Some m' are y'. Some x are y'.
21. No m are x; All m are y'. Some x' are y'.
22. All x' are m; Some y are m'. All x' are y'.
23. All m are x; No m' are y'. No x' are y'.
24. All x are m'; All m' are y. All x are y.
25. No x are m'; All m are y. No x are y'.
26. All m are x'; All y are m. All y are x'.
27. All x are m; No m are y'. All x are y.
28. All x are m; No y' are m'. All x are y.
29. No x' are m; No m' are y'. No x' are y'.
30. All x are m; All m are y'. All x are y'.
31. All x' are m'; No y' are m'. All x' are y.
32. No x are m; No y' are m'. No x are y'.
33. All m are x'; All y' are m. All y' are x'.
34. All x are m'; Some y are m'. Some y are x.
35. Some x are m; All m are y. Some x are y.
36. All m are x'; All y are m. All y are x'.
37. No m are x'; All m are y'. Some x are y'.
38. No x are m; No m are y'. No x are y'.
39. No m are x; Some m are y'. Some x' are y'.
40. No m are x'; Some y are m. Some x are y.
Sec. 7. EX7
_Trios of Concrete Propositions, proposed as Syllogisms: to be
examined._
1. No doctors are enthusiastic;
You are enthusiastic.
You are not a doctor.
2. Dictionaries are useful;
Useful books are valuable.
Dictionaries are valuable.
3. No misers are unselfish;
None but misers save egg-shells.
No unselfish people save egg-shells.
4. Some epicures are ungenerous;
All my uncles are generous.
My uncles are not epicures.
pg108
5. Gold is heavy;
Nothing but gold will silence him.
Nothing light will silence him.
6. Some healthy people are fat;
No unhealthy people are strong.
Some fat people are not strong.
7. "I saw it in a newspaper."
"All newspapers tell lies."
It was a lie.
8. Some cravats are not artistic;
I admire anything artistic.
There are some cravats that I do not admire.
9. His songs never last an hour;
A song, that lasts an hour, is tedious.
His songs are never tedious.
10. Some candles give very little light;
Candles are _meant_ to give light.
Some things, that are meant to give light, give very little.
11. All, who are anxious to learn, work hard;
Some of these boys work hard.
Some of these boys are anxious to learn.
12. All lions are fierce;
Some lions do not drink coffee.
Some creatures that drink coffee are not fierce.
13. No misers are generous;
Some old men are ungenerous.
Some old men are misers.
14. No fossil can be crossed in love;
An oyster may be crossed in love.
Oysters are not fossils.
15. All uneducated people are shallow;
Students are all educated.
No students are shallow.
16. All young lambs jump;
No young animals are healthy, unless they jump.
All young lambs are healthy.
17. Ill-managed business is unprofitable;
Railways are never ill-managed.
All railways are profitable.
18. No Professors are ignorant;
All ignorant people are vain.
No professors are vain.
pg109
19. A prudent man shuns hyaenas;
No banker is imprudent.
No banker fails to shun hyaenas.
20. All wasps are unfriendly;
No puppies are unfriendly.
Puppies are not wasps.
21. No Jews are honest;
Some Gentiles are rich.
Some rich people are dishonest.
22. No idlers win fame;
Some painters are not idle.
Some painters win fame.
23. No monkeys are soldiers;
All monkeys are mischievous.
Some mischievous creatures are not soldiers.
24. All these bonbons are chocolate-creams;
All these bonbons are delicious.
Chocolate-creams are delicious.
25. No muffins are wholesome;
All buns are unwholesome.
Buns are not muffins.
26. Some unauthorised reports are false;
All authorised reports are trustworthy.
Some false reports are not trustworthy.
27. Some pillows are soft;
No pokers are soft.
Some pokers are not pillows.
28. Improbable stories are not easily believed;
None of his stories are probable.
None of his stories are easily believed.
29. No thieves are honest;
Some dishonest people are found out.
Some thieves are found out.
30. No muffins are wholesome;
All puffy food is unwholesome.
All muffins are puffy.
31. No birds, except peacocks, are proud of their tails;
Some birds, that are proud of their tails, cannot sing.
Some peacocks cannot sing.
32. Warmth relieves pain;
Nothing, that does not relieve pain, is useful in toothache.
Warmth is useful in toothache.
pg110
33. No bankrupts are rich;
Some merchants are not bankrupts.
Some merchants are rich.
34. Bores are dreaded;
No bore is ever begged to prolong his visit.
No one, who is dreaded, is ever begged to prolong his visit.
35. All wise men walk on their feet;
All unwise men walk on their hands.
No man walks on both.
36. No wheelbarrows are comfortable;
No uncomfortable vehicles are popular.
No wheelbarrows are popular.
37. No frogs are poetical;
Some ducks are unpoetical.
Some ducks are not frogs.
38. No emperors are dentists;
All dentists are dreaded by children.
No emperors are dreaded by children.
39. Sugar is sweet;
Salt is not sweet.
Salt is not sugar.
40. Every eagle can fly;
Some pigs cannot fly.
Some pigs are not eagles.
Sec. 8. EX8
_Sets of Abstract Propositions, proposed as Premisses for Soriteses:
Conclusions to be found._
[N.B. At the end of this Section instructions are given for
varying these Examples.]
1.
1. No c are d;
2. All a are d;
3. All b are c.
2.
1. All d are b;
2. No a are c';
3. No b are c.
3.
1. No b are a;
2. No c are d';
3. All d are b.
4.
1. No b are c;
2. All a are b;
3. No c' are d.
5.
1. All b' are a';
2. No b are c;
3. No a' are d.
6.
1. All a are b';
2. No b' are c;
3. All d are a.
7.
1. No d are b';
2. All b are a;
3. No c are d'.
8.
1. No b' are d;
2. No a' are b;
3. All c are d.
pg111
9.
1. All b' are a;
2. No a are d;
3. All b are c.
10.
1. No c are d;
2. All b are c;
3. No a are d'.
11.
1. No b are c;
2. All d are a;
3. All c' are a'.
12.
1. No c are b';
2. All c' are d';
3. All b are a.
13.
1. All d are e;
2. All c are a;
3. No b are d';
4. All e are a'.
14.
1. All e are b;
2. All a are e;
3. All d are b';
4. All a' are c;
15.
1. No b' are d;
2. All e are c;
3. All b are a;
4. All d' are c'.
16.
1. No a' are e;
2. All d are c';
3. All a are b;
4. All e' are d.
17.
1. All d are c;
2. All a are e;
3. No b are d';
4. All c are e'.
18.
1. All a are b;
2. All d are e;
3. All a' are c';
4. No b are e.
19.
1. No b are c;
2. All e are h;
3. All a are b;
4. No d are h;
5. All e' are c.
20.
1. No d are h';
2. No c are e;
3. All h are b;
4. No a are d';
5. No b are e'.
21.
1. All b are a;
2. No d are h;
3. No c are e;
4. No a are h';
5. All c' are b.
22.
1. All e are d';
2. No b' are h';
3. All c' are d;
4. All a are e;
5. No c are h.
23.
1. All b' are a';
2. No d are e';
3. All h are b';
4. No c are e;
5. All d' are a.
24.
1. All h' are k';
2. No b' are a;
3. All c are d;
4. All e are h';
5. No d are k';
6. No b are c'.
25.
1. All a are d;
2. All k are b;
3. All e are h;
4. No a' are b;
5. All d are c;
6. All h are k.
26.
1. All a' are h;
2. No d' are k';
3. All e are b';
4. No h are k;
5. All a are e;
6. No b' are d.
27.
1. All c are d';
2. No h are b;
3. All a' are k;
4. No c are e';
5. All b' are d;
6. No a are c'.
28.
1. No a' are k;
2. All e are b;
3. No h are k';
4. No d' are c;
5. No a are b;
6. All c' are h.
29.
1. No e are k;
2. No b' are m;
3. No a are c';
4. All h' are e;
5. All d are k;
6. No c are b;
7. All d' are l;
8. No h are m'.
30.
1. All n are m;
2. All a' are e;
3. No c' are l;
4. All k are r';
5. No a are h';
6. No d are l';
7. No c are n';
8. All e are b;
9. All m are r;
10. All h are d.
[N.B. In each Example, in Sections 8 and 9, it is possible to
begin with _any_ Premiss, at pleasure, and thus to get as many
different Solutions (all of course yielding the _same_ Complete
Conclusion) as there are Premisses in the Example. Hence Sec. 8
really contains 129 different Examples, and Sec. 9 contains 273.]
pg112
Sec. 9. EX9
_Sets of Concrete Propositions, proposed as Premisses for Soriteses:
Conclusions to be found._
=1.=
(1) Babies are illogical;
(2) Nobody is despised who can manage a crocodile;
(3) Illogical persons are despised.
Univ. "persons"; a = able to manage a crocodile; b = babies;
c = despised; d = logical.
=2.=
(1) My saucepans are the only things I have that are made of tin;
(2) I find all _your_ presents very useful;
(3) None of my saucepans are of the slightest use.
Univ. "things of mine"; a = made of tin; b = my saucepans; c = useful;
d = your presents.
=3.=
(1) No potatoes of mine, that are new, have been boiled;
(2) All my potatoes in this dish are fit to eat;
(3) No unboiled potatoes of mine are fit to eat.
Univ. "my potatoes"; a = boiled; b = eatable; c = in this dish; d = new.
=4.=
(1) There are no Jews in the kitchen;
(2) No Gentiles say "shpoonj";
(3) My servants are all in the kitchen.
Univ. "persons"; a = in the kitchen; b = Jews; c = my servants;
d = saying "shpoonj."
=5.=
(1) No ducks waltz;
(2) No officers ever decline to waltz;
(3) All my poultry are ducks.
Univ. "creatures"; a = ducks; b = my poultry; c = officers; d = willing
to waltz.
=6.=
(1) Every one who is sane can do Logic;
(2) No lunatics are fit to serve on a jury;
(3) None of _your_ sons can do Logic.
Univ. "persons"; a = able to do Logic; b = fit to serve on a jury;
c = sane; d = your sons.
pg113
=7.=
(1) There are no pencils of mine in this box;
(2) No sugar-plums of mine are cigars;
(3) The whole of my property, that is not in this box, consists of
cigars.
Univ. "things of mine"; a = cigars; b = in this box; c = pencils;
d = sugar-plums.
=8.=
(1) No experienced person is incompetent;
(2) Jenkins is always blundering;
(3) No competent person is always blundering.
Univ. "persons"; a = always blundering; b = competent; c = experienced;
d = Jenkins.
=9.=
(1) No terriers wander among the signs of the zodiac;
(2) Nothing, that does not wander among the signs of the zodiac, is a
comet;
(3) Nothing but a terrier has a curly tail.
Univ. "things"; a = comets; b = curly-tailed; c = terriers;
d = wandering among the signs of the zodiac.
=10.=
(1) No one takes in the _Times_, unless he is well-educated;
(2) No hedge-hogs can read;
(3) Those who cannot read are not well-educated.
Univ. "creatures"; a = able to read; b = hedge-hogs; c = taking in the
Times; d = well-educated.
=11.=
(1) All puddings are nice;
(2) This dish is a pudding;
(3) No nice things are wholesome.
Univ. "things"; a = nice; b = puddings; c = this dish; d = wholesome.
=12.=
(1) My gardener is well worth listening to on military subjects;
(2) No one can remember the battle of Waterloo, unless he is very old;
(3) Nobody is really worth listening to on military subjects, unless he
can remember the battle of Waterloo.
Univ. "persons"; a = able to remember the battle of Waterloo; b = my
gardener; c = well worth listening to on military subjects; d = very
old.
pg114
=13.=
(1) All humming-birds are richly coloured;
(2) No large birds live on honey;
(3) Birds that do not live on honey are dull in colour.
Univ. "birds"; a = humming-birds; b = large; c = living on honey;
d = richly coloured.
=14.=
(1) No Gentiles have hooked noses;
(2) A man who is a good hand at a bargain always makes money;
(3) No Jew is ever a bad hand at a bargain.
Univ. "persons"; a = good hands at a bargain; b = hook-nosed; c = Jews;
d = making money.
=15.=
(1) All ducks in this village, that are branded 'B,' belong to Mrs.
Bond;
(2) Ducks in this village never wear lace collars, unless they are
branded 'B';
(3) Mrs. Bond has no gray ducks in this village.
Univ. "ducks in this village"; a = belonging to Mrs. Bond; b = branded
'B'; c = gray; d = wearing lace-collars.
=16.=
(1) All the old articles in this cupboard are cracked;
(2) No jug in this cupboard is new;
(3) Nothing in this cupboard, that is cracked, will hold water.
Univ. "things in this cupboard"; a = able to hold water; b = cracked;
c = jugs; d = old.
=17.=
(1) All unripe fruit is unwholesome;
(2) All these apples are wholesome;
(3) No fruit, grown in the shade, is ripe.
Univ. "fruit"; a = grown in the shade; b = ripe; c = these apples;
d = wholesome.
=18.=
(1) Puppies, that will not lie still, are always grateful for the loan
of a skipping-rope;
(2) A lame puppy would not say "thank you" if you offered to lend it a
skipping-rope.
(3) None but lame puppies ever care to do worsted-work.
Univ. "puppies"; a = caring to do worsted-work; b = grateful for the
loan of a skipping-rope; c = lame; d = willing to lie still.
pg115
=19.=
(1) No name in this list is unsuitable for the hero of a romance;
(2) Names beginning with a vowel are always melodious;
(3) No name is suitable for the hero of a romance, if it begins with a
consonant.
Univ. "names"; a = beginning with a vowel; b = in this list;
c = melodious; d = suitable for the hero of a romance.
=20.=
(1) All members of the House of Commons have perfect self-command;
(2) No M.P., who wears a coronet, should ride in a donkey-race;
(3) All members of the House of Lords wear coronets.
Univ. "M.P.'s"; a = belonging to the House of Commons; b = having
perfect self-command; c = one who may ride in a donkey-race; d = wearing
a coronet.
=21.=
(1) No goods in this shop, that have been bought and paid for, are still
on sale;
(2) None of the goods may be carried away, unless labeled "sold";
(3) None of the goods are labeled "sold," unless they have been bought
and paid for.
Univ. "goods in this shop"; a = allowed to be carried away; b = bought
and paid for; c = labeled "sold"; d = on sale.
=22.=
(1) No acrobatic feats, that are not announced in the bills of a circus,
are ever attempted there;
(2) No acrobatic feat is possible, if it involves turning a quadruple
somersault;
(3) No impossible acrobatic feat is ever announced in a circus bill.
Univ. "acrobatic feats"; a = announced in the bills of a circus;
b = attempted in a circus; c = involving the turning of a quadruple
somersault; d = possible.
=23.=
(1) Nobody, who really appreciates Beethoven, fails to keep silence
while the Moonlight-Sonata is being played;
(2) Guinea-pigs are hopelessly ignorant of music;
(3) No one, who is hopelessly ignorant of music, ever keeps silence
while the Moonlight-Sonata is being played.
Univ. "creatures"; a = guinea-pigs; b = hopelessly ignorant of music;
c = keeping silence while the Moonlight-Sonata is being played;
d = really appreciating Beethoven.
pg116
=24.=
(1) Coloured flowers are always scented;
(2) I dislike flowers that are not grown in the open air;
(3) No flowers grown in the open air are colourless.
Univ. "flowers"; a = coloured; b = grown in the open air; c = liked by
me; d = scented.
=25.=
(1) Showy talkers think too much of themselves;
(2) No really well-informed people are bad company;
(3) People who think too much of themselves are not good company.
Univ. "persons"; a = good company; b = really well-informed; c = showy
talkers; d = thinking too much of one's self.
=26.=
(1) No boys under 12 are admitted to this school as boarders;
(2) All the industrious boys have red hair;
(3) None of the day-boys learn Greek;
(4) None but those under 12 are idle.
Univ. "boys in this school"; a = boarders; b = industrious; c = learning
Greek; d = red-haired; e = under 12.
=27.=
(1) The only articles of food, that my doctor allows me, are such as are
not very rich;
(2) Nothing that agrees with me is unsuitable for supper;
(3) Wedding-cake is always very rich;
(4) My doctor allows me all articles of food that are suitable for
supper.
Univ. "articles of food"; a = agreeing with me; b = allowed by my
doctor; c = suitable for supper; d = very rich; e = wedding-cake.
=28.=
(1) No discussions in our Debating-Club are likely to rouse the British
Lion, so long as they are checked when they become too noisy;
(2) Discussions, unwisely conducted, endanger the peacefulness of our
Debating-Club;
(3) Discussions, that go on while Tomkins is in the Chair, are likely to
rouse the British Lion;
(4) Discussions in our Debating-Club, when wisely conducted, are always
checked when they become too noisy.
Univ. "discussions in our Debating-Club"; a = checked when too noisy;
b = dangerous to the peacefulness of our Debating-Club; c = going on
while Tomkins is in the chair; d = likely to rouse the British Lion;
e = wisely conducted.
pg117
=29.=
(1) All my sons are slim;
(2) No child of mine is healthy who takes no exercise;
(3) All gluttons, who are children of mine, are fat;
(4) No daughter of mine takes any exercise.
Univ. "my children"; a = fat; b = gluttons; c = healthy; d = sons;
e = taking exercise.
=30.=
(1) Things sold in the street are of no great value;
(2) Nothing but rubbish can be had for a song;
(3) Eggs of the Great Auk are very valuable;
(4) It is only what is sold in the street that is really _rubbish_.
Univ. "things"; a = able to be had for a song; b = eggs of the Great
Auk; c = rubbish; d = sold in the street; e = very valuable.
=31.=
(1) No books sold here have gilt edges, except what are in the front
shop;
(2) All the _authorised_ editions have red labels;
(3) All the books with red labels are priced at 5s. and upwards;
(4) None but _authorised_ editions are ever placed in the front shop.
Univ. "books sold here"; a = authorised editions; b = gilt-edged;
c = having red labels; d = in the front shop; e = priced at 5s. and
upwards.
=32.=
(1) Remedies for bleeding, which fail to check it, are a mockery;
(2) Tincture of Calendula is not to be despised;
(3) Remedies, which will check the bleeding when you cut your finger,
are useful;
(4) All mock remedies for bleeding are despicable.
Univ. "remedies for bleeding"; a = able to check bleeding;
b = despicable; c = mockeries; d = Tincture of Calendula; e = useful
when you cut your finger.
=33.=
(1) None of the unnoticed things, met with at sea, are mermaids;
(2) Things entered in the log, as met with at sea, are sure to be worth
remembering;
(3) I have never met with anything worth remembering, when on a voyage;
(4) Things met with at sea, that are noticed, are sure to be recorded in
the log;
Univ. "things met with at sea"; a = entered in log; b = mermaids;
c = met with by me; d = noticed; e = worth remembering.
pg118
=34.=
(1) The only books in this library, that I do _not_ recommend for
reading, are unhealthy in tone;
(2) The bound books are all well-written;
(3) All the romances are healthy in tone;
(4) I do not recommend you to read any of the unbound books.
Univ. "books in this library"; a = bound; b = healthy in tone;
c = recommended by me; d = romances; e = well-written.
=35.=
(1) No birds, except ostriches, are 9 feet high;
(2) There are no birds in this aviary that belong to any one but _me_;
(3) No ostrich lives on mince-pies;
(4) I have no birds less than 9 feet high.
Univ. "birds"; a = in this aviary; b = living on mince-pies; c = my;
d = 9 feet high; e = ostriches.
=36.=
(1) A plum-pudding, that is not really solid, is mere porridge;
(2) Every plum-pudding, served at my table, has been boiled in a cloth;
(3) A plum-pudding that is mere porridge is indistinguishable from soup;
(4) No plum-puddings are really solid, except what are served at _my_
table.
Univ. "plum-puddings"; a = boiled in a cloth; b = distinguishable from
soup; c = mere porridge; d = really solid; e = served at my table.
=37.=
(1) No interesting poems are unpopular among people of real taste;
(2) No modern poetry is free from affectation;
(3) All _your_ poems are on the subject of soap-bubbles;
(4) No affected poetry is popular among people of real taste;
(5) No ancient poem is on the subject of soap-bubbles.
Univ. "poems"; a = affected; b = ancient; c = interesting; d = on the
subject of soap-bubbles; e = popular among people of real taste;
h = written by you.
=38.=
(1) All the fruit at this Show, that fails to get a prize, is the
property of the Committee;
(2) None of my peaches have got prizes;
(3) None of the fruit, sold off in the evening, is unripe;
(4) None of the ripe fruit has been grown in a hot-house;
(5) All fruit, that belongs to the Committee, is sold off in the
evening.
Univ. "fruit at this Show"; a = belonging to the Committee; b = getting
prizes; c = grown in a hot-house; d = my peaches; e = ripe; h = sold off
in the evening.
pg119
=39.=
(1) Promise-breakers are untrustworthy;
(2) Wine-drinkers are very communicative;
(3) A man who keeps his promises is honest;
(4) No teetotalers are pawnbrokers;
(5) One can always trust a very communicative person.
Univ. "persons"; a = honest; b = pawnbrokers; c = promise-breakers;
d = trustworthy; e = very communicative; h = wine-drinkers.
=40.=
(1) No kitten, that loves fish, is unteachable;
(2) No kitten without a tail will play with a gorilla;
(3) Kittens with whiskers always love fish;
(4) No teachable kitten has green eyes;
(5) No kittens have tails unless they have whiskers.
Univ. "kittens"; a = green-eyed; b = loving fish; c = tailed;
d = teachable; e = whiskered; h = willing to play with a gorilla.
=41.=
(1) All the Eton men in this College play cricket;
(2) None but the Scholars dine at the higher table;
(3) None of the cricketers row;
(4) _My_ friends in this College all come from Eton;
(5) All the Scholars are rowing-men.
Univ. "men in this College"; a = cricketers; b = dining at the higher
table; c = Etonians; d = my friends; e = rowing-men; h = Scholars.
=42.=
(1) There is no box of mine here that I dare open;
(2) My writing-desk is made of rose-wood;
(3) All my boxes are painted, except what are here;
(4) There is no box of mine that I dare not open, unless it is full of
live scorpions;
(5) All my rose-wood boxes are unpainted.
Univ. "my boxes"; a = boxes that I dare open; b = full of live
scorpions; c = here; d = made of rose-wood; e = painted;
h = writing-desks.
=43.=
(1) Gentiles have no objection to pork;
(2) Nobody who admires pigsties ever reads Hogg's poems;
(3) No Mandarin knows Hebrew;
(4) Every one, who does not object to pork, admires pigsties;
(5) No Jew is ignorant of Hebrew.
Univ. "persons"; a = admiring pigsties; b = Jews; c = knowing Hebrew;
d = Mandarins; e = objecting to pork; h = reading Hogg's poems.
pg120
=44.=
(1) All writers, who understand human nature, are clever;
(2) No one is a true poet unless he can stir the hearts of men;
(3) Shakespeare wrote "Hamlet";
(4) No writer, who does not understand human nature, can stir the hearts
of men;
(5) None but a true poet could have written "Hamlet.";
Univ. "writers"; a = able to stir the hearts of men; b = clever;
c = Shakespeare; d = true poets; e = understanding human nature;
h = writer of 'Hamlet.'
=45.=
(1) I despise anything that cannot be used as a bridge;
(2) Everything, that is worth writing an ode to, would be a welcome gift
to me;
(3) A rainbow will not bear the weight of a wheel-barrow;
(4) Whatever can be used as a bridge will bear the weight of a
wheel-barrow;
(5) I would not take, as a gift, a thing that I despise.
Univ. "things"; a = able to bear the weight of a wheel-barrow;
b = acceptable to me; c = despised by me; d = rainbows; e = useful as a
bridge; h = worth writing an ode to.
=46.=
(1) When I work a Logic-example without grumbling, you may be sure
it is one that I can understand;
(2) These Soriteses are not arranged in regular order, like the examples
I am used to;
(3) No easy example ever make my head ache;
(4) I ca'n't understand examples that are not arranged in regular order,
like those I am used to;
(5) I never grumble at an example, unless it gives me a headache.
Univ. "Logic-examples worked by me"; a = arranged in regular order, like
the examples I am used to; b = easy; c = grumbled at by me; d = making
my head ache; e = these Soriteses; h = understood by me.
=47.=
(1) Every idea of mine, that cannot be expressed as a Syllogism, is
really ridiculous;
(2) None of my ideas about Bath-buns are worth writing down;
(3) No idea of mine, that fails to come true, can be expressed as a
Syllogism;
(4) I never have any really ridiculous idea, that I do not at once refer
to my solicitor;
(5) My dreams are all about Bath-buns;
(6) I never refer any idea of mine to my solicitor, unless it is worth
writing down.
Univ. "my ideas"; a = able to be expressed as a Syllogism; b = about
Bath-buns; c = coming true; d = dreams; e = really ridiculous
h = referred to my solicitor; k = worth writing down.
pg121
=48.=
(1) None of the pictures here, except the battle-pieces, are valuable;
(2) None of the unframed ones are varnished;
(3) All the battle-pieces are painted in oils;
(4) All those that have been sold are valuable;
(5) All the English ones are varnished;
(6) All those in frames have been sold.
Univ. "the pictures here"; a = battle-pieces; b = English; c = framed;
d = oil-paintings; e = sold; h = valuable; k = varnished.
=49.=
(1) Animals, that do not kick, are always unexcitable;
(2) Donkeys have no horns;
(3) A buffalo can always toss one over a gate;
(4) No animals that kick are easy to swallow;
(5) No hornless animal can toss one over a gate;
(6) All animals are excitable, except buffaloes.
Univ. "animals"; a = able to toss one over a gate; b = buffaloes;
c = donkeys; d = easy to swallow; e = excitable; h = horned;
k = kicking.
=50.=
(1) No one, who is going to a party, ever fails to brush his hair;
(2) No one looks fascinating, if he is untidy;
(3) Opium-eaters have no self-command;
(4) Every one, who has brushed his hair, looks fascinating;
(5) No one wears white kid gloves, unless he is going to a party;
(6) A man is always untidy, if he has no self-command.
Univ. "persons"; a = going to a party; b = having brushed one's hair;
c = having self-command; d = looking fascinating; e = opium-eaters;
h = tidy; k = wearing white kid gloves.
=51.=
(1) No husband, who is always giving his wife new dresses, can be a
cross-grained man;
(2) A methodical husband always comes home for his tea;
(3) No one, who hangs up his hat on the gas-jet, can be a man that is
kept in proper order by his wife;
(4) A good husband is always giving his wife new dresses;
(5) No husband can fail to be cross-grained, if his wife does not keep
him in proper order;
(6) An unmethodical husband always hangs up his hat on the gas-jet.
Univ. "husbands"; a = always coming home for his tea; b = always giving
his wife new dresses; c = cross-grained; d = good; e = hanging up his
hat on the gas-jet; h = kept in proper order; k = methodical.
pg122
=52.=
(1) Everything, not absolutely ugly, may be kept in a drawing-room;
(2) Nothing, that is encrusted with salt, is ever quite dry;
(3) Nothing should be kept in a drawing-room, unless it is free from
damp;
(4) Bathing-machines are always kept near the sea;
(5) Nothing, that is made of mother-of-pearl, can be absolutely ugly;
(6) Whatever is kept near the sea gets encrusted with salt.
Univ. "things"; a = absolutely ugly; b = bathing-machines; c = encrusted
with salt; d = kept near the sea; e = made of mother-of-pearl; h = quite
dry; k = things that may be kept in a drawing-room.
=53.=
(1) I call no day "unlucky," when Robinson is civil to me;
(2) Wednesdays are always cloudy;
(3) When people take umbrellas, the day never turns out fine;
(4) The only days when Robinson is uncivil to me are Wednesdays;
(5) Everybody takes his umbrella with him when it is raining;
(6) My "lucky" days always turn out fine.
Univ. "days"; a = called by me 'lucky'; b = cloudy; c = days when people
take umbrellas; d = days when Robinson is civil to me; e = rainy;
h = turning out fine; k = Wednesdays.
=54.=
(1) No shark ever doubts that it is well fitted out;
(2) A fish, that cannot dance a minuet, is contemptible;
(3) No fish is quite certain that it is well fitted out, unless it has
three rows of teeth;
(4) All fishes, except sharks, are kind to children;
(5) No heavy fish can dance a minuet;
(6) A fish with three rows of teeth is not to be despised.
Univ. "fishes"; a = able to dance a minuet; b = certain that he is well
fitted out; c = contemptible; d = having 3 rows of teeth; e = heavy;
h = kind to children; k = sharks.
=55.=
(1) All the human race, except my footmen, have a certain amount of
common-sense;
(2) No one, who lives on barley-sugar, can be anything but a mere baby;
(3) None but a hop-scotch player knows what real happiness is;
(4) No mere baby has a grain of common sense;
(5) No engine-driver ever plays hop-scotch;
(6) No footman of mine is ignorant of what true happiness is.
Univ. "human beings"; a = engine-drivers; b = having common sense;
c = hop-scotch players; d = knowing what real happiness is; e = living
on barley-sugar; h = mere babies; k = my footmen.
pg123
=56.=
(1) I trust every animal that belongs to me;
(2) Dogs gnaw bones;
(3) I admit no animals into my study, unless they will beg when told to
do so;
(4) All the animals in the yard are mine;
(5) I admit every animal, that I trust, into my study;
(6) The only animals, that are really willing to beg when told to do so,
are dogs.
Univ. "animals"; a = admitted to my study; b = animals that I trust;
c = dogs; d = gnawing bones; e = in the yard; h = my; k = willing to beg
when told.
=57.=
(1) Animals are always mortally offended if I fail to notice them;
(2) The only animals that belong to _me_ are in that field;
(3) No animal can guess a conundrum, unless it has been properly trained
in a Board-School;
(4) None of the animals in that field are badgers;
(5) When an animal is mortally offended, it always rushes about wildly
and howls;
(6) I never notice any animal, unless it belongs to me;
(7) No animal, that has been properly trained in a Board-School, ever
rushes about wildly and howls.
Univ. "animals"; a = able to guess a conundrum; b = badgers; c = in that
field; d = mortally offended; e = my; h = noticed by me; k = properly
trained in a Board-School; l = rushing about wildly and howling.
=58.=
(1) I never put a cheque, received by me, on that file, unless I am
anxious about it;
(2) All the cheques received by me, that are not marked with a cross,
are payable to bearer;
(3) None of them are ever brought back to me, unless they have been
dishonoured at the Bank;
(4) All of them, that are marked with a cross, are for amounts of over
L100;
(5) All of them, that are not on that file, are marked "not negotiable";
(6) No cheque of yours, received by me, has ever been dishonoured;
(7) I am never anxious about a cheque, received by me, unless it should
happen to be brought back to me;
(8) None of the cheques received by me, that are marked "not
negotiable," are for amounts of over L100.
Univ. "cheques received by me"; a = brought back to me; b = cheques that
I am anxious about; c = honoured; d = marked with a cross; e = marked
'not negotiable'; h = on that file; k = over L100; l = payable to
bearer; m = your.
pg124
=59.=
(1) All the dated letters in this room are written on blue paper;
(2) None of them are in black ink, except those that are written in the
third person;
(3) I have not filed any of them that I can read;
(4) None of them, that are written on one sheet, are undated;
(5) All of them, that are not crossed, are in black ink;
(6) All of them, written by Brown, begin with "Dear Sir";
(7) All of them, written on blue paper, are filed;
(8) None of them, written on more than one sheet, are crossed;
(9) None of them, that begin with "Dear Sir," are written in the
third person.
Univ. "letters in this room"; a = beginning with "Dear Sir";
b = crossed; c = dated; d = filed; e = in black ink; h = in third
person; k = letters that I can read; l = on blue paper; m = on one
sheet; n = written by Brown.
=60.=
(1) The only animals in this house are cats;
(2) Every animal is suitable for a pet, that loves to gaze at the moon;
(3) When I detest an animal, I avoid it;
(4) No animals are carnivorous, unless they prowl at night;
(5) No cats fails to kill mice;
(6) No animals ever take to me, except what are in this house;
(7) Kangaroos are not suitable for pets;
(8) None but carnivora kill mice;
(9) I detest animals that do not take to me;
(10) Animals, that prowl at night, always love to gaze at the moon.
Univ. "animals"; a = avoided by me; b = carnivora; c = cats;
d = detested by me; e = in this house; h = kangaroos; k = killing mice;
l = loving to gaze at the moon; m = prowling at night; n = suitable for
pets; r = taking to me.
pg125
CHAPTER II.
_ANSWERS._
_Answers to Sec. 1._ AN1
1. "All" _Sign of Quantity._
"persons represented by the Name 'I'" (or "I's") _Subject._
"are" _Copula._
"persons who have been out for a walk" _Predicate._
or, more briefly,
"All | 'I's | are | persons who have been out for a walk".
2. "All | 'I's | are | persons who feel better".
3. "No | persons who are not 'John' | are | persons who have read the
letter".
4. "No | Members of the Class 'you and I' | are | old persons".
5. "No | fat creatures | are | creatures that run well".
6. "No | not-brave persons | are | persons deserving of the fair".
7. "No | not-pale persons | are | persons who look poetical".
8. "Some | judges | are | persons who lose their tempers".
9. "All | 'I's | are | persons who do not neglect important business".
10. "All | difficult things | are | things that need attention".
11. "All | unwholesome things | are | things that should be avoided".
12. "All | laws passed last week | are | laws relating to excise".
13. "All | logical studies | are | things that puzzle me".
14. "No | persons in the house | are | Jews".
15. "Some | not well-cooked dishes | are | unwholesome dishes".
16. "All | unexciting books | are | books that make one drowsy".
17. "All | men who know what they're about | are | men who can detect
a sharper".
18. "All | Members of the Class 'you and I' | are | persons who know
what they're about".
19. "Some | bald persons | are | persons accustomed to wear wigs".
20. "All | fully occupied persons | are | persons who do not talk about
their grievances".
21. "No | riddles that can be solved | are | riddles that interest me".
pg126
_Answers to Sec. 2._ AN2
1
.---------------.
|(O) | |
| .---|---. |
| |(O)|(O)| |
|---|---|---|---|
| | | | |
| .---|---. |
|(O) | |
.---------------.
2
.---------------.
|(I) | (O)|
| .---|---. |
| | | | |
|---|---|---|---|
| | | | |
| .---|---. |
|(O) | (O)|
.---------------.
3
.---------------.
| | |
| .---|---. |
| |(O)| | |
|---|---|---|---|
| |(O)|(I)| |
| .---|---. |
| | |
.---------------.
4
.---------------.
|(O) | |
| .---|---. |
| | (I) | |
|---|---|---|(I)|
| |(O)|(O)| |
| .---|---. |
|(O) | |
.---------------.
5
.---------------.
|(O) | (I)|
| .---|---. |
| | | | |
|---|---|---|---|
| | | | |
| .---|---. |
|(O) | (O)|
.---------------.
6
.---------------.
| | |
| .---|---. |
| | |(O)| |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
| (I) |
.---------------.
7
.---------------.
|(O) | (O)|
| .---|---. |
| |(I)|(O)| |
|---|---|---|---|
| | |(O)| |
| .---|---. |
| | (I)|
.---------------.
8
.---------------.
| | |
| .---|---. |
| |(O)| | |
|---|---|---|---|
| |(O)| | |
| .---|---. |
| (I) |
.---------------.
9
.---------------.
| | |
| .---|---. |
| |(O)|(O)| |
|---|---|---|---|
| |(O)|(I)| |
| .---|---. |
| | |
.---------------.
10
.---------------.
|(O) | |
| .---|---. |
| | | | |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
|(O) | |
.---------------.
11
.---------------.
| | |
| .---|---. |
| |(O)| | |
|---|---|---|---|
| |(O)| | |
| .---|---. |
|(O) | (O)|
.---------------.
12
.---------------.
| | (O)|
| .---|---. |
| | (I) | |
|---|---|(I)|---|
| | | | |
| .---|---. |
| | (O)|
.---------------.
13
.---------------.
| | |
| .---|---. |
| | |(O)| |
|---|---|---|---|
| |(I)|(O)| |
| .---|---. |
|(O) | (O)|
.---------------.
14
.---------------.
| (I) |
| .---|---. |
| | |(O)| |
|---|(I)|---|---|
| | |(O)| |
| .---|---. |
| | |
.---------------.
15
.---------------.
|(O) | |
| .---|---. |
| | | | |
|---|(I)|---|---|
| | | | |
| .---|---. |
|(O) | (O)|
.---------------.
16
.---------------.
| (I) |
| .---|---. |
| |(O)|(O)| |
|---|---|---|---|
| |(O)| | |
| .---|---. |
| | |
.---------------.
17
.---------------.
|(I) | (O)|
| .---|---. |
| | | | |
|---|---|---|---|
| | | | |
| .---|---. |
| | (O)|
.---------------.
18
.---------------.
| (I) |
| .---|---. |
| |(O)|(O)| |
|---|---|---|(I)|
| | | | |
| .---|---. |
| | |
.---------------.
19
.---------------.
| | |
| .---|---. |
| | |(I)| |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
| | |
.---------------.
20
.---------------.
| | |
| .---|---. |
| |(I)| | |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
| | |
.---------------.
21
.---------------.
| | (O)|
| .---|---. |
| | | | |
|---|---|(I)|---|
| | | | |
| .---|---. |
|(I) | (O)|
.---------------.
22
.---------------.
| | |
| .---|---. |
| |(O)|(O)| |
|---|---|---|---|
| |(I)| | |
| .---|---. |
| | |
.---------------.
23
.---------------.
|(O) | (O)|
| .---|---. |
| |(O)| | |
|---|---|---|---|
| |(O)| | |
| .---|---. |
|(I) | |
.---------------.
24
.---------------.
| | (O)|
| .---|---. |
| | (I) | |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
| | (O)|
.---------------.
pg127
25
.---------------.
| | |
| .---|---. |
| |(I)|(O)| |
|---|---|---|---|
| | |(O)| |
| .---|---. |
| | |
.---------------.
26
.---------------.
|(O) | (O)|
| .---|---. |
| | | | |
|---|---|---|---|
| | | | |
| .---|---. |
|(I) | |
.---------------.
27
.---------------.
| | (O)|
| .---|---. |
| | | | |
|---|---|---|---|
| | (I) | |
| .---|---. |
| | (O)|
.---------------.
28
.---------------.
|(O) | (O)|
| .---|---. |
| |(O)| | |
|---|---|(I)|---|
| |(O)| | |
| .---|---. |
| | |
.---------------.
29
.---------------.
| | |
| .---|---. |
| | |(O)| |
|---|---|---|---|
| |(O)|(O)| |
| .---|---. |
| | |
.---------------.
30
.---------------.
| | |
| .---|---. |
| |(O)|(O)| |
|---|---|---|(I)|
| | | | |
| .---|---. |
| | |
.---------------.
31
.---------------.
|(I) | (O)|
| .---|---. |
| | | | |
|---|---|(I)|---|
| | | | |
| .---|---. |
| | (O)|
.---------------.
32
.---------------.
|(O) | (I)|
| .---|---. |
| |(O)|(O)| |
|---|---|---|---|
| |(I)| | |
| .---|---. |
|(O) | |
.---------------.
_Answers to Sec. 3._ AN3
1. Some xy exist, or some x are y, or some y are x.
2. No information.
3. All y' are x'.
4. No xy exist, &c.
5. All y' are x.
6. All x' are y.
7. All x are y.
8. All x' are y', and all y are x.
9. All x' are y'.
10. All x are y'.
11. No information.
12. Some x'y' exist, &c.
13. Some xy' exist, &c.
14. No xy' exist, &c.
15. Some xy exist, &c.
16. All y are x.
17. All x' are y, and all y' are x.
18. All x are y', and all y are x'.
19. All x are y, and all y' are x'.
20. All y are x'.
_Answers to Sec. 4._ AN4
1. No x' are y'.
2. Some x' are y'.
3. Some x are y'.
4. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
5. Some x' are y'.
6. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
7. Some x are y'.
8. Some x' are y'.
9. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
10. All x are y, and all y' are x'.
11. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
12. All y are x'.
13. No x' are y.
14. No x' are y'.
15. No x are y.
16. All x are y', and all y are x'.
pg128
17. No x are y'.
18. No x are y.
19. Some x are y'.
20. No x are y'.
21. Some y are x'.
22. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
23. Some x are y.
24. All y are x'.
25. Some y are x'.
26. All y are x.
27. All x are y, and all y' are x'.
28. Some y are x'.
29. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
30. Some y are x'.
31. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
32. No x are y'.
33. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
34. Some x are y.
35. All y are x'.
36. Some y are x'.
37. Some x are y'.
38. No x are y.
39. Some x' are y'.
40. All y' are x.
41. All x are y'.
42. No x are y.
_Answers to Sec. 5._ AN5
1. Somebody who has been out for a walk is feeling better.
2. No one but John knows what the letter is about.
3. You and I like walking.
4. Honesty is sometimes the best policy.
5. Some greyhounds are not fat.
6. Some brave persons get their deserts.
7. Some rich persons are not Esquimaux.
8. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
9. John is ill.
10. Some things, that are not umbrellas, should be left behind on a
journey.
11. No music is worth paying for, unless it causes vibration in the
air.
12. Some holidays are tiresome.
13. Englishmen are not Frenchmen.
14. No photograph of a lady is satisfactory.
15. No one looks poetical unless he is phlegmatic.
16. Some thin persons are not cheerful.
17. Some judges do not exercise self-control.
18. Pigs are not fed on barley-water.
19. Some black rabbits are not old.
pg129
20. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
21. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
22. Some lessons need attention.
23. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
24. No one, who forgets a promise, fails to do mischief.
25. Some greedy creatures cannot fly.
26. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
27. No bride-cakes are things that need not be avoided.
28. John is happy.
29. Some people, who are not gamblers, are not philosophers.
30. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
31. None of my lodgers write poetry.
32. Senna is not nice.
33. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
34. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
35. Logic is unintelligible.
36. Some wild creatures are fat.
37. All wasps are unwelcome.
38. All black rabbits are young.
39. Some hard-boiled things can be cracked.
40. No antelopes fail to delight the eye.
41. All well-fed canaries are cheerful.
42. Some poetry is not producible at will.
43. No country infested by dragons fails to be fascinating.
44. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
45. Some picturesque things are not made of sugar.
46. No children can sit still.
47. Some cats cannot whistle.
48. You are terrible.
49. Some oysters are not amusing.
50. Nobody in the house has a beard a yard long.
51. Some ill-fed canaries are unhappy.
52. My sisters cannot sing.
53. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
54. Some rich things are nice.
55. My cousins are none of them judges, and judges are none of them
cousins of mine.
56. Something wearisome is not eagerly wished for.
57. Senna is nasty.
58. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
59. Niggers are not any of them tall.
60. Some obstinate persons are not philosophers.
61. John is happy.
62. Some unwholesome dishes are not present here (i.e. cannot be
spoken of as "these").
63. No books suit feverish patients unless they make one drowsy.
64. Some greedy creatures cannot fly.
65. You and I can detect a sharper.
66. Some dreams are not lambs.
pg130
67. No lizard needs a hairbrush.
68. Some things, that may escape notice, are not battles.
69. My cousins are not any of them judges.
70. Some hard-boiled things can be cracked.
71. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
72. She is unpopular.
73. Some people, who wear wigs, are not children of yours.
74. No lobsters expect impossibilities.
75. No nightmare is eagerly desired.
76. Some nice things are not plumcakes.
77. Some kinds of jam need not be shunned.
78. All ducks are ungraceful.
79. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
80. No man, who begs in the street, should fail to keep accounts.
81. Some savage creatures are not spiders.
82. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
83. No travelers, who do not carry plenty of small change, fail to lose
their luggage.
84. [No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.]
85. Judges are none of them cousins of mine.
86. All my lodgers are sane.
87. Those who are busy are contented, and discontented people are not
busy.
88. None of my cousins are judges.
89. No nightingale dislikes sugar.
90. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
91. Some excuses are not clear explanations.
92. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
93. No kind deed need cause scruple.
94. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
95. [No Concl. Fallacy of Like Eliminands not asserted to exist.]
96. No cheats are trustworthy.
97. No clever child of mine is greedy.
98. Some things, that are meant to amuse, are not Acts of Parliament.
99. No tour, that is ever forgotten, is worth writing a book about.
100. No obedient child of mine is contented.
101. Your visit does not annoy me.
_Answers to Sec. 6._ AN6
1. Conclusion right.
2. No Concl. Fallacy of Like Eliminands not asserted to exist.
3. Concl. right.
4. Concl. right.
5. Concl. right.
6. No Concl. Fallacy of Like Eliminands not asserted to exist.
7. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
8. Concl. right.
9. Concl. right.
10. Concl. right.
11. Concl. right.
12. Concl. right.
13. Concl. right.
14. Concl. right.
15. Concl. right.
pg131
16. No Concl. Fallacy of Like Eliminands not asserted to exist.
17. Concl. right.
18. Concl. right.
19. Concl. right.
20. Concl. right.
21. Concl. right.
22. Concl. wrong: the right one is "Some x are y."
23. Concl. right.
24. Concl. right.
25. Concl. right.
26. Concl. right.
27. Concl. right.
28. No Concl. Fallacy of Like Eliminands not asserted to exist.
29. Concl. right.
30. Concl. right.
31. Concl. right.
32. Concl. right.
33. Concl. right.
34. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
35. Concl. right.
36. Concl. right.
37. Concl. right.
38. No Concl. Fallacy of Like Eliminands not asserted to exist.
39. Concl. right.
40. Concl. right.
_Answers to Sec. 7._ AN7
1. Concl. right.
2. Concl. right.
3. Concl. right.
4. Concl. wrong: right one is "Some epicures are not uncles of mine."
5. Concl. right.
6. No Concl. Fallacy of Unlike Eliminands with an Entity-Premiss.
7. Concl. wrong: right one is "The publication, in which I saw it,
tells lies."