Deductive Logic
by
St. George Stock
Part 5 out of 6
is equal to
All cases of A not being B are cases of C being D.
This is a case of C being D.
§ 769. So far as regards the consequent, the two species of complex
reasoning hitherto discussed are identical both in appearance and
reality. The apparent difference of procedure in the case of the
antecedent, namely, that it is affirmed in the partly conjunctive, but
denied in the disjunctive syllogism, is due merely to the fact that in
the disjunctive proposition the truth of the consequent is involved in
the falsity of the antecedent, so that the antecedent being
necessarily negative, to deny it in appearance is in reality to assert
it.
CHAPTER XXV.
_The Disjunctive Syllogism regarded as an Immediate Inference_.
§ 770. If no stress be laid on the transition from disjunctive
hypothesis to fact, the disjunctive syllogism will run with the same
facility as its predecessor into the moulds of immediate inference.
§ 771.
_Denial of Antecedent_. Subalternation.
Either A is B or C is D, Every case of A not being B
is a case of C being D.
.'. A not being B, C is D. .'. Some case of A not being B
is a case of C being D.
§ 772.
_Denial of Consequent_. Conversion by Contraposition
+ Subalternation.
Either A is B or C is D. All cases of A not being B
are cases of C being D.
.'. C not being D, A is B .'. All cases of C not being D are
cases of A being B.
.'. Some case of C not being D is
a case of A being B.
§ 773. Similarly the two invalid types of disjunctive syllogism will
be found to coincide with fallacies of immediate inference.
§ 774.
_Affirmation of Antecedent_. Contraposition without
Conversion.
Either A is B or C is D. All cases of A not being B are
cases of C being D.
.'. A being B, C is not D .'. All cases of A being B are
cases of C not being D.
§ 775. The affirmation of the antecedent thus comes under the
formula--
All not-A is B,
.'. All A is not-B,
a form of inference which cannot hold except where A and B are known
to be incompatible. Who, for instance, would assent to this?--
All non-boating men play cricket.
.'. All boating men are non-cricketers.
§ 776.
_Affirmation of Consequent_. Simple Conversion of A.
Either A is B or C is D. All cases of A not being B are
cases of C being D.
.'.C being D, A is not B. .'. All cases of C being D are
cases of A not being B.
§ 777. We may however argue in this way--
Conversion of A per accidens.
Either A is B or C is D. All cases of A not being B
are cases of C being D.
.'. C being D, A is sometimes B. .'. Some cases of C being D are
cases of A not being B.
The men who pass this examination must have either talent or industry.
.'. Granting that they are industrious, they may be without talent.
CHAPTER XXVI.
_Of the Mixed Form of Complex Syllogism_.
§ 778. Under this head are included all syllogisms in which a
conjunctive is combined with a disjunctive premiss. The best known
form is
_The Dilemma_.
§ 779. The Dilemma may be defined as--
A complex syllogism, having for its major premiss a conjunctive
proposition with more than one antecedent, or more than one
consequent, or both, which (antecedent or consequent) the minor
premiss disjunctively affirms or denies.
§ 780. It will facilitate the comprehension of the dilemma, if the
following three points are borne in mind--
(1) that the dilemma conforms to the canon of the partly conjunctive
syllogism, and therefore a valid conclusion can be obtained only by
affirming the antecedent or denying the consequent;
(2) that the minor premiss must be disjunctive;
(3) that if only the antecedent be more than one, the conclusion
will be a simple proposition; but if both antecedent and consequent
be more than one, the conclusion will itself be disjunctive.
§ 781. The dilemma, it will be seen, differs from the partly
conjunctive syllogism chiefly in the fact of having a disjunctive
affirmation of the antecedent or denial of the consequent in the
minor, instead of a simple one. It is this which constitutes the
essence of the dilemma, and which determines its possible
varieties. For if only the antecedent or only the consequent be more
than one, we must, in order to obtain a disjunctive minor, affirm the
antecedent or deny the consequent respectively; whereas, if there be
more than one of both, it is open to us to take either course. This
gives us four types of dilemma.
§ 782.
(1). _Simple Constructive._
If A is B or C is D, E is F.
Either A is B or C is D.
.'. E is F.
(2). _Simple Destructive._
If A is B, C is D and E is F.
Either C is not D or E is not F.
.'. A is not B.
(3). _Complex Constructive._
If A is B, C is D; and if E is F, G is H.
Either A is B or E is F.
.'. Either C is D or G is H.
(4). _Complex Destructive_.
If A is B, C is D; and if E is F, G is H.
Either C is not D or G is not H.
.'. Either A is not B or E is not F.
§ 783.
(1). _Simple Constructive_.
If she sinks or if she swims, there will be an end of her.
She must either sink or swim.
.'. There will be an end of her.
(2). _Simple Destructive_.
If I go to Town, I must pay for my ticket and pay my hotel bill.
Either I cannot pay for my ticket or I cannot pay my hotel bill.
.'. I cannot go to Town.
(3). _Complex Constructive_.
If I stay in this room, I shall be burnt to death, and if I jump
out of the window, I shall break my neck.
I must either stay in the room or jump out of the window.
.'. I must either be burnt to death or break my neck.
(4). _Complex Destructive_.
If he were clever, he would see his mistake; and
if he were candid, he would acknowledge it.
Either he does not see his mistake or he will not acknowledge it.
.'. Either he is not clever or he is not candid.
§ 784. It must be noticed that the simple destructive dilemma would
not admit of a disjunctive consequent. If we said,
If A is B, either C is D or E is F,
Either C is not D or E is not F,
we should not be denying the consequent. For 'E is not F' would make
it true that C is D, and 'C is not D' would make it true that E is F;
so that in either case we should have one of the alternatives true,
which is just what the disjunctive form 'Either C is D or E is F'
insists upon.
§ 785. In the case of the complex constructive dilemma the several
members, instead of being distributively assigned to one another, may
be connected together as a whole--thus--
If either A is B or E is F, either C is D or G is H.
Either A is B or E is F.
.'. Either C is D or G is H.
In this shape the likeness of the dilemma to the partly conjunctive
syllogism is more immediately recognisable. The major premiss in this
shape is vaguer than in the former. For each antecedent has now a
disjunctive choice of consequents, instead of being limited to
one. This vagueness, however, does not affect the conclusion. For, so
long as the conclusion is established, it does not matter from which
members of the major its own members flow.
§ 786. It must be carefully noticed that we cannot treat the complex
destructive dilemma in the same way.
If either A is B or E is F, either C is D or G is H.
Either C is not D or G is not H.
Since the consequents are no longer connected individually with the
antecedents, a disjunctive denial of them leaves it still possible for
the antecedent as a whole to be true. For 'C is not D' makes it true
that G is H, and 'G is not H' makes it true that C is D. In either
case then one is true, which is all that was demanded by the
consequent of the major. Hence the consequent has not really been
denied.
§ 787. For the sake of simplicity we have limited the examples to the
case of two antecedents or consequents. But we may have as many of
either as we please, so as to have a Trilemma, a Tetralemma, and so
on.
TRILEMMA.
If A is B, C is D; and if E is F, G is H; and if K is L, M is N.
Either A is B or E is F or K is L.
.'. Either C is D or G is H or K is L.
§ 788. Having seen what the true dilemma is, we shall now examine some
forms of reasoning which resemble dilemmas without being so.
§ 789. This, for instance, is not a dilemma--
If A is B or if E is F, C is D.
But A is B and E is F.
.'. C is D.
If he observes the sabbath or if he refuses to eat pork, he is a
Jew.
But he both observes the sabbath and refuses to eat pork.
.'. He is a Jew.
What we have here is a combination of two partly conjunctive
syllogisms with the same conclusion, which would have been established
by either of them singly. The proof is redundant.
§ 790. Neither is the following a dilemma--
If A is B, C is D and E is F.
Neither C is D nor E is F.
.'. A is not B.
If this triangle is equilateral, its sides and its angles will be
equal.
But neither its sides nor its angles are equal.
.'. It is not equilateral.
This is another combination of two conjunctive syllogisms, both
pointing to the same conclusion. The proof is again redundant. In this
case we have the consequent denied in both, whereas in the former we
had the antecedent affirmed. It is only for convenience that such
arguments as these are thrown into the form of a single
syllogism. Their real distinctness may be seen from the fact that we
here deny each proposition separately, thus making two independent
statements--C is not D and E is not F. But in the true instance of the
simple destructive dilemma, what we deny is not the truth of the two
propositions contained in the consequent, but their compatibility; in
other words we make a disjunctive denial.
§ 791. Nor yet is the following a dilemma--
If A is B, either C is D or E is F.
Neither C is D nor E is F.
.'. A is not B.
If the barometer falls there will be either wind or rain.
There is neither wind nor rain.
.'. The barometer has not fallen.
What we have here is simply a conjunctive major with the consequent
denied in the minor. In the consequent of the major it is asserted
that the two propositions, 'C is D' and 'E is F' cannot both be false;
and in the minor this is denied by the assertion that they are both
false.
§ 792. A dilemma is said to be rebutted or retorted, when another
dilemma is made out proving an opposite conclusion. If the dilemma be
a sound one, and its premisses true, this is of course impossible, and
any appearance of contradiction that may present itself on first sight
must vanish on inspection. The most usual mode of rebutting a dilemma
is by transposing and denying the consequents in the major--
If A is B, C is D; and if E is F, G is H.
Either A is B or E is F.
.'. Either C is D or G is H.
The same rebutted--
If A is B, G is not H; and if E is F, C is not D.
Either A is B or E is F.
.'. Either G is not H or C is not D.
= Either C is not D or G is not H.
§ 793. Under this form comes the dilemma addressed by the Athenian
mother to her son--'Do not enter public life: for, if you say what is
just, men will hate you; and, if you say what is unjust, the gods will
hate you' to which the following retort was made--'I ought to enter
public life: for, if 1 say what is just, the gods will love me; and,
if 1 say what is unjust, men will love me.' But the two conclusions
here are quite compatible. A man must, on the given premisses, be both
hated and loved, whatever course he takes. So far indeed are two
propositions of the form
Either C is D or G is H,
and Either C is not D or G is not H,
from being incompatible, that they express precisely the same thing
when contradictory alternatives have been selected, e.g.--
Either a triangle is equilateral or non-equilateral.
Either a triangle is non-equilateral or equilateral.
§ 794. Equally illusory is the famous instance of rebutting a dilemma
contained in the story of Protagoras and Euathlus
(Aul. Gell. Noct. Alt. v. 10), Euathlus was a pupil of Protagoras in
rhetoric. He paid half the fee demanded by his preceptor before
receiving lessons, and agreed to pay the remainder when he won his
first case. But as he never proceeded to practise at the bar, it
became evident that he meant to bilk his tutor. Accordingly Protagoras
himself instituted a law-suit against him, and in the preliminary
proceedings before the jurors propounded to him the following
dilemma--'Most foolish young man, whatever be the issue of this suit,
you must pay me what I claim: for, if the verdict be given in your
favour, you are bound by our bargain; and if it be given against you,
you are bound by the decision of the jurors.' The pupil, however, was
equal to the occasion, and rebutted the dilemma as follows. 'Most
sapient master, whatever be the issue of this suit, I shall not pay
you what you claim: for, if the verdict be given in my favour, I am
absolved by the decision of the jurors; and, if it be given against
me, I am absolved by our bargain.' The jurors are said to have been so
puzzled by the conflicting plausibility of the arguments that they
adjourned the case till the Greek Kalends. It is evident, however,
that a grave injustice was thus done to Protagoras. His dilemma was
really invincible. In the counter-dilemma of Euathlus we are meant to
infer that Protagoras would actually lose his fee, instead of merely
getting it in one way rather than another. In either case he would
both get and lose his fee, in the sense of getting it on one plea, and
not getting it on another: but in neither case would he actually lose
it.
§ 795. If a dilemma is correct in form, the conclusion of course
rigorously follows: but a material fallacy often underlies this form
of argument in the tacit assumption that the alternatives offered in
the minor constitute an exhaustive division. Thus the dilemma 'If pain
is severe, it will be brief; and if it last long it will be slight,'
&c., leaves out of sight the unfortunate fact that pain may both be
severe and of long continuance. Again the following dilemma--
If students are idle, examinations are unavailing; and, if
they are industrious, examinations are superfluous,
Students are either idle or industrious,
.'. Examinations are either unavailing or superfluous,
is valid enough, so far as the form is concerned. But the person who
used it would doubtless mean to imply that students could be
exhaustively divided into the idle and the industrious. No deductive
conclusion can go further than its premisses; so that all that the
above conclusion can in strictness be taken to mean is that
examinations are unavailing, when students are idle, and superfluous,
when they are industrious--which is simply a reassertion as a matter
of fact of what was previously given as a pure hypothesis.
CHAPTER XXVII.
_Of the Reduction of the Dilemma._
§ 796. As the dilemma is only a peculiar variety of the partly
conjunctive syllogism, we should naturally expect to find it reducible
in the same way to the form of a simple syllogism. And such is in fact
the case. The constructive dilemma conforms to the first figure and
the destructive to the second.
1) _Simple Constructive Dilemma_.
Barbara.
If A is B or if E is F, C is D. All cases of either A being B or E
being F are cases of C being D.
Either A is B or E is F. All actual cases are cases of either
A being B OP E being F.
.'. C is D. .'. All actual cases are cases of C
being D.
(2) _Simple Destructive_.
Camstres.
If A is B, C is D and E is F. All cases of A being B are cases of
C being D and E being F.
Either C is not D or E is not F. No actual cases are cases of C being
D and E being F.
.'. A is not B. .'. No actual cases are cases of A
being B.
(3) _Complex Constructive_.
Barbara.
If A is B, C is D; and if E is F, All cases of either A being B or
G is H. being F are cases of either C being
D or G being H.
Either A is B or E is F. All actual cases are cases of either A
being B or E being F.
.'. Either C is D or G is H. .'. All actual cases are cases of either C
being D or G being H.
(4) _Complex Destructive_.
If A is B, C is D; and if E is F, All cases of A being B and E being F
G is H. are cases of C being D and G
being H.
Either C is not D Or G is No actual cases are cases of C being
not H D and G being H.
Either A is not B or E is No actual cases are cases of A being
not F. B and E being F.
§ 797. There is nothing to prevent our having Darii, instead of
Barbara, in the constructive form, and Baroko, instead of Camestres,
in the destructive. As in the case of the partly conjunctive syllogism
the remaining moods of the first and second figure are obtained by
taking a negative proposition as the consequent of the major premiss,
e.g.--
_Simple Constructive_. Celarent or Ferio.
If A is B or if E is F, C is not D No cases of either A being B or E
being F are cases of C being D.
Either A is B or E is F. All (or some) actual cases are cases of
either A being B or E being F
.'. C is not D. .'. All (or some) actual cases are not
cases of C being D.
CHAPTER XXVIII.
_Of the Dilemma regarded as an Immediate Inference._
§ 798. Like the partly conjunctive syllogism, the dilemma can be
expressed under the forms of immediate inference. As before, the
conclusion in the constructive type resolves itself into the
subalternate of the major itself, and in the destructive type into the
subalternate of its contrapositive. The simple constructive dilemma,
for instance, may be read as follows--
If either A is B or E is F, C is D,
.'. Either A being B or E being F, C is D,
which is equivalent to
Every case of either A being B or E being F is a case of C being D.
.'. Some case of either A being B or E being F is a case of C being D.
The descent here from 'every' to 'some' takes the place of the
transition from hypothesis to fact.
§ 799. Again the complex destructive may be read thus--
If A is B, C is D; and if E is F, G is H,
.'. It not being true that C is D and G is H, it is not
true that A is B and E is F,
which may be resolved into two steps of immediate inference, namely,
conversion by contraposition followed by subalternation--
All cases of A being B and E being F are cases of C being D and G
being H.
.'. Whatever is not a case of C being D and G being H is not a case
of A being B and E being F.
.'. Some case which is not one of C being D and G being H is not a
case of A being B and E being F.
CHAPTER XXIX.
_Of Trains of Reasoning._
§ 800. The formal logician is only concerned to examine whether the
conclusion duly follows from the premisses: he need not concern
himself with the truth or falsity of his data. But the premisses of
one syllogism may themselves be conclusions deduced from other
syllogisms, the premisses of which may in their turn have been
established by yet earlier syllogisms. When syllogisms are thus linked
together we have what is called a Train of Reasoning.
§ 801. It is plain that all truths cannot be established by
reasoning. For the attempt to do so would involve us in an infinite
regress, wherein the number of syllogisms required would increase at
each step in a geometrical ratio. To establish the premisses of a
given syllogism we should require two preceding syllogisms; to
establish their premisses, four; at the next step backwards, eight; at
the next, sixteen; and so on ad infinitum. Thus the very possibility
of reasoning implies truths that are known to us prior to all
reasoning; and, however long a train of reasoning may be, we must
ultimately come to truths which are either self-evident or are taken
for granted.
§ 802. Any syllogism which establishes one of the premisses of another
is called in reference to that other a Pro-syllogism, while a
syllogism which has for one of its premisses the conclusion of another
syllogism is called in reference to that other an Epi-syllogism.
_The Epicheirema_.
§ 803. The name Epicheirema is given to a syllogism with one or both
of its premisses supported by a reason. Thus the following is a
double epicheirema--
All B is A, for it is E.
All C is B, for it is F.
.'. All C is A.
All virtue is praiseworthy, for it promotes the general welfare.
Generosity is a virtue, for it prompts men to postpone self to others.
.'. Generosity is praiseworthy.
§ 804. An epicheirema is said to be of the first or second order
according as the major or minor premiss is thus supported. The double
epicheirema is a combination of the two orders.
§ 805. An epicheirema, it will be seen, consists of one syllogism
fully expressed together with one, or, it may be, two enthymemes (§
557). In the above instance, if the reasoning which supports the
premisses were set forth at full length, we should have, in place of
the enthymemes, the two following pro-syllogisms--
(i) All E is A.
All B is E.
.'. All B is A.
Whatever promotes the general welfare is praiseworthy.
Every virtue promotes the general welfare.
.'. Every virtue is praiseworthy.
(2) All F is B.
All C is F.
.'. All C is B.
Whatever prompts men to postpone self to others is a virtue.
Generosity prompts men to postpone self to others.
.'. Generosity is a virtue.
§ 806. The enthymemes in the instance above given are both of the
first order, having the major premiss suppressed. But there is
nothing to prevent one or both of them from being of the second
order--
All B is A, because all F is.
All C is B, because all F is.
.'. All C is A.
All Mahometans are fanatics, because all Monotheists are.
These men are Mahometans, because all Persians are.
.'. These men are fanatics.
Here it is the minor premiss in each syllogism that is suppressed,
namely,
(1) All Mahometans are Monotheists.
(2) These men are Persians.
_The Sorites_.
§ 807. The Sorites is the neatest and most compendious form that can
be assumed by a train of reasoning.
§ 808. It is sometimes more appropriately called the chain-argument,
and map be defined as--
A train of reasoning, in which one premiss of each epi-syllogism is
supported by a pro-syllogism, the other being taken for granted.
This is its inner essence.
§ 809. In its outward form it may be described as--A series of
propositions, each of which has one term in common with that which
preceded it, while in the conclusion one of the terms in the last
proposition becomes either subject or predicate to one of the terms in
the first.
§ 810. A sorites may be either--
(1) Progressive,
or (2) Regressive.
_Progressive Sorites_.
All A is B.
All B is C.
All C is D.
All D is E.
.'. All A is E.
_Regressive Sorites_.
All D is E.
All C is D.
All B is C.
All A is B.
.'. All A is E.
§ 811. The usual form is the progressive; so that the sorites is
commonly described as a series of propositions in which the predicate
of each becomes the subject of the next, while in the conclusion the
last predicate is affirmed or denied of the first subject. The
regressive form, however, exactly reverses these attributes; and would
require to be described as a series of propositions, in which the
subject of each becomes the predicate of the next, while in the
conclusion the first predicate is affirmed or denied of the last
subject.
§ 812. The regressive sorites, it will be observed, consists of the
same propositions as the progressive one, only written in reverse
order. Why then, it may be asked, do we give a special name to it,
though we do not consider a syllogism different, if the minor premiss
happens to precede the major? It is because the sorites is not a mere
series of propositions, but a compressed train of reasoning; and the
two trains of reasoning may be resolved into their component
syllogisms in such a manner as to exhibit a real difference between
them.
§ 813. The Progressive Sorites is a train of reasoning in which the
minor premiss of each epi-syllogism is supported by a pro-syllogism,
while the major is taken for granted.
§ 814. The Regressive Sorites is a train of reasoning in which the
major premiss of each epi-syllogism is supported by a pro-syllogism,
while the minor is taken for granted.
_Progressive Sorites_.
(i) All B is C.
All A is B.
.'. All A is C.
(2) All C is D.
All A is C.
.'. All A is D.
(3) All D is E.
All A is D.
.'. All A is E.
_Regressive Sorites_.
(1) All D is E.
All C is D.
.'. All C is E.
(2) All C is E.
All B is C.
.'. All B is E.
(3) All B is E.
All A is B.
.'. All A is E.
§ 815. Here is a concrete example of the two kinds of sorites,
resolved each into its component syllogisms--
_Progressive Sorites_.
All Bideford men are Devonshire men.
All Devonshire men are Englishmen.
All Englishmen are Teutons.
All Teutons are Aryans.
.'. All Bideford men are Aryans.
(1) All Devonshire men are Englishmen.
All Bideford men are Devonshire men.
.'. All Bideford men are Englishmen.
(2) All Englishmen are Teutons.
All Bideford men are Englishmen.
.'. All Bideford men are Teutons.
(3) All Teutons are Aryans.
All Bideford men are Teutons.
.'. All Bideford men are Aryans.
_Regressive Sorites._
All Teutons are Aryans.
All Englishmen are Teutons.
All Devonshiremen are Englishmen.
All Bideford men are Devonshiremen.
.'. All Bideford men are Aryans.
(1) All Teutons are Aryans.
All Englishmen are Teutons.
.'. All Englishmen are Aryans.
(2) All Englishmen are Aryans.
All Devonshiremen are Englishmen.
.'. All Devonshiremen are Aryans.
(3) All Devonshiremen are Aryans.
All Bideford men are Devonshiremen.
.'. All Bideford men are Aryans.
§ 816. When expanded, the sorites is found to contain as many
syllogisms as there are propositions intermediate between the first
and the last. This is evident also on inspection by counting the
number of middle terms.
§ 817. In expanding the progressive form we have to commence with the
second proposition of the sorites as the major premiss of the first
syllogism. In the progressive form the subject of the conclusion is
the same in all the syllogisms; in the regressive form the predicate
is the same. In both the same series of means, or middle terms, is
employed, the difference lying in the extremes that are compared with
one another through them.
[Illustration]
§ 818. It is apparent from the figure that in the progressive form we
work from within outwards, in the regressive form from without
inwards. In the former we first employ the term 'Devonshiremen' as a
mean to connect 'Bideford men' with 'Englishmen'; next we employ
'Englishmen' as a mean to connect the same subject 'Bideford men' with
the wider term 'Teutons'; and, lastly, we employ 'Teutons' as a mean
to connect the original subject 'Bideford men' with the ultimate
predicate 'Ayrans.'
§ 819. Reversely, in the regressive form we first use 'Teutons' as a
mean whereby to bring 'Englishmen' under 'Aryans'; next we use
'Englishmen' as a mean whereby to bring 'Devonshiremen' under the dame
predicate 'Aryans'; and, lastly, we use 'Devonshiremen' as a mean
whereby to bring the ultimate subject 'Bideford men' under the
original predicate 'Aryans.'
§ 820. A sorites may be either Regular or Irregular.
§ 821. In the regular form the terms which connect each proposition in
the series with its predecessor, that is to say, the middle terms,
maintain a fixed relative position; so that, if the middle term be
subject in one, it will always be predicate in the other, and vice
versâ. In the irregular form this symmetrical arrangement is violated.
§ 822. The syllogisms which compose a regular sorites, whether
progressive or regressive, will always be in the first figure.
In the irregular sorites the syllogisms may fall into different
figures.
§ 823. For the regular sorites the following rules may
be laid down.
(1) Only one premiss can be particular, namely, the first, if the
sorites be progressive, the last, if it be regressive.
(2) Only one premiss can be negative, namely, the last, if the
sorites be progressive, the first, if it be regressive.
§ 824. _Proof of the Rules for the Regular Sorites_.
(1) In the progressive sorites the proposition which stands first is
the only one which appears as a minor premiss in the expanded
form. Each of the others is used in its turn as a major. If any
proposition, therefore, but the first were particular, there would
be a particular major, which involves undistributed middle, if the
minor be affirmative, as it must be in the first figure.
In the regressive sorites, if any proposition except the last were
particular, we should have a particular conclusion in the syllogism
in which it occurred as a premiss, and so a particular major in the
next syllogism, which again is inadmissible, as involving
undistributed middle.
(2) In the progressive sorites, if any premiss before the last were
negative, we should have a negative conclusion in the syllogism in
which it occurs. This would necessitate a negative minor in the next
syllogism, which is inadmissible in the first figure, as involving
illicit process of the major.
In the regressive sorites the proposition which stands first is the
only one which appears as a major premiss in the expanded form.
Each of the others is used in its turn as a minor. If any premiss,
therefore, but the first were negative, we should have a negative
minor in the first figure, which involves illicit process of the
major.
§ 825. The rules above given do not apply to the irregular sorites,
except so far as that only one premiss can be particular and only one
negative, which follows from the general rules of syllogism. But there
is nothing to prevent any one premiss from being particular or any one
premiss from being negative, as the subjoined examples will show. Both
the instances chosen belong to the progressive order of sorites.
(1) Barbara.
All B is A.
All C is B.
All C is A.
All B is A.
All C is B.
Some C is D.
All D is E
.'. Some A is E
[Illustration]
(2) Disamis.
Some C is D.
All C is A.
Some A is D.
(3) Darii.
All D is E
Some A is D.
Some A is E.
(1) Barbara.
All B is C.
All A is B.
All A is C.
All A is B.
All B is C.
No D is C.
All E is D.
.'. No A is E.
[Illustration]
(2) Cesare.
No D is C.
All A is C.
.'. No A is D.
(3) Camestres.
All E is D.
No A is D.
.'. No A is E.
§ 826. A chain argument may be composed consisting
of conjunctive instead of simple propositions. This is
subject to the same laws as the simple sorites, to which
it is immediately reducible.
_Progressive._ _Regressive._
If A is B, C is D. If E is F, G is H.
If C is D, E is F. If C is D, E is F.
If E is F, G is H. If A is B, C is D.
.'. If A is B, G is H. .'. If A is B, G is H.
CHAPTER XXX.
_Of Fallacies_.
§ 827. After examining the conditions on which correct thoughts
depend, it is expedient to classify some of the most familiar forms of
error. It is by the treatment of the Fallacies that logic chiefly
vindicates its claim to be considered a practical rather than a
speculative science. To explain and give a name to fallacies is like
setting up so many sign-posts on the various turns which it is
possible to take off the road of truth.
§ 828. By a fallacy is meant a piece of reasoning which appears to
establish a conclusion without really doing so. The term applies both
to the legitimate deduction of a conclusion from false premisses and
to the illegitimate deduction of a conclusion from any
premisses. There are errors incidental to conception and judgement,
which might well be brought under the name; but the fallacies with
which we shall concern ourselves are confined to errors connected with
inference.
§ 829. When any inference leads to a false conclusion, the error may
have arisen either in the thought itself or in the signs by which the
thought is conveyed. The main sources of fallacy then are confined to
two--
(1) thought,
(2) language.
§ 830. This is the basis of Aristotle's division of fallacies, which
has not yet been superseded. Fallacies, according to him, are either
in the language or outside of it. Outside of language there is no
source of error but thought. For things themselves do not deceive us,
but error arises owing to a misinterpretation of things by the
mind. Thought, however, may err either in its form or in its
matter. The former is the case where there is some violation of the
laws of thought; the latter whenever thought disagrees with its
object. Hence we arrive at the important distinction between Formal
and Material fallacies, both of which, however, fall under the same
negative head of fallacies other than those of language.
| In the language
| (in the signs of thought)
|
Fallacy -| |--In the Form.
|--Outside the language -|
| (in the thought itself) |
|
|--in the Matter.
§ 831. There are then three heads to which fallacies may be
referred-namely, Formal Fallacies, Fallacies of Language, which are
commonly known as Fallacies of Ambiguity, and, lastly, Material
Fallacies.
§ 832. Aristotle himself only goes so far as the first step in the
division of fallacies, being content to class them according as they
are in the language or outside of it. After that he proceeds at once
to enumerate the infimæ species under each of the two main heads. We
shall presently imitate this procedure for reasons of expediency. For
the whole phraseology of the subject is derived from Aristotle's
treatise on Sophistical Refutations, and we must either keep to his
method or break away from tradition altogether. Sufficient confusion
has already arisen from retaining Aristotle's language while
neglecting his meaning.
§ 833. Modern writers on logic do not approach fallacies from the same
point of view as Aristotle. Their object is to discover the most
fertile sources of error in solitary reasoning; his was to enumerate
the various tricks of refutation which could be employed by a sophist
in controversy. Aristotle's classification is an appendix to the Art
of Dialectic.
§ 834. Another cause of confusion in this part of logic is the
identification of Aristotle's two-fold division of fallacies, commonly
known under the titles of In dictione and Extra diotionem, with the
division into Logical and Material, which is based on quite a
different principle.
§ 835. Aristotle's division perhaps allows an undue importance to
language, in making that the principle of division, and so throwing
formal and material fallacies under a common head. Accordingly another
classification has been adopted, which concentrates attention from the
first upon the process of thought, which ought certainly to be of
primary importance in the eyes of the logician. This classification
is as follows.
§ 836. Whenever in the course of our reasoning we are involved in
error, either the conclusion follows from the premisses or it does
not. If it does not, the fault must lie in the process of reasoning,
and we have then what is called a Logical Fallacy. If, on the other
hand, the conclusion does follow from the premisses, the fault must
lie in the premisses themselves, and we then have what is called a
Material Fallacy. Sometimes, however, the conclusion will appear to
follow from the premisses until the meaning of the terms is examined,
when it will be found that the appearance is deceptive owing to some
ambiguity in the language. Such fallacies as these are, strictly
speaking, non-logical, since the meaning of words is extraneous to the
science which deals with thought. But they are called
Semi-logical. Thus we arrive by a different road at the same three
heads as before, namely, (1) Formal or Purely Logical Fallacies, (2)
Semi-logical Fallacies or Fallacies of Ambiguity, (3) Material
Fallacies.
§ 837. For the sake of distinctness we will place the two divisions
side by side, before we proceed to enumerate the infimae species.
|--In the language
| (Fallacy of Ambiguity)
Fallacy-|
| |--In the Form.
|--Outside the language -|
|
|--In the Matter.
|--Formal or purely logical.
|--Logical -|
Fallacy-| |--Semi-logical
| (Fallacy of Ambiguity).
|--Material
838. Of one of these three heads, namely, formal fallacies, it is not
necessary to say much, as they have been amply treated of in the
preceding pages. A formal fallacy arises from the breach of any of the
general rules of syllogism. Consequently it would be a formal fallacy
to present as a syllogism anything which had more or less than two
premisses. Under the latter variety comes what is called 'a woman's
reason,' which asserts upon its own evidence something which requires
to be proved. Schoolboys also have been known to resort to this form
of argument--'You're a fool.' 'Why?' 'Because you are.' When the
conclusion thus merely reasserts one of the premisses, the other must
be either absent or irrelevant. If, on the other hand, there are more
than two premisses, either there is more than one syllogism or the
superfluous premiss is no premiss at all, but a proposition irrelevant
to the conclusion.
839. The remaining rules of the syllogism are more able to be broken
than the first; so that the following scheme presents the varieties of
formal fallacy which are commonly enumerated--
|--Four Terms.
Formal Fallacy-|--Undistributed Middle.
|--Illicit Process.
|--Negative Premisses and Conclusion.
§ 840. The Fallacy of Four Terms is a violation of the second of the
general rules of syllogism (§ 582). Here is a palpable instance of
it--
All men who write books are authors.
All educated men could write books.
.'. All educated men are authors.
Here the middle term is altered in the minor premiss to the
destruction of the argument. The difference between the actual writing
of books and the power to write them is precisely the difference
between one who is an author and one who is not.
§ 841. Since a syllogism consists of three terms, each of which is
used twice over, it would be possible to have an apparent syllogism
with as many as six terms in it. The true name for the fallacy
therefore is the Fallacy of More than Three Terms. But it is rare to
find an attempted syllogism which has more than four terms in it, just
as we are seldom tendered a line as an hexameter, which has more than
seven feet.
§ 842. The Fallacies of Undistributed Middle and Illicit Process have
been treated of under §§ 585, 586. The heading 'Negative Premisses
and Conclusion' covers violations of the three general rules of
syllogism relating to negative premisses (§§ 590-593). Here is an
instance of the particular form of the fallacy which consists in the
attempt to extract an affirmative conclusion out of two negative
premisses--
All salmon are fish, for neither salmon nor fish belong to the class
mammalia.
The accident of a conclusion being true often helps to conceal the
fact that it is illegitimately arrived at. The formal fallacies which
have just been enumerated find no place in Aristotle's division. The
reason is plain. His object was to enumerate the various modes in
which a sophist might snatch an apparent victory, whereas by openly
violating any of the laws of syllogism a disputant would be simply
courting defeat.
§ 843. We now revert to Aristotle's classification of fallacies, or
rather of Modes of Refutation. We will take the species he enumerates
in their order, and notice how modern usage has departed from the
original meaning of the terms. Let it be borne in mind that, when the
deception was not in the language, Aristotle did not trouble himself
to determine whether it lay in the matter or in the form of thought.
§ 844. The following scheme presents the Aristotelian classification
to the eye at a glance:--
| |--Equivocation.
| |--Amphiboly.
|--In the language -|--Composition.
| |--Division.
| |--Accent.
| |--Figure of Speech.
Modes of -|
Refutation. | |--Accident.
| |--A dicto secundum quid.
| |--Ignoratio Elenchi.
|--Outside the language -|--Consequent.
| |--Petitio Principii.
| |--Non causa pro causa.
| |--Many Questions.
[Footnote: for "In the language": The Greek is [Greek: para ten lexin],
the exact meaning of which is; 'due to the statement.']
§ 845. The Fallacy of Equivocation [Greek: òmonumía] consists in an
ambiguous use of any of the three terms of a syllogism. If, for
instance, anyone were to argue thus--
No human being is made of paper,
All pages are human beings,
.'. No pages are made of paper--
the conclusion would appear paradoxical, if the minor term were there
taken in a different sense from that which it bore in its proper
premiss. This therefore would be an instance of the fallacy of
Equivocal Minor.
§ 846. For a glaring instance of the fallacy of Equivocal Major, we
may take the following--
No courageous creature flies,
The eagle is a courageous creature,
.'. The eagle does not fly--
the conclusion here becomes unsound only by the major being taken
ambiguously.
§ 847. It is, however, to the middle term that an ambiguity most
frequently attaches. In this case the fallacy of equivocation assumes
the special name of the Fallacy of Ambiguous Middle. Take as an
instance the following--
Faith is a moral virtue.
To believe in the Book of Mormon is faith.
.'. To believe in the Book of Mormon is a moral virtue.
Here the premisses singly might be granted; but the conclusion would
probably be felt to be unsatisfactory. Nor is the reason far to
seek. It is evident that belief in a book cannot be faith in any sense
in which that quality can rightly be pronounced to be a moral virtue.
§ 848. The Fallacy of Amphiboly ([Greek: ámphibolía]) is an ambiguity
attaching to the construction of a proposition rather than to the
terms of which it is composed. One of Aristotle's examples is this--
[Greek: tò boúlesthai labeîn me toùs polemíous]
which may be interpreted to mean either 'the fact of my wishing to
take the enemy,' or 'the fact of the enemies' wishing to take me.' The
classical languages are especially liable to this fallacy owing to the
oblique construction in which the accusative becomes subject to the
verb. Thus in Latin we have the oracle given to Pyrrhus (though of
course, if delivered at all, it must have been in Greek)--
Aio te, AEacida, Romanos vincere posse.
Pyrrhus the Romans shall, I say, subdue (Whately),
[Footnote: Cicero, De Divinatione, ii. § 116; Quintilian,
Inst. Orat. vii 9, § 6.]
which Pyrrhus, as the story runs, interpreted to mean that he could
conquer the Romans, whereas the oracle subsequently explained to him
that the real meaning was that the Romans could conquer him. Similar
to this, as Shakspeare makes the Duke of York point out, is the
witch's prophecy in Henry VI (Second Part, Act i, sc. 4),
The duke yet lives that Henry shall depose.
An instance of amphiboly may be read on the walls of Windsor
Castle--Hoc fecit Wykeham. The king mas incensed with the bishop for
daring to record that he made the tower, but the latter adroitly
replied that what he really meant to indicate was that the tower was
the making of him. To the same head may be referred the famous
sentence--'I will wear no clothes to distinguish me from my Christian
brethren.'
§ 849. The Fallacy of Composition [Greek: diaíresis] is likewise a
case of ambiguous construction. It consists, as expounded by
Aristotle, in taking words together which ought to be taken
separately, e.g.
'Is it possible for a man who is not writing to write?'
'Of course it is.'
'Then it is possible for a man to write without writing.'
And again--
'Can you carry this, that, and the other?' 'Yes.'
'Then you can carry this, that, and the other,'--
a fallacy against which horses would protest, if they could.
§ 850. It is doubtless this last example which has led to a convenient
misuse of the term 'fallacy of composition' among modern writers, by
whom it is defined to consist in arguing from the distributive to the
collective use of a term.
§ 851. The Fallacy of Division ([Greek: diaíresis]), on the other hand,
consists in taking words separately which ought to be taken together,
e.g.
[Greek: ègó s' êteka doûlon ônt' èleúteron [Footnote: Evidently the
original of the line in Terence's _Andria_, 37,--feci ex servo
ut esses libertus mihi.],
where the separation of [Greek: doûlon] from [Greek: ôntra] would lead
to an interpretation exactly contrary to what is intended.
And again--
[Greek: pentékont' àndrôn èkatòn lípe dîos Àchilleús],
where the separation of [Greek: àndrôn] from [Greek: èkatòn] leads to
a ludicrous error.
Any reader whose youth may have been nourished on 'The Fairchild
Family' may possibly recollect a sentence which ran somewhat on this
wise--'Henry,' said Mr. Fairchild, 'is this true? Are you a thief and
a liar too?' But I am afraid he will miss the keen delight which can
be extracted at a certain age from turning the tables upon
Mr. Fairchild thus--Henry said, 'Mr. Fairchild, is this true? Are
_you_ a thief and a liar too?'
§ 852. The fallacy of division has been accommodated by modern writers
to the meaning which they have assigned to the fallacy of
composition. So that by the 'fallacy of division' is now meant arguing
from the collective to the distributive use of a term. Further, it is
laid down that when the middle term is used distributively in the
major premiss and collectively in the minor, we have the fallacy of
composition; whereas, when the middle term is used collectively in the
major premiss and distributively in the minor, we have the fallacy of
division. Thus the first of the two examples appended would be
composition and the second division.
(1) Two and three are odd and even.
Five is two and three.
.'. Five is odd and even.
(2) The Germans are an intellectual people.
Hans and Fritz are Germans.
.'. They are intellectual people.
§ 853. As the possibility of this sort of ambiguity is not confined to
the middle term, it seems desirable to add that when either the major
or minor term is used distributively in the premiss and collectively
in the conclusion, we have the fallacy of composition, and in the
converse case the fallacy of division. Here is an instance of the
latter kind in which the minor term is at fault--
Anything over a hundredweight is too heavy to lift.
These sacks (collectively) are over a hundredweight.
.'. These sacks (distributively) are too heavy to lift.
§ 854. The ambiguity of the word 'all,' which has been before
commented upon (§ 119), is a great assistance in the English language
to the pair of fallacies just spoken of.
§ 835. The Fallacy of Accent ([Greek: prosodía]) is neither more nor
less than a mistake in Greek accentuation. As an instance Aristotle
gives Iliad xxiii. 328, where the ancient copies of Homer made
nonsense of the words [Greek: tò mèn oú katapútetai ómbro] by writing
[Greek: oû] with the circumflex in place of [Greek: oú] with the acute
accent. [Footnote: This goes to show that the ancient Greeks did not
distinguish in pronunciation between the rough and smooth breathing
any more than their modern representatives.] Aristotle remarks that
the fallacy is one which cannot easily occur in verbal argument, but
rather in writing and poetry.
§ 856. Modern writers explain the fallacy of accent to be the mistake
of laying the stress upon the wrong part of a sentence. Thus when the
country parson reads out, 'Thou shall not bear false witness
_against_ thy neighbour,' with a strong emphasis upon the word
'against,' his ignorant audience leap [sic] to the conclusion that it
is not amiss to tell lies provided they be in favour of one's
neighbour.
§ 857. The Fallacy of Figure of Speech [Greek: tò schêma tês léxeos]
results from any confusion of grammatical forms, as between the
different genders of nouns or the different voices of verbs, or their
use as transitive or intransitive, e.g. [Greek: úgiaínein] has the
same grammatical form as [Greek: témnein] or [Greek: oìkodomeîn], but
the former is intransitive, while the latter are transitive. A sophism
of this kind is put into the mouth of Socrates by Aristophanes in the
Clouds (670-80). The philosopher is there represented as arguing that
[Greek: kápdopos] must be masculine because [Greek: Kleónumos] is. On
the surface this is connected with language, but it is essentially a
fallacy of false analogy.
§ 858. To this head may be referred what is known as the Fallacy of
Paronymous Terms. This is a species of equivocation which consists in
slipping from the use of one part of speech to that of another, which
is derived from the same source, but has a different meaning. Thus
this fallacy would be committed if, starting from the fact that there
is a certain probability that a hand at whist will consist of thirteen
trumps, one were to proceed to argue that it was probable, or that he
had proved it.
§ 859. We turn now to the tricks of refutation which lie outside the
language, whether the deception be due to the assumption of a false
premiss or to some unsoundness in the reasoning.
§ 860. The first on the list is the Fallacy of Accident ([Greek: tò
sumbebekós]). This fallacy consists in confounding an essential with
an accidental difference, which is not allowable, since many things
are the same in essence, while they differ in accidents. Here is the
sort of example that Aristotle gives--
'Is Plato different from Socrates ?' 'Yes.' 'Is Socrates a man ?'
'Yes.' 'Then Plato is different from man.'
To this we answer--No: the difference of accidents between Plato and
Socrates does not go so deep as to affect the underlying essence. To
put the thing more plainly, the fallacy lies in assuming that whatever
is different from a given subject must be different from it in all
respects, so that it is impossible for them to have a common
predicate. Here Socrates and Plato, though different from one another,
are not so different but that they have the common predicate 'man.'
The attempt to prove that they have not involves an illicit process of
the major.
§ 861. The next fallacy suffers from the want of a convenient name. It
is called by Aristotle [Greek: tò áplos tóde ê pê légestai kaì mè
kupíos] or, more briefly, [Greek: tò áplôs ê mé], or [Greek: tò pê kaí
áplôs], and by the Latin writers 'Fallacia a dicto secundum quid ad
dictum simpliciter.' It consists in taking what is said in a
particular respect as though it held true without any restriction,
e.g., that because the nonexistent ([Greek: tò mè ôn]) is a matter of
opinion, that therefore the non-existent is, or again that because the
existent ([Greek: tò ôn]) is not a man, that therefore the existent is
not. Or again, if an Indian, who as a whole is black, has white teeth,
we should be committing this species of fallacy in declaring him to be
both white and not-white. For he is only white in a certain respect
([Greek: pê]), but not absolutely ([Greek: àplôs]). More
difficulty, says Aristotle, may arise when opposite qualities exist in
a thing in about an equal degree. When, for instance, a thing is half
white and half black, are we to say that it is white or black? This
question the philosopher propounds, but does not answer. The force of
it lies in the implied attack on the Law of Contradiction. It would
seem in such a case that a thing may be both white and not-white at
the same time. The fact is--so subtle are the ambiguities of
language--that even such a question as 'Is a thing white or
not-white?' straightforward, as it seems, is not really a fair one. We
are entitled sometimes to take the bull by the horns, and answer with
the adventurous interlocutor in one of Plato's dialogues--'Both and
neither.' It may be both in a certain respect, and yet neither
absolutely.
§ 862. The same sort of difficulties attach to the Law of Excluded
Middle, and may be met in the same way. It might, for instance, be
urged that it could not be said with truth of the statue seen by
Nebuchadnezzar in his dream either that it was made of gold or that it
was not made of gold: but the apparent plausibility of the objection
would be due merely to the ambiguity of language. It is not true, on
the one hand, that it was made of gold (in the sense of being composed
entirely of that metal); and it is not true, on the other, that it was
not made of gold (in the sense of no gold at all entering into its
composition). But let the ambiguous proposition be split up into its
two meanings, and the stringency of the Law of Excluded Middle will at
once appear--
(1) It must either have been composed entirely of gold or not.
(2) Either gold must have entered into its composition or not.
§ 863. By some writers this fallacy is treated as the converse of the
last, the fallacy of accident being assimilated to it under the title
of the 'Fallacia a dicto simpliciter ad dictum secundum quid.' In this
sense the two fallacies may be defined thus.
The Fallacy of Accident consists in assuming that what holds true as a
general rule will hold true under some special circumstances which may
entirely alter the case. The Converse Fallacy of Accident consists in
assuming that what holds true under some special circumstances must
hold true as a general rule.
The man who, acting on the assumption that alcohol is a poison,
refuses to take it when he is ordered to do so by the doctor, is
guilty of the fallacy of accident; the man who, having had it
prescribed for him when he was ill, continues to take it morning,
noon, and night, commits the converse fallacy.
§ 864. There ought to be added a third head to cover the fallacy of
arguing from one special case to another.
§ 865. The next fallacy is Ignoratio Elenchi [Greek: èlégchou
âgnoia]. This fallacy arises when by reasoning valid in itself one
establishes a conclusion other than what is required to upset the
adversary's assertion. It is due to an inadequate conception of the
true nature of refutation. Aristotle therefore is at the pains to
define refutation at full length, thus--
'A refutation [Greek: êlegchos] is the denial of one and the same--not
name, but thing, and by means, not of a synonymous term, but of the
same term, as a necessary consequence from the data, without
assumption of the point originally at issue, in the same respect, and
in the same relation, and in the same way, and at the same time.'
The ELENCHUS then is the exact contradictory of the opponent's
assertion under the terms of the law of contradiction. To establish by
a syllogism, or series of syllogisms, any other proposition, however
slightly different, is to commit this fallacy. Even if the substance
of the contradiction be established, it is not enough unless the
identical words of the opponent are employed in the
contradictory. Thus if his thesis asserts or denies something about
[Greek: lópion], it is not enough for you to prove the contradictory
with regard to [Greek: ìmátion]. There will be need of a further
question and answer to identify the two, though they are admittedly
synonymous. Such was the rigour with which the rules of the game of
dialectic were enforced among the Greeks!
§ 866. Under the head of Ignoratio Elenchi it has become usual to
speak of various forme of argument which have been labelled by the
Latin writers under such names as 'argumentum ad hominem,' 'ad
populum,' 'ad verecundiam,' 'ad ignorantiam,' 'ad baculum'--all of
them opposed to the 'argumentum ad rem' or 'ad judicium.'
§ 867. By the 'argumentum ad hominem' was perhaps meant a piece of
reasoning which availed to silence a particular person, without
touching the truth of the question. Thus a quotation from Scripture
is sufficient to stop the mouth of a believer in the inspiration of
the Bible. Hume's Essay on Miracles is a noteworthy instance of the
'argumentum ad hominem' in this sense of the term. He insists strongly
on the evidence for certain miracles which he knew that the prejudices
of his hearers would prevent their ever accepting, and then asks
triumphantly if these miracles, which are declared to have taken place
in an enlightened age in the full glare of publicity, are palpably
imposture, what credence can be attached to accounts of extraordinary
occurrences of remote antiquity, and connected with an obscure corner
of the globe? The 'argumentum ad judicium' would take miracles as a
whole, and endeavour to sift the amount of truth which may lie in the
accounts we have of them in every age. [Footnote: On this subject see
the author's _Attempts at Truth_ (Trubner & Co.), pp. 46-59.]
§ 868. In ordinary discourse at the present day the term 'argumentum
ad hominem' is used for the form of irrelevancy which consists in
attacking the character of the opponent instead of combating his
arguments, as illustrated in the well-known instructions to a
barrister--'No case: abuse the plaintiff's attorney.'
§ 869. The 'argumentum ad populum' consists in an appeal to the
passions of one's audience. An appeal to passion, or to give it a less
question-begging name, to feeling, is not necessarily amiss. The heart
of man is the instrument upon which the rhetorician plays, and he has
to answer for the harmony or the discord that comes of his
performance.
§ 870. The 'argumentum ad verecundiam' is an appeal to the feeling of
reverence or shame. It is an argument much used by the old to the
young and by Conservatives to Radicals.
§ 871. The 'argumentum ad ignorantiam' consists simply in trading on
the ignorance of the person addressed, so that it covers any kind of
fallacy that is likely to prove effective with the hearer.
§ 872. The 'argumentum ad baculum' is unquestionably a form of
irrelevancy. To knock a man down when he differs from you in opinion
may prove your strength, but hardly your logic.
A sub-variety of this form of irrelevancy was exhibited lately at a
socialist lecture in Oxford, at which an undergraduate, unable or
unwilling to meet the arguments of the speaker, uncorked a bottle,
which had the effect of instantaneously dispersing the audience. This
might be set down as the 'argumentum ad nasum.'
§ 873. We now come to the Fallacy of the Consequent, a term which has
been more hopelessly abused than any. What Aristotle meant by it was
simply the assertion of the consequent in a conjunctive proposition,
which amounts to the same thing as the simple conversion of A (§ 489),
and is a fallacy of distribution. Aristotle's example is this--
If it has rained, the ground is wet.
.'. If the ground is wet, it has rained.
This fallacy, he tells us, is often employed in rhetoric in dealing
with presumptive evidence. Thus a speaker, wanting to prove that a man
is an adulterer, will argue that he is a showy dresser, and has been
seen about at nights. Both these things however may be the case, and
yet the charge not be true.
§ 874. The Fallacy of Petitio or Assumptio Principii [Greek: tò èn
àrchê aìteîstai or lambánein] to which we now come, consists in an
unfair assumption of the point at issue. The word [Greek: aìteîstai],
in Aristotle's name for it points to the Greek method of dialectic by
means of question and answer. This fact is rather disguised by the
mysterious phrase 'begging the question.' The fallacy would be
committed when you asked your opponent to grant, overtly or covertly,
the very proposition originally propounded for discussion.
§ 875. As the question of the precise nature of this fallacy is of
some importance we will take the words of Aristotle himself
(Top. viii. 13. §§ 2, 3). 'People seem to beg the question in five
ways. First and most glaringly, when one takes for granted the very
thing that has to be proved. This by itself does not readily escape
detection, but in the case of "synonyms," that is, where the name and
the definition have the same meaning, it does so more
easily. [Footnote: Some light is thrown upon this obscure passage by a
comparison with Cat. I. § 3, where 'synonym' is defined. To take the
word here in its later and modern sense affords an easy
interpretation, which is countenanced by Alexander Aphrodisiensis, but
it is flat against the usage of Aristotle, who elsewhere gives the
name 'synonym,' not to two names for the same thing, but to two things
going under the same name. See Trendelenberg on the passage.]
Secondly, when one assumes universally that which has to be proved in
particular, as, if a man undertaking to prove that there is one
science of contraries, were to assume that there is one science of
opposites generally. For he seems to be taking for granted along with
several other things what he ought to have proved by itself.
Thirdly, when one assumes the particulars where the universal has to
be proved; for in so doing a man is taking for granted separately what
he was bound to prove along with several other things. Again, when
one assumes the question at issue by splitting it up, for instance,
if, when the point to be proved is that the art of medicine deals with
health and disease, one were to take each by itself for granted.
Lastly, if one were to take for granted one of a pair of necessary
consequences, as that the side is incommensurable with the diagonal,
when it is required to prove that the diagonal is incommensurable with
the side.'
§ 876. To sum up briefly, we may beg the question in five ways--
(1) By simply asking the opponent to grant the point which requires
to be proved;
(2) by asking him to grant some more general truth which involves
it;
(3) by asking him to grant the particular truths which it involves;
(4) by asking him to grant the component parts of it in detail;
(5) by asking him to grant a necessary consequence of it.
§ 877. The first of these five ways, namely, that of begging the
question straight off, lands us in the formal fallacy already spoken
of (§ 838), which violates the first of the general rules of
syllogism, inasmuch as a conclusion is derived from a single premiss,
to wit, itself.
§ 878. The second, strange to say, gives us a sound syllogism in
Barbara, a fact which countenances the blasphemers of the syllogism in
the charge they bring against it of containing in itself a petitio
principii. Certainly Aristotle's expression might have been more
guarded. But it is clear that his quarrel is with the matter, not with
the form in such an argument. The fallacy consists in assuming a
proposition which the opponent would be entitled to deny. Elsewhere
Aristotle tells us that the fallacy arises when a truth not evident by
its own light is taken to be so. [Footnote: [Greek: Ôtan tò mè dí
aùtoû gnostòn dí aùtoû tis èpicheiraê deiknúnai, tót' aìteîtai tò èx
àrchês.]. Anal. Pr. II. 16. § I ad fin.]
§ 879. The third gives us an inductio per enumerationem simplicem, a
mode of argument which would of course be unfair as against an
opponent who was denying the universal.
§ 880. The fourth is a more prolix form of the first.
§ 881. The fifth rests on Immediate Inference by Relation (§ 534).
§ 882. Under the head of petitio principii comes the fallacy of
Arguing in a Circle, which is incidental to a train of reasoning. In
its most compressed form it may be represented thus--
(1) B is A.
C is B.
.'. C is A.
(2) C is A.
B is C.
.'. B is A.
§ 883. The Fallacy of Non causa pro causa ([Greek: tò mè aîtion] or
[Greek: aîtoin]) is another, the name of which has led to a complete
misinterpretation. It consists in importing a contradiction into the
discussion, and then fathering it on the position controverted. Such
arguments, says Aristotle, often impose upon the users of them
themselves. The instance he gives is too recondite to be of general
interest.
§ 884. Lastly, the Fallacy of Many Questions ([Greek: tò tà déo
èrotémata ên poieîn]) is a deceptive form of interrogation, when a
single answer is demanded to what is not really a single question. In
dialectical discussions the respondent was limited to a simple 'yes'
or 'no'; and in this fallacy the question is so framed as that either
answer would seem to imply the acceptance of a proposition which would
be repudiated. The old stock instance will do as well as
another--'Come now, sir, answer "yes" or "no." Have you left off
beating your mother yet?' Either answer leads to an apparent
admission of impiety.
A late Senior Proctor once enraged a man at a fair with this form of
fallacy. The man was exhibiting a blue horse; and the distinguished
stranger asked him--'With what did you paint your horse?'
EXERCISES.
These exercises should be supplemented by direct questions upon the
text, which it is easy for the student or the teacher to supply for
himself.
PART I.
CHAPTER I.
Classify the following words according as they are categorematic,
syncategorematic or acategorematic;--
come peradventure why
through inordinately pshaw
therefore circumspect puss
grand inasmuch stop
touch sameness back
cage disconsolate candle.
CHAPTER II.
Classify the following things according as they are substances,
qualities or relations;--
God likeness weight
blueness grass imposition
ocean introduction thinness
man air spirit
Socrates raillery heat
mortality plum fire.
CHAPTER III.
1. Give six instances each of-attribute, abstract, singular,
privative, equivocal and relative terms.
2. Select from the following list of words such as are terms, and
state whether they are (1) abstract or concrete, (2) singular or
common, (3) univocal or equivocal:--
van table however
enter decidedly tiresome
very butt Solomon
infection bluff Czar
short although Caesarism
distance elderly Nihilist.
3. Which of the following words are abstract terms?--
quadruped event through
hate desirability thorough
fact expressly thoroughness
faction wish light
inconvenient will garden
inconvenience volition grind.
4. Refer the following terms to their proper place under each of the
divisions in the scheme:--
horse husband London
free lump empty
liberty rational capital
impotent reason Capitol
impetuosity irrationality grave
impulsive double calf.
5. Give six instances each of proper names and designations.
6. Give six instances each of connotative and non-connotative terms.
7. Give the extension and intension of--
sermon animal sky
clock square gold
sport fish element
bird student fluid
art river line
gas servant language
CHAPTER IV.
Arrange the following terms in order of extension--carnivorous, thing,
matter, mammal, organism, vertebrate, cat, substance, animal.
* * * * *
PART II.
CHAPTER I.
Give a name to each of the following sentences:--
(1) Oh, that I had wings like a dove!
(2) The more, the merrier.
(3) Come rest in this bosom, my own stricken deer.
(4) Is there balm in Gilead?
(5) Hearts may be trumps.
CHAPTER II.
Analyse the following propositions into subject, copula and
predicate:--
(1) He being dead yet speaketh.
(2) There are foolish politicians.
(3) Little does he care.
(4) There is a land of pure delight.
(5) All's well that ends well.
(6) Sweet is the breath of morn.
(7) Now it came to pass that the beggar died.
(8) Who runs may read.
(9) Great is Diana of the Ephesians.
(10) Such things are.
(11) Not more than others I deserve.
(12) The day will come when Ilium's towers shall perish.
CHAPTER III.
1. Express in logical form, affixing the proper symbol:--
(1) Some swans are not white.
(2) All things are possible to them that believe.
(3) No politicians are unprincipled.
(4) Some stones float on water.
(5) The snow has melted.
(6) Eggs are edible.
(7) All kings are not wise.
(8) Moths are not butterflies.
(9) Some men are born great.
(10) Not all who are called are chosen.
(11) It is not good for man to be alone.
(12) Men of talents have been known to fail in life.
(13) 'Tis none but a madman would throw about fire.
(14) Every bullet does not kill.
(15) Amongst Unionists are Whigs.
(16) Not all truths are to be told.
(17) Not all your efforts can save him.
(18) The whale is a mammal.
(19) Cotton is grown in Cyprus.
(20) An honest man's the noblest work of God.
(21) No news is good news.
(22) No friends are like old friends.
(23) Only the ignorant affect to despise knowledge.
(24) All that trust in Him shall not be ashamed.
(25) All is not gold that glitters.
(26) The sun shines upon the evil and upon the good.
(27) Not to go on is to go back.
(28) The king, minister, and general are a pretty trio.
(29) Amongst dogs are hounds.
(30) A fool is not always wrong.
(31) Alexander was magnanimous.
(32) Food is necessary to life.
(33) There are three things to be considered,
(34) By penitence the Eternal's wrath's appeased.
(35) Money is the miser's end.
(36) Few men succeed in life.
(37) All is lost, save honour.
(38) It is mean to hit a man when he is down.
(39) Nothing but coolness could have saved him.
(40) Books are generally useful.
(41) He envies others' virtue who has none himself.
(42) Thankless are all such offices.
(43) Only doctors understand this subject.
(44) All her guesses but two were correct.
(45) All the men were twelve.
(46) Gossip is seldom charitable.
2. Give six examples of indefinite propositions, and then quantify
them according to their matter.
3. Compose three propositions of each of the following kinds:--
(1) with common terms for subjects;
(2) with abstract terms for subjects;
(3) with singular terms for predicates;
(4) with collective terms for predicates;
(5) with attributives in their subjects;
(6) with abstract terms for predicates.
CHAPTER IV.
1. Point out what terms are distributed or undistributed in the
following propositions:--
(1) The Chinese are industrious.
(2) The angle in a semi-circle is a right angle.
(3) Not one of the crew survived.
(4) The weather is sometimes not propitious.
The same exercise may be performed upon any of the propositions in the
preceding list.
2. Prove that in a negative proposition the predicate must be
distributed.
CHAPTER V.
Affix its proper symbol to each of the following propositions:--
(1) No lover he who is not always fond.
(2) There are Irishmen and Irishmen.
(3) Men only disagree,
Of creatures rational.
(4) Some wise men are poor.
(5) No Popes are some fallible beings.
(6) Some step-mothers are not unjust.
(7) The most original of the Roman poets was Lucretius.
(8) Some of the immediate inferences are all the forms of
conversion.
CHAPTER VI.
1. Give six examples of terms standing one to another as genus to
species.
2. To which of the heads of predicables would you refer the following
statements? And why?
(1) A circle is the largest space that can be contained by one line.
(2) All the angles of a square are right angles.
(3) Man alone among animals possesses the faculty of laughter.
(4) Some fungi are poisonous.
(5) Most natives of Africa are negroes.
(6) All democracies are governments.
(7) Queen Anne is dead.
CHAPTER VII.
1. Define the following terms--
Sun inn-keeper tea-pot
hope anger virtue
bread diplomacy milk
carpet man death
sincerity telescope mountain
poverty Senate novel.
2. Define the following terms as used in Political Economy--
Commodity barter value
wealth land price
money labour rent
interest capital wages
credit demand profits.
3. Criticise the following as definitions--
(1) Noon is the time when the shadows of bodies are shortest.
(2) Grammar is the science of language.
(3) Grammar is a branch of philology.
(4) Grammar is the art of speaking and writing a language with
propriety.
(5) Virtue is acting virtuously.
(6) Virtue is that line of conduct which tends to produce happiness.
(7) A dog is an animal of the canine species.
(8) Logic is the art of reasoning.
(9) Logic is the science of the investigation of truth by means of
evidence.
(10) Music is an expensive noise.
(11) The sun is the centre of the solar system.
(12) The sun is the brightest of those heavenly bodies that move
round the earth.
(13) Rust is the red desquamation of old iron.
(14) Caviare is a kind of food.
(15) Life is the opposite of death.
(16) Man is a featherless biped.
(17) Man is a rational biped.
(18) A gentleman is a person who has no visible means of
subsistence.
(19) Fame is a fancied life in others' breath.
(20) A fault is a quality productive of evil or inconvenience.
(21) An oligarchy is the supremacy of the rich in a state.
(22) A citizen is one who is qualified to exercise deliberative and
judicial functions.
(23) Length is that dimension of a solid which would be measured by
the longest line.
(24) An eccentricity is a peculiar idiosyncrasy.
(25) Deliberation is that species of investigation which is
concerned with matters of action.
(26) Memory is that which helps us to forget.
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