Part 4 out of 6
Nomen habent nullum, nee, si bene colligis, usum.
§ 630. The vowels in these lines indicate the letters of the mood. All
the special rules of the four figures can be gathered from an
inspection of them. The following points should be specially noted.
The first figure proves any kind of conclusion, and is the only one
which can prove A.
The second figure proves only negatives.
The third figure proves only particulars.
The fourth figure proves any conclusion except A.
§ 631. The first figure is called the Perfect, and the rest the
Imperfect figures. The claim of the first to be regarded as the
perfect figure may be rested on these grounds--
1. It alone conforms directly to the Dictum de Omni et Nullo.
2. It suffices to prove every kind of conclusion, and is the only
figure in which a universal affirmative proposition can be
established.
3. It is only in a mood of this figure that the major, middle and
minor terms are to be found standing in their relative order of
extension.
§ 632. The reason why a universal affirmative, which is of course
infinitely the most important form of proposition, can only be proved
in the first figure may be seen as follows.
_Proof that A can only be established in figure I._
An A conclusion necessitates both premisses being A propositions (by
Rule 7). But the minor term is distributed in the conclusion, as being
the subject of an A proposition, and must therefore be distributed in
the minor premiss, in order to which it must be the subject. Therefore
the middle term must be the predicate and is consequently
undistributed. In order therefore that the middle term may be
distributed, it must be subject in the major premiss, since that also
is an A proposition. But when the middle term is subject in the major
and predicate in the minor premiss, we have what is called the first
figure.
_Of the Special Canons of the Four Figures._
§ 633. So far we have given only a negative test of legitimacy, having
shown what moods are not invalidated by running counter to any of the
special rules of the four figures. We will now lay down special canons
for the four figures, conformity to which will serve as a positive
test of the validity of a given mood in a given figure. The special
canon of the first figure--will of course be practically equivalent to
the Dictum de Omni et Nullo. All of them will be expressed in terms of
extension, for the sake of perspicuity.
_Special Canons of the Four Figures._
§ 634. CANON. If one term wholly includes or excludes another, which
wholly or partly includes a third, the first term wholly or partly
includes or excludes the third.
(1) Total inclusion (Barbara).
All B is A.
All C is B.
.'. All C is A.
(2) Partial inclusion (Darii).
All B is A.
Some C is B.
.'. Some C is A.
(3) Total exclusion (Celarent).
No B is A.
All C is B.
.'. No C is A.
(4) Partial exclusion (Ferio).
No B is A.
Some C is B.
.'. Some C is not A.
§ 635. CANON. If one term is excluded from another, which wholly or
partly includes a third, or is included in another from which a third
is wholly or partly excluded, the first is excluded from the whole or
part of the third.
Here we have four cases, all of exclusion--
(1) Total exclusion on the ground of inclusion in an excluded term
(Cesare).
No A is B.
All C is B.
.'. No C is A.
(2) Partial exclusion on the ground of a similar partial inclusion
(Festino).
No A is B.
Some C is B.
.'. Some C is not A.
(3) Total exclusion on the ground of exclusion from an including
term (Camestres).
All A is B.
No C is B.
.'. No C is A.
(4) Partial exclusion on the ground of a similar partial exclusion
(Baroko).
All A is B.
Some C is not B.
.'. Some C is not A.
§ 636. CANON. If two terms include another term in common, or if the
first includes the whole and the second a part of the same term, or
vice versâ, the first of these two terms partly includes the second;
and if the first is excluded from the whole of a term which is wholly
or in part included in the second, or is excluded from part of a term
which is wholly included in the second, the first is excluded from
part of the second.
Here it is evident from the statement that six cases arise--
(1) Total inclusion of the same term in two others
(Darapti).
All B is A.
All B is C.
.'. some C is A.
(2) Total inclusion in the first and partial inclusion
in the second (Datisi).
All B is A.
Some B is C.
.'. some C is A.
(3) Partial inclusion in the first and total inclusion in
the second (Disamis).
Some B is A.
All B is C.
.'. some C is A.
(4) Total exclusion of the first from a term which is
wholly included in the second (Felapton).
No B is A.
All B is C.
.'. some C is not A.
(5) Total exclusion of the first from a term which is
partly included in the second (Ferison).
No B is A.
Some B is C.
.'. some C is not A.
(6) Exclusion of the first from part of a term which
is wholly included in the second (Bokardo).
Some B is not A.
All B is C.
.'. Some C is not A.
§ 637. CANON. If one term is wholly or partly included in another
which is wholly included in or excluded from a third, the third term
wholly or partly includes the first, or, in the case of total
inclusion, is wholly excluded from it; and if a term is excluded from
another which is wholly or partly included in a third, the third is
partly excluded from the first.
(1) Of the inclusion of a whole term (Bramsntip).
[Illustration]
All A is B.
All B is C.
.'. Some C is (all) A.
(2) Of the inclusion of part of a term (DIMARIS).
Some A is B.
All B is C.
.'. Some C is (some) A,
(3) Of the exclusion of a whole term (Camenes).
All A is B.
No B is C.
.'. No C is A.
(4) Partial exclusion on the ground of including
the whole of an excluded term (Fesapo).
No A is B.
All B is C.
.'. Some C is not A.
(5) Partial exclusion on the ground of including
part of an excluded term (Fresison).
No A is B.
Some B is C.
.'. Some C is not A.
§ 638. It is evident from the diagrams that in the subaltern moods the
conclusion is not drawn directly from the premisses, but is an
immediate inference from the natural conclusion. Take for instance AAI
in the first figure. The natural conclusion from these premisses is
that the minor term C is wholly contained in the major term A. But
instead of drawing this conclusion we go on to infer that something
which is contained in C, namely some C, is contained in A.
All B is A.
All C is B.
.'. all C is A.
.'. some C is A.
Similarly in EAO in figure 1, instead of arguing that the whole of C
is excluded from A, we draw a conclusion which really involves a
further inference, namely that part of C is excluded from A.
No B is A.
All C is B.
.'. no C is A.
.'. some C is not A.
§ 639. The reason why the canons have been expressed in so cumbrous a
form is to render the validity of all the moods in each figure at once
apparent from the statement. For purposes of general convenience they
admit of a much more compendious mode of expression.
§ 640. The canon of the first figure is known as the Dictum de Omni et
Nullo--
What is true (distributively) of a whole term is true of all that it
includes.
§ 641. The canon of the second figure is known as the Dictum de
Diverse--
If one term is contained in, and another excluded from a third term,
they are mutually excluded.
§ 642. The canon of the third figure is known as the Dictum de Exemplo
et de Excepto--
Two terms which contain a common part partly agree, or, if one
contains a part which the other does not, they partly differ.
§ 643. The canon of the fourth figure has had no name assigned to it,
and does not seem to admit of any simple expression. Another mode of
formulating it is as follows:--
Whatever is affirmed of a whole term may have partially affirmed of
it whatever is included in that term (Bramantip, Dimaris), and
partially denied of it whatever is excluded (Fesapo); whatever is
affirmed of part of a term may have partially denied of it whatever
is wholly excluded from that term (Fresison); and whatever is denied
of a whole term may have wholly denied of it whatever is wholly
included in that term (Camenes).
§ 644. From the point of view of intension the canons of the first
three figures may be expressed as follows.
§ 645. Canon of the first figure. Dictum de Omni et Nullo--
An attribute of an attribute of anything is an attribute of the
thing itself.
§ 646. Canon of the second figure. Dictum de Diverso--
If a subject has an attribute which a class has not, or vice versa,
the subject does not belong to the class.
§ 647. Canon of the third figure.
If a certain attribute can be affirmed of any portion of the
members of a class, it is not incompatible with the distinctive
attributes of that class.
If a certain attribute can be denied of any portion of the members
of a class, it is not inseparable from the distinctive attributes
of that class.
_Of the Special Uses of the Four Figures._
§ 648. The first figure is useful for proving the properties of a
thing.
§ 649. The second figure is useful for proving distinctions between
things.
§ 650. The third figure is useful for proving instances or exceptions.
§ 651. The fourth figure is useful for proving the species of a genus.
B is or is not A.
C is B.
.'. C is or is not A.
We prove that C has or has not the property A by predicating of it B,
which we know to possess or not to possess that property.
Luminous objects are material.
Comets are luminous.
.'. Comets are material.
No moths are butterflies.
The Death's head is a moth.
.'. The Death's head is not a butterfly.
A is B. A is not B.
C is not B. C is B.
.'. C is not A. .'. C is not A.
We establish the distinction between C and A by showing that A has an
attribute which C is devoid of, or is devoid of an attribute which C
has.
All fishes are cold-blooded.
A whale is not cold-blooded.
.'. A whale is not a fish.
No fishes give milk.
A whale gives milk.
.'. A whale is not a fish.
B is A. B is not A.
B is C. B is C.
.'. Some C is A. .'. Some C is not A.
We produce instances of C being A by showing that C and A meet, at all
events partially, in B. Thus if we wish to produce an instance of the
compatibility of great learning with original powers of thought, we
might say
Sir William Hamilton was an original thinker.
Sir William Hamilton was a man of great learning.
.'. Some men of great learning are original thinkers.
Or we might urge an exception to the supposed rule about Scotchmen
being deficient in humour under the same figure, thus--
Sir Walter Scott was not deficient in humour.
Sir Walter Scott was a Scotchman.
.'. Some Scotchmen are not deficient in humour.
All A is B, No A is B.
All B is C. All B is C.
.'. Some C is A .'.Some C is not A.
We show here that A is or is not a species of C by showing that A
falls, or does not fall, under the class B, which itself falls under
C. Thus--
All whales are mammals.
All mammals are warm-blooded.
.'. Some warm-blooded animals are whales.
No whales are fishes.
All fishes are cold-blooded.
.'. Some cold-blooded animals are not whales.
_Of the Syllogism with three figures._
§ 656. It will be remembered that in beginning to treat of figure (§
565) we pointed out that there were either four or three ligures
possible according as the conclusion was assumed to be known or
not. For, if the conclusion be not known, we cannot distinguish
between the major and the minor term, nor, consequently, between one
premiss and another. On this view the first and the fourth figures are
the same, being that arrangement of the syllogism in which the middle
term occupies a different position in one premiss from what it does in
the other. We will now proceed to constitute the legitimate moods and
figures of the syllogism irrespective of the conclusion.
§ 657. When the conclusion is set out of sight, the number of possible
moods is the same as the number of combinations that can be made of
the four things, A, E, I, O, taken two together, without restriction
as to repetition. These are the following 16:--
AA EA IA OA
AE -EE- IE -OE-
AI EI -II- -OI-
AO -EO- -IO- -OO-
of which seven may be neglected as violating the general rules of the
syllogism, thus leaving us with nine valid moods--
AA. AE. AI. AO. EA. EI. IA. IE. OA.
§ 658. We will now put these nine moods successively into the three
figures. By so doing it will become apparent how far they are valid in
each.
§ 659. Let it be premised that
when the extreme in the premiss that stands first is predicate in
the conclusion, we are said to have a Direct Mood;
when the extreme in the premiss that stands second is predicate in
the conclusion, we are said to have an Indirect Mood.
_Mood AA._
All B is A.
All C is B.
.'. All C is A, or Some A is C, (Barbara & Bramantip).
_Mood AE._
All B is A.
No C is B.
.'. Illicit Process, or Some A is not C, (Fesapo).
_Mood AI._
All B is A.
Some C is B.
.'. Some C is A, or Some A is C. (Darii & Disamis).
_Mood AO._
All B is A.
Some C is not B.
.'. Illicit Process, (Ferio).
_Mood EA._
No B is A.
All C is B.
.'. No C is A, or No A is C, (Celarent & Camenes).
_Mood EI._
No B is A.
Some C is B.
.'. Some C is not A, or Illicit Process.
_Mood IA._
Some B is A.
All C is B.
.'. Undistributed Middle.
_Mood IE._
Some B is C. Some B is not A.
No A is B. All C is B.
.'. Illicit Process, or Some C is not A, (Fresison).
_Mood OA._
Some B is not A.
All C is B.
.'. Undistributed Middle.
§ 661. Thus we are left with six valid moods, which yield four direct
conclusions and five indirect ones, corresponding to the four moods of
the original first figure and the five moods of the original fourth,
which appear now as indirect moods of the first figure.
§ 662. But why, it maybe asked, should not the moods of the first
figure equally well be regarded as indirect moods of the fourth? For
this reason-that all the moods of the fourth figure can be elicited
out of premisses in which the terms stand in the order of the first,
whereas the converse is not the case. If, while retaining the quantity
and quality of the above premisses, i. e. the mood, we were in each
case to transpose the terms, we should find that we were left with
five valid moods instead of six, since AI in the reverse order of the
terms involves undistributed middle; and, though we should have
Celarent indirect to Camenes, and Darii to Dimaris, we should never
arrive at the conclusion of Barbara or have anything exactly
equivalent to Ferio. In place of Barbara, Bramantip would yield as an
indirect mood only the subaltern AAI in the first figure. Both Fesapo
and Fresison would result in an illicit process, if we attempted to
extract the conclusion of Ferio from them as an indirect mood. The
nearest approach we could make to Ferio would be the mood EAO in the
first figure, which may be elicited indirectly from the premisses of
CAMENES, being subaltern to CELARENT. For these reasons the moods of
the fourth figure are rightly to be regarded as indirect moods of the
first, and not vice versâ.
_Mood AA._
All A is B.
All C is B.
.'. Undistributed Middle.
_Mood AE._
All A is B.
No C is B.
.'. No C is A, or No A is C, (Camestres & Cesare).
_Mood AI._
All A is B.
Some C is B.
.'. Undistributed Middle.
_Mood AO._
All A is B.
Some C is not B.
.'. Some C is not A, (Baroko), or Illicit Process.
_Mood EA._
No A is B.
All C is B.
.'. No C is A, or No A is C, (Cesare & Carnestres).
_Mood EI_
No A is B.
Some C is B.
.'. Some C is not A, (Festino), or Illicit Process.
_Mood IA._
Some A is B.
All C is B.
.'. Undistributed Middle.
_Mood IE._
Some A is B.
No C is B.
.'. Illicit Process, or Some A is not C, (Festino).
_Mood OA._
Some A is not B.
All C is B.
.'. Illicit Process, or Some A is not C, (Baroko).
§ 664. Here again we have six valid moods, which yield four direct
conclusions corresponding to Cesare, CARNESTRES, FESTINO and
BAROKO. The same four are repeated in the indirect moods.
_Mood AA._
All B is A.
All B is C.
.'. Some C is A, or Some A is C, (Darapti).
_Mood AE._
All B is A.
No B is C.
.'. Illicit Process, or Some A is not C, (Felapton).
_Mood AI._
All B is A,
Some B is C.
.'. Some C is A, or Some A is C, (Datisi & Disamis).
_Mood AO._
All B is A.
Some B is not C.
.'. Illicit Process, Or Some A is not C, (Bokardo).
_Mood EA._
No B is A.
All B is C.
.'. Some C is not A, (Felapton), or Illicit Process.
_Mood EI._
No B is A.
Some B is C.
.'. Some C is not A, (Ferison), or Illicit Process.
_Mood IA._
Some B is A.
All B is C.
.'. Some C is A, Or Some A is C, (Disamis & Datisi).
_Mood IE._
Some B is A.
No B is C.
.'. Illicit Process, or Some A is not C, (Ferison).
_Mood QA._
Some B is not A.
All B is C.
.'. Some C is not A, (Bokardo), or Illicit Process.
§ 666. In this figure every mood is valid, either directly or
indirectly. We have six direct moods, answering to Darapti, Disamis,
Datisi, Felapton, Bokardo and Ferison, which are simply repeated by
the indirect moods, except in the case of Darapti, which yields a
conclusion not provided for in the mnemonic lines. Darapti, though
going under one name, has as much right to be considered two moods as
Disamis and Datisi.
§ 667. We revert now to the standpoint of the old logicians, who
regarded the Dictum de Omni et Nullo as the principle of all
syllogistic reasoning. From this point of view the essence of mediate
inference consists in showing that a special case, or class of cases,
comes under a general rule. But a great deal of our ordinary reasoning
does not conform to this type. It was therefore judged necessary to
show that it might by a little manipulation be brought into conformity
with it. This process is called Reduction.
§ 668. Reduction is of two kinds--
(2) Indirect or Ad Impossibile.
§ 669. The problem of direct, or ostensive, reduction is this--
Given any mood in one of the imperfect figures (II, III and IV) how
to alter the form of the premisses so as to arrive at the same
conclusion in the perfect figure, or at one from which it can be
immediately inferred. The alteration of the premisses is effected by
means of immediate inference and, where necessary, of transposition.
§ 670. The problem of indirect reduction, or reductio (per
deductionem) ad impossibile, is this--Given any mood in one of the
imperfect figures, to show by means of a syllogism in the perfect
figure that its conclusion cannot be false.
§ 671. The object of reduction is to extend the sanction of the Dictum
de Omni et Nullo to the imperfect figures, which do not obviously
conform to it.
§ 672. The mood required to be reduced is called the Reducend; that to
which it conforms, when reduced, is called the Reduct.
_Direct or Ostensive Reduction._
§ 673. In the ordinary form of direct reduction, the only kind of
immediate inference employed is conversion, either simple or by
limitation; but the aid of permutation and of conversion by negation
and by contraposition may also be resorted to.
§ 674. There are two moods, Baroko and Bokardo, which cannot be
reduced ostensively except by the employment of some of the means last
mentioned. Accordingly, before the introduction of permutation into
the scheme of logic, it was necessary to have recourse to some other
expedient, in order to demonstrate the validity of these two
moods. Indirect reduction was therefore devised with a special view to
the requirements of Baroko and Bokardo: but the method, as will be
seen, is equally applicable to all the moods of the imperfect figures.
§ 675. The mnemonic lines, 'Barbara, Celarent, etc., provide complete
directions for the ostensive reduction of all the moods of the second,
third, and fourth figures to the first, with the exception of Baroko
and Bokardo. The application of them is a mere mechanical trick, which
will best be learned by seeing the process performed.
§ 676. Let it be understood that the initial consonant of each name of
a figured mood indicates that the reduct will be that mood which
begins with the same letter. Thus the B of Bramantip indicates that
Bramantip, when reduced, will become Barbara.
§ 677. Where m appears in the name of a reducend, me shall have to
take as major that premiss which before was minor, and vice versa-in
other words, to transpose the premisses, m stands for mutatio or
metathesis.
§ 678. s, when it follows one of the premisses of a reducend,
indicates that the premiss in question must be simply converted; when
it follows the conclusion, as in Disamis, it indicates that the
conclusion arrived at in the first figure is not identical in form
with the original conclusion, but capable of being inferred from it by
simple conversion. Hence s in the middle of a name indicates something
to be done to the original premiss, while s at the end indicates
something to be done to the new conclusion.
§ 679. P indicates conversion per accidens, and what has just been
said of s applies, mutatis mutandis, to p.
§ 680. k may be taken for the present to indicate that Baroko and
Bokardo cannot be reduced ostensively.
Cesare. \ / Celarent.
No A is B. \ = / No B is A.
All C is B. / \ All C is B.
.'. No C is A. / \ .'. No C is A.
Camestres. \ / Celarent.
All A is B. \ = / No B is C.
No C is B. / \ All A is B.
.'. No C is A. / \ .'. No A is C.
.'. No C is A.
Festino. Ferio.
No A is B. \ / No B is A.
Some C is B. | = | Some C is B.
.'. Some C is not A./ \ .'. Some C is not A.
[Baroko]
Darapti. \ / Darii.
All B is A. \ = / All B is A.
All B is C. / \ Some C is B.
.'. Some C is A. / \ Some C is A.
Disamis. \ / Darii.
Some B is A. \ = / All B is C.
All B is C. / \ Some A is B.
.'. Some C is A. / \ .'. Some A is C.
.'. Some C is A.
Datisi. \ / Darii.
All B is A. \ = / All B is A.
Some B is C. / \ Some C is B.
.'. Some C is A. / \ .'. Some C is A.
Felapton. \ / Ferio.
No B is A. \ = / No B is A.
All B is C. / \ Some C is B.
.'. Some C is not-A. / \ .'. Some C is not-A.
Ferison. \ / Ferio.
No B is A. \ = / No B is A.
Some B is C. / \ Some C is B
.'. Some C is not A. / \ .'. Some C is not A.
Bramantip. \ / Barbara.
All A is B. \ = / All B is C.
All B is C. / \ All A is B.
.. Some C is A. / \ .. All A is C.
.'. Some C is A.
Camenes Celarent
All A is B \ / No B is C.
No B is C. | = | All A is B.
.. No C is A./ \ .'. No A is C.
.'. No C is A.
Dimaris. Darii.
Some A is B. \ / All B is C.
All B is C. | = | Some A is B.
.'. Some C is A./ \ .'. Some A is C.
.'. Some C is A.
Fesapo. Ferio.
No A is B. \ / No B is A.
All B is C. | = | Some C is B.
.'. Some C is not A./ \ .'. Some C is not A.
Fresison. Ferio.
No A is B. \ / No B is A.
Some B is C. | = | Some C is B.
.'. Some C is not A./ \ .'. Some C is not A.
§ 684. The reason why Baroko and Bokardo cannot be reduced ostensively
by the aid of mere conversion becomes plain on an inspection of
them. In both it is necessary, if we are to obtain the first figure,
that the position of the middle term should be changed in one
premiss. But the premisses of both consist of A and 0 propositions, of
which A admits only of conversion by limitation, the effect of which
would be to produce two particular premisses, while 0 does not admit
of conversion at all,
It is clear then that the 0 proposition must cease to be 0 before we
can get any further. Here permutation comes to our aid; while
conversion by negation enables us to convert the A proposition,
without loss of quantity, and to elicit the precise conclusion we
require out of the reduct of Boltardo.
(Baroko) Fanoao. Ferio.
All A is B. \ / No not-B is A.
Some C is not-B. | = | Some C is not-B.
.'. Some C is not-A./ \ .'. Some C is not-A.
(Bokardo) Donamon. Darii.
Some B is not-A. \ / All B is C.
All B is C. | = | Some not-A is B
.'. Some C is not-A./ \ .'. Some not-A is C.
.'. Some C is not-A.
§ 685. In the new symbols, Fanoao and Donamon, [pi] has been
adopted as a symbol for permutation; n signifies conversion by
negation. In Donamon the first n stands for a process which resolves
itself into permutation followed by simple conversion, the second for
one which resolves itself into simple conversion followed by
permutation, according to the extended meaning which we have given to
the term 'conversion by negation.' If it be thought desirable to
distinguish these two processes, the ugly symbol Do[pi]samos[pi] may
be adopted in place of Donamon.
§ 686. The foregoing method, which may be called Reduction by
Negation, is no less applicable to the other moods of the second
figure than to Baroko. The symbols which result from providing for its
application would make the second of the mnemonic lines run thus--
Benare[pi], Cane[pi]e, Denilo[pi], Fano[pi]o secundae.
§ 687. The only other combination of mood and figure in which it will
be found available is Camenes, whose name it changes to Canene.
(Cesare) Benarea. Barbara.
No A is B. \ / All B is not-A.
All C is B. | = | All C is B.
.'. No C is A. / \ .'. All C is not-A.
.'. No C is A.
(Camestres) Cane[pi]e. Celarent.
All A is B. \ / No not-B is A.
No C is B. | = | All C is not-B.
.'. No C is A. / \ .'. No C is A.
(Festino) Denilo[pi]. Darii.
No A is B. \ / All B is not-A.
Some C is B. | = | Some C is B.
.'. Some C is not A./ \ .'. Some C is not-A.
.'. Some C is not A.
(Camenes) Canene. Celarent.
All A is B. \ / No not-B is A.
No B is C. | = | All C is not-B.
.'. No C is A. / \ .'. No C is A.
§ 689. The following will serve as a concrete instance of Cane[pi]e
reduced to the first figure.
All things of which we have a perfect idea are perceptions.
A substance is not a perception.
.'. A substance is not a thing of which we have a perfect idea.
When brought into Celarent this becomes--
No not-perception is a thing of which we have a perfect idea.
A substance is a not-perception.
.'. No substance is a thing of which we have a perfect idea.
§ 690. We may also bring it, if we please, into Barbara, by permuting
the major premiss once more, so as to obtain the contrapositive of the
original--
All not-perceptions are things of which we have an imperfect idea.
All substances are not-perceptions.
.'. All substances are things of which we have an imperfect idea.
§ 691. We will apply this method to Baroko.
All A is B. All fishes are oviparous.
Some C is not B. Some marine animals are not oviparous.
.'. Some C is not A. .'. Some marine animals are not fishes.
§ 692. The reasoning in such a syllogism is evidently conclusive: but
it does not conform, as it stands, to the first figure, nor
(permutation apart) can its premisses be twisted into conformity with
it. But though we cannot prove the conclusion true in the first
figure, we can employ that figure to prove that it cannot be false, by
showing that the supposition of its falsity would involve a
contradiction of one of the original premisses, which are true ex
hypothesi.
§ 693. If possible, let the conclusion 'Some C is not A' be
false. Then its contradictory 'All C is A' must be true. Combining
this as minor with the original major, we obtain premisses in the
first figure,
All A is B, All fishes are oviparous,
All C is A, All marine animals are fishes,
All C is B, All marine animals are oviparous.
But this conclusion conflicts with the original minor, 'Some C is not
B,' being its contradictory. But the original minor is ex hypothesi
true. Therefore the new conclusion is false. Therefore it must either
be wrongly drawn or else one or both of its premisses must be false.
But it is not wrongly drawn; since it is drawn in the first figure, to
which the Dictum de Omni et Nullo applies. Therefore the fault must
lie in the premisses. But the major premiss, being the same with that
of the original syllogism, is ex hypothesi true. Therefore the minor
premiss, 'All C is A,' is false. But this being false, its
contradictory must be true. Now its contradictory is the original
conclusion, 'Some C is not A,' which is therefore proved to be true,
since it cannot be false.
§ 694. It is convenient to represent the two syllogisms in
juxtaposition thus--
Baroko. Barbara.
All A is B. All A is B.
Some C is not B. \/ All C is A.
.'. Some C is not A. /\ All C is B.
§ 695. The lines indicate the propositions which conflict with one
another. The initial consonant of the names Baroko and Eokardo
indicates that the indirect reduct will be Barbara. The k indicates
that the O proposition, which it follows, is to be dropped out in the
new syllogism, and its place supplied by the contradictory of the old
conclusion.
§ 696. In Bokardo the two syllogisms will stand thus--
Bokardo. Barbara.
Some B is not A. \ / All C is A.
All B is C. X All B is C.
.'. Some C is not A./ \ .'. All B is A.
§ 697. The method of indirect reduction, though invented with a
special view to Baroko and Bokardo, is applicable to all the moods of
the imperfect figures. The following modification of the mnemonic
lines contains directions for performing the process in every
case:--Barbara, Celarent, Darii, Ferioque prioris; Felake, Dareke,
Celiko, Baroko secundae; Tertia Cakaci, Cikari, Fakini, Bekaco,
Bokardo, Dekilon habet; quarta insuper addit Cakapi, Daseke, Cikasi,
Cepako, Cesďkon.
§ 698. The c which appears in two moods of the third figure, Cakaci
and Bekaco, signifies that the new conclusion is the contrary, instead
of, as usual, the contradictory of the discarded premiss.
§ 699. The letters s and p, which appear only in the fourth figure,
signify that the new conclusion does not conflict directly with the
discarded premiss, but with its converse, either simple or per
accidens, as the case may be.
§ 700. l, n and r are meaningless, as in the original lines.
_Of Immediate Inference as applied to Complex Propositions._
§ 701. So far we have treated of inference, or reasoning, whether
mediate or immediate, solely as applied to simple propositions. But it
will be remembered that we divided propositions into simple and
complex. I t becomes incumbent upon us therefore to consider the laws
of inference as applied to complex propositions. Inasmuch however as
every complex proposition is reducible to a simple one, it is evident
that the same laws of inference must apply to both.
§ 702. We must first make good this initial statement as to the
essential identity underlying the difference of form between simple
and complex propositions.
§ 703. Complex propositions are either Conjunctive or Disjunctive (§
214).
§ 704. Conjunctive propositions may assume any of the four forms, A,
E, I, O, as follows--
(A) If A is B, C is always D.
(E) If A is B, C is never D.
(I) If A is B, C is sometimes D.
(O) If A is B, C is sometimes not D.
§ 705. These admit of being read in the form of simple propositions,
thus--
(A) If A is B, C is always D = All cases of A being B are cases of C
being D. (Every AB is a CD.)
(E) If A is B, C is never D = No cases of A being B are cases of C
being D. (No AB is a CD.)
(I) If A is B, C is sometimes D = Some cases of A being B are cases
of C being D. (Some AB's are CD's.)
(O) If A is B, C is sometimes not D = Some cases of A being B are
not cases of C being D. (Some AB's are not CD's.)
§ 706. Or, to take concrete examples,
(A) If kings are ambitious, their subjects always suffer.
= All cases of ambitious kings are cases of subjects suffering.
(E) If the wind is in the south, the river never freezes.
= No cases of wind in the south are cases of the river freezing.
(I) If a man plays recklessly, the luck sometimes goes against him.
= Some cases of reckless playing are cases of going against one.
(O) If a novel has merit, the public sometimes do not buy it.
= Some cases of novels with merit are not cases of the public buying.
§ 707. We have seen already that the disjunctive differs from the
conjunctive proposition in this, that in the conjunctive the truth
of the antecedent involves the truth of the consequent, whereas in the
disjunctive the falsity of the antecedent involves the truth of the
consequent. The disjunctive proposition therefore
may be reduced to a conjunctive
and so to a simple proposition with a negative term for subject.
All cases of A not being B are cases of C being D.
(Every not-AB is a CD.)
§ 708. It is true that the disjunctive proposition, more than any
other form, except U, seems to convey two statements in one
breath. Yet it ought not, any more than the E proposition, to be
regarded as conveying both with equal directness. The proposition 'No
A is B' is not considered to assert directly, but only implicitly,
that 'No B is A.' In the same way the form 'Either A is B or C is D'
ought to be interpreted as meaning directly no more than this, 'If A
is not B, C is D.' It asserts indeed by implication also that 'If C is
not D, A is B.' But this is an immediate inference, being, as we shall
presently see, the contrapositive of the original. When we say 'So and
so is either a knave or a fool,' what we are directly asserting is
that, if he be not found to be a knave, he will be found to be a
fool. By implication we make the further statement that, if he be not
cleared of folly, he will stand condemned of knavery. This inference
is so immediate that it seems indistinguishable from the former
proposition: but since the two members of a complex proposition play
the part of subject and predicate, to say that the two statements are
identical would amount to asserting that the same proposition can have
two subjects and two predicates. From this point of view it becomes
clear that there is no difference but one of expression between the
disjunctive and the conjunctive proposition. The disjunctive is
merely a peculiar way of stating a conjunctive proposition with a
negative antecedent.
§ 709. Conversion of Complex Propositions.
A / If A is B, C is always D.
\ .'. If C is D, A is sometimes B.
E / If A is B, C is never D.
\ .'. If C is D, A is never B.
I / If A is S, C is sometimes D.
\ .'. If C is D, A is sometimes B.
§ 710. Exactly the same rules of conversion apply to conjunctive as to
simple propositions.
§ 711. A can only be converted per accidens, as above.
is equivalent to the simple proposition
'All cases of A being B are cases of C being D.'
'Some cases of C being D are cases of A being B,'
which, when thrown back into the conjunctive form, becomes
'If C is D, A is sometimes B.'
§ 712. This expression must not be misunderstood as though it
contained any reference to actual existence. The meaning might be
better conveyed by the form
But it is perhaps as well to retain the other, as it serves to
emphasize the fact that formal logic is concerned only with the
connection of ideas.
§ 713. A concrete instance will render the point under discussion
clearer. The example we took before of an A proposition in the
conjunctive form--
'If kings are ambitious, their subjects always suffer'
'If subjects suffer, it may be that their kings are ambitious,'
i.e. among the possible causes of suffering on the part of subjects is
to be found the ambition of their rulers, even if every actual case
should be referred to some other cause. It is in this sense only that
the inference is a necessary one. But then this is the only sense
which formal logic is competent to recognise. To judge of conformity
to fact is no part of its province. From 'Every AB is a CD' it follows
that ' Some CD's are AB's' with exactly the same necessity as that
with which 'Some B is A' follows from 'All A is B.' In the latter case
also neither proposition may at all conform to fact. From 'All
centaurs are animals' it follows necessarily that 'Some animals are
centaurs': but as a matter of fact this is not true at all.
§ 714. The E and the I proposition may be converted simply, as above.
§ 715. O cannot be converted at all. From the proposition
'If a man runs a race, he sometimes does not win it,'
it certainly does not follow that
'If a man wins a race, he sometimes does not run it.'
§ 716. There is a common but erroneous notion that all conditional
propositions are to be regarded as affirmative. Thus it has been
asserted that, even when we say that 'If the night becomes cloudy,
there will be no dew,' the proposition is not to be regarded as
negative, on the ground that what we affirm is a relation between the
cloudiness of night and the absence of dew. This is a possible, but
wholly unnecessary, mode of regarding the proposition. It is precisely
on a par with Hobbes's theory of the copula in a simple proposition
being always affirmative. It is true that it may always be so
represented at the cost of employing a negative term; and the same is
the case here.
§ 717. There is no way of converting a disjunctive proposition except
by reducing it to the conjunctive form.
§ 718. _Permutation of Complex Propositions_.
(A) If A is B, C is always D.
.'. If A is B, C is never not-D. (E)
(E) If A is B, C is never D.
.'. If A is B, C is always not-D. (A)
(I) If A is B, C is sometimes D.
.'. If A is B, C is sometimes not not-D. (O)
(O) If A is B, C is sometimes not D.
.'. If A is B, C is sometimes not-D. (I)
(A) If a mother loves her children, she is always kind to them.
.'. If a mother loves her children, she is never unkind to
them. (E)
(E) If a man tells lies, his friends never trust him.
.'. If a man tells lies, his friends always distrust him. (A)
(I) If strangers are confident, savage dogs are sometimes friendly.
.'. If strangers are confident, savage dogs are sometimes not
unfriendly. (O)
(O) If a measure is good, its author is sometimes not popular.
.'. If a measure is good, its author is sometimes
unpopular. (I)
§ 720. The disjunctive proposition may be permuted as it stands
without being reduced to the conjunctive form.
Either A is B or C is D.
.'. Either A is B or C is not not-D.
Either a sinner must repent or he will be damned.
.'. Either a sinner must repent or he will not be saved.
§ 721. _Conversion by Negation of Complex Propositions._
(A) If A is B, C is always D.
.'. If C is not-D, A is never B. (E)
(E) If A is B, C is never D.
.'. If C is D, A is always not-B. (A)
(I) If A is B, C is sometimes D.
.'. If C is D, A is sometimes not not-B. (O)
(O) If A is B, C is sometimes not D.
.'. If C is not-D, A is sometimes B. (I)
(E per acc.) If A is B, C is never D.
.'. If C is not-D, A is sometimes B. (I)
(A per ace.) If A is B, C is always D.
.'. If C is D, A is sometimes not not-D. (O)
(A) If a man is a smoker, he always drinks.
.'. If a man is a total abstainer, he never smokes. (E)
(E) If a man merely does his duty, no one ever thanks him.
.'. If people thank a man, he has always done more than his
duty. (A)
(I) If a statesman is patriotic, he sometimes adheres to a party.
.'. If a statesman adheres to a party, he is sometimes not
unpatriotic. (O)
(O) If a book has merit, it sometimes does not sell.
.'. If a book fails to sell, it sometimes has merit. (I)
(E per acc.) If the wind is high, rain never falls.
.'. If rain falls, the wind is sometimes high. (I)
(A per acc.) If a thing is common, it is always cheap.
.'. If a thing is cheap, it is sometimes not uncommon. (O)
§ 723. When applied to disjunctive propositions, the distinctive
features of conversion by negation are still discernible. In each of
the following forms of inference the converse differs in quality from
the convertend and has the contradictory of one of the original terms
(§ 515).
(A) Either A is B or C is always D.
.'. Either C is D or A is never not-B. (E)
(E) Either A is B or C is never D.
.'. Either C is not-D or A is always B. (A)
(I) Either A is B or C is sometimes D.
.'. Either C is not-D or A is sometimes not B. (O)
(O) Either A is B or C is sometimes not D.
.'. Either C is D or A is sometimes not-B. (I)
(A) Either miracles are possible or every ancient historian is
untrustworthy.
.'. Either ancient historians are untrustworthy or miracles are
not impossible. (E)
(E) Either the tide must turn or the vessel can not make the port.
.'. Either the vessel cannot make the port or the tide must
turn. (A)
(1) Either he aims too high or the cartridges are sometimes bad.
.'. Either the cartridges are not bad or he sometimes does not
aim too high. (0)
(O) Either care must be taken or telegrams will sometimes not be
correct.
.'. Either telegrams are correct or carelessness is sometimes
shown. (1)
§ 726. In the above examples the converse of E looks as if it had
undergone no change but the mere transposition of the
alternative. This appearance arises from mentally reading the E as an
A proposition: but, if it were so taken, the result would be its
contrapositive, and not its converse by negation.
§ 727. The converse of I is a little difficult to grasp. It becomes
easier if we reduce it to the equivalent conjunctive--
'If the cartridges are bad, he sometimes does not aim too high.'
Here, as elsewhere, 'sometimes' must not be taken to mean more than
'it may be that.'
§ 728. _Conversion by Contraposition of Complex Propositions._
As applied to conjunctive propositions conversion by contraposition
assumes the following forms--
(A) If A is B, C is always D.
.'. If C is not-D, A is always not-B.
(O) If A is B, C is sometimes not D.
.'. If C is not-D, A is sometimes not not-B.
(A) If a man is honest, he is always truthful.
.'. If a man is untruthful, he is always dishonest.
(O) If a man is hasty, he is sometimes not malevolent.
.'. If a man is benevolent, he is sometimes not unhasty.
§ 729. As applied to disjunctive propositions conversion by
contraposition consists simply in transposing the two alternatives.
(A) Either A is B or C is D.
.'. Either C is D or A is B.
For, when reduced to the conjunctive shape, the reasoning would run
thus--
If A is not B, C is D.
.'. If C is not D, A is B.
All not-A is B.
.'. All not-B is A.
Similarly in the case of the O proposition
(O) Either A is B or C is sometimes not D.
.'. Either C is D or A is sometimes not B.
§ 730. On comparing these results with the converse by negation of
each of the same propositions, A and 0, the reader will see that they
differ from them, as was to be expected, only in being permuted. The
validity of the inference may be tested, both here and in the case of
conversion by negation, by reducing the disjunctive proposition to the
conjunctive, and so to the simple form, then performing the process as
in simple propositions, and finally throwing the converse, when so
obtained, back into the disjunctive form. We will show in this manner
that the above is really the contrapositive of the 0 proposition.
(O) Either A is B or C is sometimes not D.
= If A is not B, C is sometimes not D.
= Some cases of A not being B are not cases of C being D. (Some A is
not B.)
= Some cases of C not being D are not cases of A being B. (Some
not-B is not not-A.)
= If C is not D, A is sometimes not B.
= Either C is D or A is sometimes not B.
§ 731. A Complex Syllogism is one which is composed, in whole or part,
of complex propositions.
§ 732. Though there are only two kinds of complex proposition, there
are three varieties of complex syllogism. For we may have
(1) a syllogism in which the only kind of complex proposition
employed is the conjunctive;
(2) a syllogism in which the only kind of complex proposition
employed is the disjunctive;
(3) a syllogism which has one premiss conjunctive and the other
disjunctive.
The chief instance of the third kind is that known as the Dilemma.
Syllogism
___________________|_______________
| |
Simple Complex
(Categorical) (Conditional)
_____________________|_______________
| | |
Conjunctive Disjunctive Dilemma
(Hypothetical)
§ 733. The Conjunctive Syllogism has one or both premisses conjunctive
propositions: but if only one is conjunctive, the other must be a
simple one.
§ 734. Where both premisses are conjunctive, the conclusion will be of
the same character; where only one is conjunctive, the conclusion will
be a simple proposition.
§ 735. Of these two kinds of conjunctive syllogisms we will first take
that which consists throughout of conjunctive propositions.
_The Wholly Conjunctive Syllogism_.
§ 736. Wholly conjunctive syllogisms do not differ essentially from
simple ones, to which they are immediately reducible. They admit of
being constructed in every mood and figure, and the moods of the
imperfect figures may be brought into the first by following the
ordinary rules of reduction. For instance--
If A is B, C is never D. \ / If C is D, A is never B.
If E is F, C is always D. | = | If E is F, C is always D.
.'. If E is F, A is never B. / \ .'. If E is F, A is never B.
If it is day, the stars never shine.\ /If the stars shine, it is never day.
If it is night, the stars always \=/ If it is night, the stars always
shine. / \ shine.
.'. If it is night, it is never day / \.'. If it is night, it is never day.
Disamis. Darii.
If C is D, A is sometimes B. \ / If C is D, E is always F.
If C is D, E is always F. | = | If A is B, C is sometimes D.
If E is F, A is sometimes B. / \ .'. If A is B, E is sometimes F.
.'. If E is F, A is sometimes B.
If she goes, I sometimes go. \ / If she goes, he always goes,
If she goes, he always goes. | = | If I go, she sometimes goes.
.'. If he goes, I sometimes go. / \ .'. If I go, he sometimes goes.
.'. If he goes, I sometimes go.
_The Partly Conjunctive Syllogism._
§ 737. It is this kind which is usually meant when the Conjunctive or
Hypothetical Syllogism is spoken of.
§ 738. Of the two premisses, one conjunctive and one simple, the
conjunctive is considered to be the major, and the simple premiss the
minor. For the conjunctive premiss lays down a certain relation to
hold between two propositions as a matter of theory, which is applied
in the minor to a matter of fact.
§ 739. Taking a conjunctive proposition as a major premiss, there are
four simple minors possible. For we may either assert or deny the
antecedent or the consequent of the conjunctive.
Constructive Mood. Destructive Mood.
(1) If A is B, C is D. (2) If A is B, C is D.
A is B. C is not D.
.'. C is D. .'. A is not B.
(3) If A is B, C is D. (4) If A is B, C is D.
A is not B. C is D.
No conclusion. No conclusion.
§ 740. When we take as a minor 'A is not B ' (3), it is clear that we
can get no conclusion. For to say that C is D whenever A is B gives us
no right to deny that C can be D in the absence of that
condition. What we have predicated has been merely inclusion of the
case AB in the case CD.
§ 741. Again, when we take as a minor, 'C is D' (4), we can get no
universal conclusion. For though A being B is declared to involve as a
consequence C being D, yet it is possible for C to be D under other
circumstances, or from other causes. Granting the truth of the
proposition 'If the sky falls, we shall catch larks,' it by no means
follows that there are no other conditions under which this result can
be attained.
§ 742. From a consideration of the above four cases we elicit the
following
_Canon of the Conjunctive Syllogism._
To affirm the antecedent is to affirm the consequent, and to deny the
consequent is to deny the antecedent: but from denying the antecedent
or affirming the consequent no conclusion follows.
§ 743. There is a case, however, in which we can legitimately deny the
antecedent and affirm the consequent of a conjunctive proposition,
namely, when the relation predicated between the antecedent and the
consequent is not that of inclusion but of coincidence--where in fact
the conjunctive proposition conforms to the type u.
_Denial of the Antecedent_.
If you repent, then only are you forgiven.
You do not repent.
.'. You are not forgiven.
_Affirmation of the Consequent_.
If you repent, then only are you forgiven.
You are forgiven.
.'. You repent.
_Of the Reduction of the Partly Conjunctive Syllogism._
§ 744. Such syllogisms as those just treated of, if syllogisms they
are to be called, have a major and a middle term visible to the eye,
but appear to be destitute of a minor. The missing minor term is
however supposed to be latent in the transition from the conjunctive
to the simple form of proposition. When we say 'A is B,' we are taken
to mean, 'As a matter of fact, A is B' or 'The actual state of the
case is that A is B.' The insertion therefore of some such expression
as 'The case in hand,' or 'This case,' is, on this view, all that is
wanted to complete the form of the syllogism. When reduced in this
manner to the simple type of argument, it will be found that the
constructive conjunctive conforms to the first figure and the
destructive conjunctive to the second.
_Constructive Mood_. _Barbara_.
If A is B, C is D. \ / All cases of A being B are cases of
\ = / C being D.
A is B. / \ This is a case of A being B.
.'. C is D. / \ .'. This is a case of C being D.
_Destructive Mood._ Camestres.
If A is B, C is D. \ / All cases of A being B are cases of
\ = / C being D.
C is not D. / \ This is not a case of C being D.
.'. A is not B. / \ .'. This is not a case of A being B.
§ 745. It is apparent from the position of the middle term that the
constructive conjunctive must fall into the first figure and the
destructive conjunctive into the second. There is no reason, however,
why they should be confined to the two moods, Barbara and
Carnestres. If the inference is universal, whether as general or
singular, the mood is Barbara or Carnestres; if it is particular, the
mood is Darii or Baroko.
Barbara. Camestres.
If A is B, C is always D. \ If A is B, C is always D. \
A is always B. \ C is never D. \
.'. C is always D. \ .'. A is never B. \
| |
If A is B, C is always D. / If A is B, C is always D. /
A is in this case B. / C is not in this case D. /
.'. C is in this case D. / .'. A is not in this case B. /
If A is B, C is always D. If A is B, C is never D.
A is sometimes B. C is sometimes not D.
.'. C is sometimes D. .'. A is sometimes not B.
§ 746. The remaining moods of the first and second figure are obtained
by taking a negative proposition as the consequent in the major
premiss.
Celarent. Ferio.
If A is B, C is never D. If A is B, C is never D.
A is always B. A is sometimes B.
.'. C is never D. .'. C is sometimes not D.
_Cesare_. Festino.
If A is B, C is never D. If A is B, C is never D.
C is always D. C is sometimes D.
.'. A is never B. .'. A is sometimes not B.
§ 747. As the partly conjunctive syllogism is thus reducible to the
simple form, it follows that violations of its laws must correspond
with violations of the laws of simple syllogism. By our throwing the
illicit moods into the simple form it will become apparent what
fallacies are involved in them.
If A is B, C is D. \ / All cases of A being B are cases of C
\ = / being D.
A is not B. / \ This is not a case of A being B.
.'. C is not D. / \ .'. This is not a case of C being D.
Here we see that the denial of the antecedent amounts to illicit
process of the major term.
§ 7481 _Affirmation of Consequent_.
If A is B, C is D. \ / All Cases of A being B are cases of C
| = | being D.
C is D. / \ This is a case of C being D.
Here we see that the affirmation of the consequent amounts to
undistributed middle.
§ 749. If we confine ourselves to the special rules of the four
figures, we see that denial of the antecedent involves a negative
minor in the first figure, and affirmation of the consequent two
affirmative premisses in the second. Or, if the consequent in the
major premiss were itself negative, the affirmation of it would amount
to the fallacy of two negative premisses. Thus--
If A is B, C is not D. \ / No cases of A being B are cases of C
| = | being D.
C is not D. / \ This is not a case of C being D.
§ 750. The positive side of the canon of the conjunctive
syllogism--'To affirm the antecedent is to affirm the consequent,'
corresponds with the Dictum de Omni. For whereas something (viz. C
being D) is affirmed in the major of all conceivable cases of A being
B, the same is affirmed in the conclusion of something which is
included therein, namely, 'this case,' or 'some cases,' or even 'all
actual cases.'
§ 751. The negative side--'to deny the consequent is to deny the
antecedent'--corresponds with the Dictum de Diverse (§ 643). For
whereas in the major all conceivable cases of A being B are included
in C being D, in the minor 'this case,' or 'some cases,' or even 'all
actual cases' of C being D, are excluded from the same notion.
§ 752. The special characteristic of the partly conjunctive syllogism
lies in the transition from hypothesis to fact. We might lay down as
the appropriate axiom of this form of argument, that 'What is true in
the abstract is true--in the concrete,' or 'What is true in theory is
also true in fact,' a proposition which is apt to be neglected or
denied. But this does not vitally distinguish it from the ordinary
syllogism. For though in the latter we think rather of the transition
from a general truth to a particular application of it, yet at bottom
a general truth is nothing but a hypothesis resting upon a slender
basis of observed fact. The proposition 'A is B' may be expressed in
the form 'If A is, B is.' To say that 'All men are mortal' may be
interpreted to mean that 'If we find in any subject the attributes of
humanity, the attributes of mortality are sure to accompany them.'
_Of the Partly Conjunctive Syllogism regarded as an Immediate
Inference_.
§ 753. It is the assertion of fact in the minor premiss, where we have
the application of an abstract principle to a concrete instance, which
alone entitles the partly conjunctive syllogism to be regarded as a
syllogism at all. Apart from this the forms of semi-conjunctive
reasoning run at once into the moulds of immediate inference.
§ 754. The constructive mood will then be read in this way--
If A is B, C is D,
.'. A being B, C is D.
reducing itself to an instance of immediate inference by subaltern
opposition--
Every case of A being B, is a case of C being D.
.'. Some particular case of A being B is a case of C being D.
§ 755. Again, the destructive conjunctive will read as follows--
If A is B, C is D,
.'. C not being D, A is not B.
All cases of A being B are cases of C being D.
.'. Whatever is not a case of C being D is not a case of A being B.
.'. Some particular case of C not being D is not a case of A being
B.
But what is this but an immediate inference by contraposition, coming
under the formula
All A is B,
.'. All not-B is not-A,
and followed by Subalternation?
§ 756. The fallacy of affirming the consequent becomes by this mode of
treatment an instance of the vice of immediate inference known as the
simple conversion of an A proposition. 'If A is B, C is D' is not
convertible with 'If C is D, A is B' any more than 'All A is B' is
convertible with 'All B is A.'
§ 757. We may however argue in this way
If A is B, C is D,
C is D,
.'. A may be B,
which is equivalent to saying,
When A is B, C is always D,
.'. When C is D, A is sometimes B,
and falls under the legitimate form of conversion of A per accidens--
All cases of A being B are cases of C being D.
.'. Some cases of C being D are cases of A being B.
§ 758. The fallacy of denying the antecedent assumes the following
form--
If A is B, C is D,
.'. If A is not B, C is not D,
All cases of A being B are cases of C being D.
.'. Whatever is not a case of A being B is not a case of C being D.
This is the same as to argue--
All A is B,
.'. All not-A is not-B,
an erroneous form of immediate inference for which there is no special
name, but which involves the vice of simple conversion of A, since
'All not-A is not-B' is the contrapositive, not of 'All A is B,' but
of its simple converse 'All B is A.'
§ 759. The above-mentioned form of immediate inference, however
(namely, the employment of contraposition without conversion), is
valid in the case of the U proposition; and so also is simple
conversion. Accordingly we are able, as we have seen, in dealing with
a proposition of that form, both to deny the antecedent and to assert
the consequent with impunity--
If A is B, then only C is D,
.'. A not being B, C is not D;
and again, C being D, A must be B.
_Of the Disjunctive Syllogism_.
§ 760. Roughly speaking, a Disjunctive Syllogism results from the
combination of a disjunctive with a simple premiss. As in the
preceding form, the complex proposition is regarded as the major
premiss, since it lays down a hypothesis, which is applied to fact in
the minor.
§ 761. The Disjunctive Syllogism may be exactly defined as follows--
A complex syllogism, which has for its major premiss a disjunctive
proposition, either the antecedent or consequent of which is in the
minor premiss simply affirmed or denied.
§ 762. Thus there are four types of disjunctive syllogism possible.
(1) Either A is B or C is D. (2) Either A is B or C is D.
A is not B. C is not D.
.'. C is D. .'. A is B.
Either death is annihilation or we are immortal.
Death is not annihilation.
.'. We are immortal.
Either the water is shallow or the boys will be drowned.
The boys are not drowned.
.'. The water is shallow.
(3) Either A is B or C is D. (4) Either A is B or C is D.
A is B. C is D.
.'. C is not D. .'. A is not B.
§ 763. Of these four, however, it is only the constructive moods that
are formally conclusive. The validity of the two destructive moods is
contingent upon the kind of alternatives selected. If these are such
as necessarily to exclude one another, the conclusion will hold, but
not otherwise. They are of course mutually exclusive whenever they
embody the result of a correct logical division, as 'Triangles are
either equilateral, isosceles or scalene.' Here, if we affirm one of
the members, we are justified in denying the rest. When the major thus
contains the dividing members of a genus, it may more fitly be
symbolized under the formula, 'A is either B or C.' But as this admits
of being read in the shape, 'Either A is B or A is C,' we retain the
wider expression which includes it. Any knowledge, however, which we
may have of the fact that the alternatives selected in the major are
incompatible must come to us from material sources; unless indeed we
have confined ourselves to a pair of contradictory terms (A is either
B or not-B). There can be nothing in the form of the expression to
indicate the incompatibility of the alternatives, since the same form
is employed when the alternatives are palpably compatible. When, for
instance, we say, 'A successful student must be either talented or
industrious,' we do not at all mean to assert the positive
incompatibility of talent and industry in a successful student, but
only the incompatibility of their negatives--in other words, that, if
both are absent, no student can be successful. Similarly, when it is
said, 'Either your play is bad or your luck is abominable,' there is
nothing in the form of the expression to preclude our conceiving that
both may be the case.
§ 764. There is no limit to the number of members in the disjunctive
major. But if there are only two alternatives, the conclusion will be
a simple proposition; if there are more than two, the conclusion will
itself be a disjunctive. Thus--
Either A is B or C is D or E is F or G is H.
E is not F.
.'. Either A is B or C is D or G is H.
§ 765. The Canon of the Disjunctive Syllogism may be laid down as
follows--
To deny one member is to affirm the rest, either simply or
disjunctively; but from affirming any member nothing follows.
_Of the Reduction of the Disjunctive Syllogism._
§ 766. We have seen that in the disjunctive syllogism the two
constructive moods alone are formally valid. The first of these,
namely, the denial of the antecedent, will in all cases give a simple
syllogism in the first figure; the second of them, namely, the denial
of the consequent, will in all cases give a simple syllogism in the
second figure.
_Denial of Antecedent_ = Barbara.
Either A is B or C is D.
A is not B.
.'.C is D
If A is not B, C is D.
A is not B.
.'. C is D.
All cases of A not being B are cases of C being D.
This is a case of A not being B.
.'. This is a case of C being D.
_Denial of Consequent_ = Camestres.
Either A is E or C is D.
C is not D.
.'. A is B.
If A is not B, C is D.
C is not D.
.'. A is B.
All cases of A not being B are cases of C being D.
This is not a case of C being D.
.'. This is not a case of A being B.
§ 767. The other moods of the first and second figures can be obtained
by varying the quality of the antecedent and consequent in the major
premiss and reducing the quantity of the minor.
§ 768. The invalid destructive moods correspond with the two invalid
types of the partly conjunctive syllogism, and have the same fallacies
of simple syllogism underlying them. Affirmation of the antecedent of
a disjunctive is equivalent to the semi-conjunctive fallacy of denying
the antecedent, and therefore involves the ordinary syllogistic
fallacy of illicit process of the major.
Affirmation of the consequent of a disjunctive is equivalent to the
same fallacy in the semi-conjunctive form, and therefore involves the
ordinary syllogistic fallacy of undistributed middle.
_Affirmation of Antecedent_ = _Illicit Major_.
Either A is B or C is D.
A is B.
.'. C is not D.
If A is not B, C is D.
A is B.
.'. C is not D.
All cases of A not being B are cases of C being D.
This is not a case of A not being B.
.'. This is not a case of C not being D.
_Affirmation of Consequent_ = _Undistributed Middle_.