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.
CHAPTER XV.
_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._
FIGURE 1.
§ 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.
Here four cases arise–
[Illustration]
(1) Total inclusion (Barbara).
All B is A.
All C is B.
.’. All C is A.
[Illustration]
(2) Partial inclusion (Darii).
All B is A.
Some C is B.
.’. Some C is A.
[Illustration]
(3) Total exclusion (Celarent).
No B is A.
All C is B.
.’. No C is A.
[Illustration]
(4) Partial exclusion (Ferio).
No B is A.
Some C is B.
.’. Some C is not A.
FIGURE II.
§ 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).
[Illustration]
No A is B.
All C is B.
.’. No C is A.
(2) Partial exclusion on the ground of a similar partial inclusion (Festino).
[Illustration]
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).
[Illustration]
All A is B.
No C is B.
.’. No C is A.
(4) Partial exclusion on the ground of a similar partial exclusion (Baroko).
[Illustration]
All A is B.
Some C is not B.
.’. Some C is not A.
FIGURE III.
§ 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).
[Illustration]
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).
[Illustration]
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).
[Illustration]
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).
[Illustration]
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).
[Illustration]
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).
[Illustration]
Some B is not A.
All B is C.
.’. Some C is not A.
FIGURE IV.
§ 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.
Here we have five cases–
(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).
[Illustration]
Some A is B.
All B is C.
.’. Some C is (some) A,
(3) Of the exclusion of a whole term (Camenes).
[Illustration]
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).
[Illustration]
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).
[Illustration]
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.
[Illustration]
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.
[Illustration]
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.
1. Dictum de Exemplo–
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.
2. Dictum de Excepto–
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.
CHAPTER XVI.
_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.
FIGURE 1.
§ 652.
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.
FIGURE II.
§ 653.
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.
FIGURE III.
§ 654.
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.
FIGURE IV.
§ 655.
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.
CHAPTER XVII.
_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.
§ 660. FIGURE 1.
_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.
$663. FIGURE II.
_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.
§ 665. FIGURE III.
_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.
CHAPTER XVIII.
_Of Reduction._
§ 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–
(1) Direct or Ostensive.
(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.
§ 681. FIGURE II.
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]
§ 682. FIGURE III.
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.
[Bokardo].
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.
§ 683. FIGURE IV.
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.
§ 688.
(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.
_Indirect Reduction._
§ 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,
which lead to the conclusion
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, Cesïkon.
§ 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.
CHAPTER XIX.
_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
Either A is B or C is D
may be reduced to a conjunctive
If A is not B, C is D,
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.
The original proposition
‘If A is B, C is always D’
is equivalent to the simple proposition
‘All cases of A being B are cases of C being D.’
This, when converted, becomes
‘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
‘If C is D, A may be B.’
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’
may be converted into
‘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)
§ 719.
(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)
§ 722.
(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).
§ 724.
(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)
§ 725.
(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.
which is the same in form as
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.
CHAPTER XX.
_Of Complex Syllogisms_.
§ 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)
_The Conjunctive Syllogism_.
§ 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–
Cesare. Celarent.
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.
[Illustration]
§ 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.
For example–
_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.
CHAPTER XXI.
_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. /
Darii. Baroko.
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.
_Denial of Anteceded_.
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.’
CHAPTER XXII.
_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.
which is equivalent to
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,
equivalent to–
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.
CHAPTER XXIII.
_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.
_Constructive Moods._
(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.
_Destructive Moods_.
(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.
CHAPTER XXIV.
_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
is equal to
If A is not B, C is D.
A is not B.
.’. C is D.
is equal to
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.
is equal to
If A is not B, C is D.
C is not D.
.’. A is B.
is equal to
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.
is equal to
If A is not B, C is D.
A is B.
.’. C is not D.
is equal to
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_.
Either A is B or C is D.
C is D.
is equal to
If A is not B, C is D.
C is D.