CHAPTER VIII.
PROBLEMS ON THE SYLLOGISM.
342. Bearing of the existential interpretation of propositions upon the validity of syllogistic reasonings.—We may as before take different suppositions with regard to the existential import of propositions, and proceed to consider how far the validity of the various syllogistic moods is affected by each in turn.
(1) Let every proposition be interpreted as implying the existence both of its subject and of its predicate.[421] In this case, the existence of the major, middle, and minor terms is in every case guaranteed by the premisses, and therefore no further assumption with regard to existence is required in order that the conclusion may be legitimately obtained.[422] We may regard the above supposition as that which is tacitly made in the ordinary doctrine of the syllogism.
[421] It will be observed that this is not quite the same as supposition (1) in section [156].
[422] If, however, we are to be allowed to proceed as in section [206] (where from all P is M, all S is M, we inferred some not-S is not-P) we must posit the existence not merely of the terms directly involved, but also of their contradictories.
(2) Let every proposition be interpreted as implying the existence of its subject. Under this supposition an affirmative proposition ensures the existence of its predicate also; but not so a negative proposition. It follows that any mood will be valid unless the minor term is in its premiss the predicate of a negative proposition. This cannot happen either in figure 1 or in figure 2, since in these figures the minor is always subject in its premiss; nor in figure 3, since in this figure the minor 391 premiss is always affirmative. In figure 4, the only moods with a negative minor are Camenes and its weakened form AEO. Our conclusion then is that on the given supposition every ordinarily recognised mood is valid except these two.[423]
[423] Reduction to figure 1 appears to be affected by this supposition, since it makes the contraposition of A and the conversion of E in general invalid. The contraposition of A is involved in the direct reduction of Baroco (Faksoko). The process is, however, in this particular case valid, as the existence of not-M is given by the minor premiss. The conversion of E is involved in the reduction of Cesare, Camestres, and Festino from figure 2; and of Camenes, Fesapo, and Fresison from figure 4. Since, however, one premiss must be affirmative the existence of the middle term is thereby guaranteed, and hence the simple conversion of E in the second figure, and in the major of the fourth becomes valid. Also the conversion of the conclusion resulting from the reduction of Camestres is legitimate, since the original minor term is subject in its premiss. Hence Camenes (and its weakened form) are the only moods whose reduction is rendered illegitimate by the supposition under consideration. This result agrees with that reached in the text.
(3) Let no proposition be interpreted as implying the existence either of its subject or of its predicate. Taking S, M, P, as the minor, middle, and major terms respectively, the conclusion will imply that if there is any S there is some P or not-P (according as it is affirmative or negative). Will the premisses also imply this? If so, then the syllogism is valid; but not otherwise.
It has been shewn in section [212] that a universal affirmative conclusion, All S is P, can be proved only by means of the premisses, All M is P, All S is M ; and it is clear that these premisses themselves imply that if there is any S there is some P. On our present supposition, then, a syllogism is valid if its conclusion is universal affirmative.
Again, as shewn in section [212], a universal negative conclusion, No S is P, can be proved only in the following ways:—
| (i) | No M is P (or No P is M), |
| All S is M, | |
| ⎯⎯⎯⎯ | |
| therefore, | No S is P ; |
| (ii) | All P is M, |
| No S is M (or No M is S), | |
| ⎯⎯⎯⎯ | |
| therefore, | No S is P. |
392 In (i) the minor premiss implies that if S exists then M exists, and the major premiss that if M exists then not-P exists. In (ii) the minor premiss implies that if S exists then not-M exists, and the major premiss that if not-M exists then not-P exists (as shewn in section [158]). Hence a syllogism is valid if its conclusion is universal negative.
Next, let the conclusion be particular. In figure 1, the implication of the conclusion with regard to existence is contained in the premisses themselves, since the minor term is the subject of an affirmative minor premiss, and the middle term the subject of the major premiss. In figure 2, we may consider the weakened moods disposed of in what has been already said with regard to universal conclusions; for under our present supposition subalternation is a valid process. The remaining moods with particular conclusions in this figure are Festino and Baroco. In the former, the minor premiss implies that if S exists then M exists, and the major that if M exists then not-P exists; in the latter, the minor premiss implies that if S exists then not-M exists, and the major that if not-M exists then not-P exists.
All the ordinarily recognised moods, then, of figures 1 and 2 are valid. But it is otherwise with moods yielding a particular conclusion in figures 3 and 4, with the single exception of the weakened form of Camenes (which is itself the only mood with a universal conclusion in these figures). Subalternation being a valid process, the legitimacy of the latter follows from the legitimacy of Camenes itself. But in all other cases in figures 3 and 4, the minor term is the predicate of an affirmative minor premiss. Its existence, therefore, carries no further implication of existence with it in the premisses. It does so in the conclusion. Hence all the moods of figures 3 and 4, with the exception of AEE and AEO in the latter figure, are invalid. Take, as an example, a syllogism in Darapti,—
| All M is P, | |
| All M is S, | |
| ⎯⎯⎯⎯⎯ | |
| therefore, | Some S is P. |
The conclusion implies that if S exists P exists; but 393 consistently with the premisses, S may be existent while M and P are both non-existent. An implication is, therefore, contained in the conclusion which is not justified by the premisses.
Hence on the supposition that no proposition implies the existence either of its subject or of its predicate all the ordinarily recognised moods of figures 1 and 2 are valid, but none of those of figures 3 and 4 excepting Camenes and the weakened form of Camenes.[424]
[424] An express statement concerning existence may, however, render the rejected moods legitimate. If, for instance, the existence of the middle term is expressly given, then Darapti becomes valid.
(4) Let particulars be interpreted as implying, but universals as not implying, the existence of their subjects. The legitimacy of moods with universal conclusions may be established as in the preceding case. Taking moods with particular conclusions, it is obvious that they will be valid if the minor premiss is particular, having the minor term as its subject; or if the minor premiss is particular affirmative, whether the minor term is its subject or predicate. Disamis, Bocardo, and Dimaris are also valid, since the major premiss in each case guarantees the existence of M, and the minor implies that if M exists then S exists. The above will be found to cover all the valid moods in which one premiss is particular. There remain only the moods in which from two universals we infer a particular. It is clear that all these moods must be invalid, for their conclusions will imply the existence of the minor term, and this cannot be guaranteed by the premisses.[425]
[425] Hypothetical conclusions (of the form If S exists then &c.) will of course still be legitimate.
On the supposition then that particulars imply, while universals do not imply, the existence of their subjects, the moods rendered invalid are all the weakened moods, together with Darapti, Felapton, Bramantip, and Fesapo,[426] each of which contains a strengthened premiss. More briefly, any ordinarily recognised 394 mood is on this supposition valid, unless it contains either a strengthened premiss or a weakened conclusion.[427]
[426] It will be observed that the letter p occurs in the mnemonic for each of these moods, indicating that their reduction to figure 1 involves conversion per accidens. On the supposition under discussion this process is invalid, and we may find here a confirmation of the above result.
[427] This result may be regarded as affording an additional argument in favour of the adoption of supposition (4).
343. Connexion between the truth and falsity of premisses and conclusion in a valid syllogism.—By saying that a syllogism is valid we mean that the truth of its conclusion follows from the truth of its premisses; and it is an immediate inference from this that if the conclusion is false one or both of the premisses must be false. The converse does not, however, hold good in either case. The truth of the premisses does not follow from the truth of the conclusion; nor does the falsity of the conclusion follow from the falsity of either or both of the premisses.
The above statements would probably be accepted as self-evident; still it is more satisfactory to give a formal proof of them, and such a proof is afforded by means of the three following theorems.[428]
[428] It is assumed throughout this section that our schedule of propositions does not include U. The theorems hold good, however, for the sixfold schedule, including Y and η, as well as for the ordinary fourfold schedule.
(1) Given a valid syllogism, then in no case will the combination of either premiss with the conclusion establish the other premiss.
We have to shew that if one premiss and the conclusion of a valid syllogism are taken as a new pair of premisses they do not in any case suffice to establish the other premiss.
Were it possible for them to do so, then the premiss given true would have to be affirmative, for if it were negative, the original conclusion would be negative, and combining these we should have two negative premisses which could yield no conclusion.
Also, the middle term would have to be distributed in the premiss given true. This is clear if it is not distributed in the other premiss; and since the other premiss is the conclusion of the new syllogism, if it is distributed there, it must also be distributed in the premiss given true or we should have an illicit process in the new syllogism. 395
Therefore, the premiss given true, being affirmative and distributing the middle term, cannot distribute the other term which it contains.[429] Neither therefore can this term be distributed in the original conclusion. But this is the term which will be the middle term of the new syllogism, and we shall consequently have undistributed middle.
Hence the truth of one premiss and the conclusion of a valid syllogism does not establish the truth of the other premiss; and à fortiori the truth of the conclusion cannot by itself establish the truth of both the premisses.[430]
[429] This statement, though not holding good for U, holds good for Y as well as A.
[430] Other methods of solution more or less distinct from the above might be given. A somewhat similar problem is discussed by Solly, Syllabus of Logic, pp. 123 to 126, 132 to 136. We have shewn that one premiss and the conclusion of a valid syllogism will never suffice to prove the other premiss, but it of course does not follow that they will never yield any conclusion at all; for a consideration of this question, see the following [section].
(2) The contradictories of the premisses of a valid syllogism will not in any case suffice to establish the contradictory of the original conclusion.
The premisses of the original syllogism must be either (α) both affirmative, or (β) one affirmative and one negative.
In case (α), the contradictories of the original premisses will both be negative; and from two negatives nothing follows.
In case (β), the contradictories of the original premisses will be one negative and one affirmative; and if this combination yields any conclusion, it will be negative. But the original conclusion must also be negative, and therefore its contradictory will be affirmative.
In neither case then can we establish the contradictory of the original conclusion.[431]
[431] It is possible, however, that some conclusion may be obtainable. See section [359].
(3) One premiss and the contradictory of the other premiss of a valid syllogism will not in any case suffice to establish the contradictory of the original conclusion.[432]
[432] It does not follow that one premiss and the contradictory of the other premiss of a valid syllogism will never yield any conclusion at all. See the following [section].
396 This follows at once from the first of the theorems established in this section. Let the premisses of a valid syllogism be P and Q, and the conclusion R, P and the contradictory of Q will not prove the contradictory of R ; for if they did, it would follow that P and R would prove Q ; but this has been shewn not to be the case.
We have now established by strictly formal reasoning Aristotle’s dictum that although it is not possible syllogistically to get a false conclusion from true premisses, it is quite possible to get a true conclusion from false premisses.[433] In other words, the falsity of one or both of the premisses does not establish the falsity of the conclusion of a syllogism. The second of the above theorems deals with the case in which both the premisses are false; the third with that in which one only of the premisses is false.
[433] Hamilton (Logic, I. p. 450) considers the doctrine “that if the conclusion of a syllogism be true, the premisses may be either true or false, but that if the conclusion be false, one or both of the premisses must be false” to be extralogical, if it is not absolutely erroneous. He is clearly wrong, since the doctrine in question admits of a purely formal proof.
344. Arguments from the truth of one premiss and the falsity of the other premiss in a valid syllogism, or from the falsity of one premiss to the truth of the conclusion, or from the truth of one premiss to the falsity of the conclusion.—In this section we shall consider three problems, mutually involved in one another, which are in a manner related to the theorems contained in the preceding section. It has, for example, been shewn that one premiss and the contradictory of the other premiss will not in any case suffice to establish the contradictory of the original conclusion; the object of the first of the following problems is to enquire in what cases they can establish any conclusion at all.
(i) To find a pair of valid syllogisms having a common premiss, such that the remaining premiss of the one contradicts the remaining premiss of the other.[434]
[434] This problem was suggested by the following question of Mr O’Sullivan’s, which puts the same problem in another form: Given that one premiss of a valid syllogism is false and the other true, determine generally in what cases a conclusion can be drawn from these data.
397 We have to find cases in which P and Q, P and Qʹ (the contradictory of Q) are the premisses of two valid syllogisms. In working out this problem and the problems that follow, it must be remembered that if two propositions are contradictories, they will differ in quality, and also in the distribution of their terms, so that any term distributed in either of them is undistributed in the other and vice versâ. We may, therefore, assume that Q is affirmative and Qʹ negative. Let P contain the terms X and Y, while Q and Qʹ contain the terms Y and Z, so that Y is the middle term, and X and Z the extreme terms, of each syllogism.
Since Qʹ is negative, P must be affirmative; and since Y must be undistributed either in Q or in Qʹ, it must be distributed in P.
Hence P = YaX.
Qʹ must distribute Z: for the conclusion (being negative) must distribute one term, and X is undistributed in P. It follows that Z is undistributed in Q.
Hence Q = YaZ or YiZ or ZiY ;
Qʹ = YoZ or YeZ or ZeY.
If the different possible combinations are worked out, it will be found that the following are the syllogisms satisfying the condition that if one premiss (that in black type) is retained, while the other is replaced by its contradictory, a conclusion is still obtainable:—
In figure 1: AII;
In figure 3: AAI, AAI, IAI, AII, EAO, OAO;
In figure 4: IAI, EAO.
(ii) To find a pair of valid syllogisms having a common conclusion, such that a premiss in the one contradicts a premiss in the other.
Let Q and Qʹ (which we may assume to be respectively affirmative and negative) be the premisses in question, and Pʹ the conclusion; also let Q and Qʹ contain the terms Y and Z, while Pʹ contains the terms X and Z, so that Z is the middle term, and X and Y the extreme terms, of each syllogism.
It follows immediately that Pʹ is negative; also that Y 398 must be undistributed in Pʹ, since it is necessarily undistributed either in Q or in Qʹ.
Hence Pʹ = YoX.
Since X is distributed in Pʹ it must also be distributed in the premiss which is combined with Qʹ ; and as this premiss must be affirmative, it cannot also distribute Z, which must therefore be distributed in Qʹ (and undistributed in Q).
Hence Q = YaZ or YiZ or ZiY ;
Qʹ = YoZ or YeZ or ZeY.
If the different possible combinations are worked out, it will be found that the following are the syllogisms satisfying the condition that the same conclusion is obtainable from another pair of premisses, of which one contradicts one of the original premisses (namely, that in black type):—
In figure 1: EAO, EIO;
In figure 2: EAO, AEO, EIO, AOO;
In figure 3: EIO;
In figure 4: AEO, EIO.
(iii) To find a pair of valid syllogisms having a common premiss, such that the conclusion of one contradicts the conclusion of the other.[435]
[435] This problem was suggested by the following question of Mr Panton’s, which puts the same problem in another form: If the conclusion be substituted for a premiss in a valid mood, investigate the conditions which must be fulfilled in order that the new premisses should be legitimate.
Let P be the common premiss, Q and Qʹ (respectively affirmative and negative) the contradictory conclusions; also let P contain the terms X and Y, while Q and Qʹ contain the terms Y and Z, so that X is the middle term, and Y and Z the extreme terms, of each syllogism.
Since Q is affirmative, P must be affirmative; and since either Q or Qʹ will distribute Y, P must distribute Y.
Hence P = YaX.
The premiss which, combined with P, proves Q must be affirmative and must distribute X ; it cannot therefore distribute Z, and Z must accordingly be undistributed in Q (and distributed in Qʹ). 399
Hence Q = YaZ or YiZ or ZiY ;
Qʹ = YoZ or YeZ or ZeY.
If the different possible combinations are worked out, it will be found that the following are the syllogisms satisfying the condition that the contradictory of the conclusion is obtainable, although one of the premisses (that in black type) is retained:—
In figure 1: AAA, AAI, EAE, EAO;
In figure 2: EAE, EAO, AEE;
In figure 4: AAI, AEE.[436]
[436] It will be observed that each of the above problems yields nine cases. Between them they cover all the 24 valid moods; but there are three moods (namely, EAO in figures 1 and 2 and AAI in figure 3) which occur twice over. The 15 unstrengthened and unweakened moods are equally distributed, namely, the four yielding I conclusions (together with OAO) falling under (i); the six yielding O conclusions (except OAO) under (ii); the five yielding A or E conclusions under (iii). All the moods of figure 1 (except those with an I premiss) fall under (iii); all the moods of figure 2 (except those with an E conclusion) under (ii); all the moods of figure 3 (except the one not having an A premiss) under (i).
The three sets of moods worked out above are mutually derivable from one another. Thus,
| (i) | (ii) | (iii) | ||
| P and Q ∴ R | = | Q and Rʹ ∴ Pʹ | = | Rʹ and P ∴ Qʹ |
| P and Qʹ ∴ Tʹ | = | Qʹ and T ∴ Pʹ | = | T and P ∴ Q |
In this table (i) represents the possible cases in which, one premiss being retained, the other premiss may be replaced by its contradictory. We can then deduce (ii) the cases in which, the conclusion being retained, one premiss may be replaced by its contradictory; and (iii) the cases in which, one premiss being retained, the conclusion may be replaced by its contradictory. We might of course equally well start from (ii) or from (iii), and thence deduce the two others.
Comparing the first syllogism of (i) with the second syllogism of (iii) and vice versâ, we see further that (i) gives the cases in which, one premiss being retained, the conclusion may be replaced by the other premiss; and that (iii) gives the cases in which, one premiss being retained, the other premiss may be replaced by the conclusion.
400 The following is another method of stating and solving all three problems: To determine in what cases it is possible to obtain two incompatible trios of propositions, each trio containing three and only three terms and each including a proposition which is identical with a proposition in the other and also a proposition which is the contradictory of a proposition in the other.
Let the propositions be P, Q, Rʹ and P, Qʹ, T ; and let P contain the terms X and Y ; Q and Qʹ the terms Y and Z ; R and T, the terms Z and X. Suppose Q to be affirmative, and Qʹ negative.
Then since one of each trio of propositions must be negative, and not more than one can be so (as shewn in section [214]), P and T must be affirmative, and Rʹ negative.
Again, since each of the terms X, Y, Z must be distributed once at least in each trio of propositions (as shewn in section 214), and since Y must be undistributed either in Q or in Qʹ, Y must be distributed in P.
Hence P = YaX.
X, being undistributed in P, must be distributed in Rʹ and T.
Hence T = XaZ.
Z, being undistributed in T, must be distributed in Qʹ, and therefore undistributed in Q, and distributed in Rʹ.
Hence Q = YaZ or YiZ or ZiY ;
Qʹ = YoZ or YeZ or ZeY ;
Rʹ = XeZ or ZeX.
We have then the following solution of our problem:—
| YaZ, YaZ or YiZ or ZiY, XeZ or ZeX ; |
| YaZ, YoZ or YeZ or ZeY, XaZ. |
345. Numerical Moods of the Syllogism.[437]—The following are examples of numerical moods in the different figures of the syllogism:—401
[437] This section was suggested by the following question of Mr Johnson’s:—“Shew the validity of the following syllogisms: (i) All M’s are P’s, At least n S’s are M’s, therefore, At least n S’s are P’s; (ii) All P’s are M’s, Less than n S’s are M’s, therefore, Less than n S’s are P’s; (iii) Less than n M’s are P’s, At least n M’s are S’s, therefore, Some S’s are not P’s. Deduce from the above the ordinary non-numerical moods of the first three figures.”
The above moods may be established as follows:—
(i) From All M’s are P’s, it follows that Every S which is M is also P, and since At least n S’s are M’s, it follows further that At least n S’s are P’s.
Denoting the major premiss of (i) by A, the minor by B, and the conclusion by C, we obtain immediately the following syllogisms:—
| A, | Cʹ, | |||
| Cʹ, | B, | |||
| ⎯ | ⎯ | |||
| ∴ | Bʹ ; | ∴ | Aʹ ; |
and these are respectively equivalent to (iv) and (vii).
(v) is obtainable from (iv) by transposing the premisses and converting the conclusion;
(ii) from (v) by converting the major premiss;
(iii) from (vii) by converting the minor premiss;
(vi) from (iii) by converting the major premiss;
(viii) from (i) by converting the minor premiss;
(ix) from (viii) by transposing the premisses and converting the conclusion;
(x) from (i) by transposing the premisses and converting the conclusion;
(xi) from (iv) by converting the minor premiss;
(xii) from (vii) by converting the major premiss.
The ordinary non-numerical moods of the different figures may be deduced from the above results as follows:—
Figure 1. (i) Putting n = total number of S’s, we have MaP, SaM, ∴ SaP, that is, Barbara ; and putting n = 1, we have MaP, SiM, ∴ SiP, that is, Darii.
(ii) Putting n = 1, MeP, SaM, ∴ SeP (Celarent).
(iii) Putting n = 1, MeP, SiM, ∴ SoP (Ferio).
AAI and EAO follow à fortiori.
Figure 2 (iv) Putting n = total number of S’s, PaM, SoM, ∴ SoP (Baroco); putting n = 1, PaM, SeM, ∴ SeP (Camestres).
403 (v) Putting n = 1, PeM, SaM, ∴ SeP (Cesare).
(vi) Putting n = 1, PeM, SiM, ∴ SoP (Festino).
AEO and EAO follow à fortiori.
Figure 3. (vii) Putting n = total number of M’s, MoP, MaS, ∴ SoP (Bocardo); putting n = 1, MeP, MiS, ∴ SoP (Ferison).
(viii) Putting n = 1, MaP, MiS, ∴ SiP (Datisi).
(ix) Putting n = 1, MiP, MaS, ∴ SiP (Disamis).
Darapti and Felapton follow à fortiori.
Figure 4. (x) Putting n = 1, PiM, MaS, ∴ SiP (Dimaris).
(xi) Putting n = 1, PaM, MeS, ∴ SeP (Camenes).
(xii) Putting n = 1, PeM, MiS, ∴ SoP (Fresison).
Bramantip, AEO, and Fesapo follow à fortiori.
EXERCISES.
346. “Whatever P and Q may stand for, we may shew à priori that some P is Q. For All PQ is Q by the law of identity, and similarly All PQ is P ; therefore, by a syllogism in Darapti, Some P is Q.” How would you deal with this paradox? [K.]
A solution is afforded by the discussion contained in section [342]; and this example seems to shew that the enquiry—how far assumptions with regard to existence are involved in syllogistic processes—is not irrelevant or unnecessary.
347. What conclusion can be drawn from the following propositions? The members of the board were all either bondholders or shareholders, but not both; and the bondholders, as it happened, were all on the board. [V.]
We may take as our premisses:
No member of the board is both a bondholder and a shareholder,
All bondholders are members of the board;
and these premisses yield a conclusion (in Celarent),
No bondholder is both a bondholder and a shareholder,
that is, No bondholder is a shareholder.
348. The following rules were drawn up for a club:—
(i) The financial committee shall be chosen from amongst the 404 general committee; (ii) No one shall be a member both of the general and library committees, unless he be also on the financial committee; (iii) No member of the library committee shall be on the financial committee.
Is there anything self-contradictory or superfluous in these rules? [VENN, Symbolic Logic, p. 331.]
Let F = member of the financial committee,
G = member of the general committee,
L = member of the library committee.
The above rules may then be expressed symbolically as follows:—
(i) All F is G ;
(ii) If any L is G, that L is F ;
(iii) No L is F.
From (ii) and (iii) we obtain (iv) No L is G.
The rules may therefore be written in the form,
(1) All F is G,
(2) No L is G,
(3) No L is F.
But in this form (3) is deducible from (1) and (2).
Hence all that is contained in the rules as originally stated may be expressed by (1) and (2); that is, the rules as originally stated were partly superfluous, and they may be reduced to
(1) The financial committee shall be chosen from amongst the general committee;
(2) No one shall be a member both of the general and library committees.
If (ii) is interpreted as implying that there are some individuals who are on both the general and library committees, then it follows that (ii) and (iii) are inconsistent with each other.
349. Given that the middle term is distributed twice in the premisses of a syllogism, determine directly (i.e., without any reference to the mnemonic verses or the special rules of the figures) in what different moods it might possibly be. [K.]
The premisses must be either both affirmative, or one affirmative and one negative.
In the first case, both premisses being affirmative can distribute their subjects only. The middle term must, therefore, be the subject in each, and both must be universal. This limits us to the one syllogism,— 405
| All M is P, | |
| All M is S, | |
| therefore, | Some S is P. |
In the second case, one premiss being negative, the conclusion must be negative and will, therefore, distribute the major term. Hence, the major premiss must distribute the major term, and also (by hypothesis) the middle term. This condition can be fulfilled only by its being one or other of the following,—No M is P or No P is M. The major being negative, the minor must be affirmative, and in order to distribute the middle term must be All M is S.
In this case we get two syllogisms, namely,—
| No M is P, | |
| All M is S, | |
| therefore, | Some S is not P ; |
| No P is M, | |
| All M is S, | |
| therefore, | Some S is not P. |
The given condition limits us, therefore, to three syllogisms (one affirmative and two negative); and by reference to the mnemonic verses we may identify these with Darapti and Felapton in figure 3, and Fesapo in figure 4.
350. If the major premiss and the conclusion of a valid syllogism agree in quantity, but differ in quality, find the mood and figure. [T.]
Since we cannot have a negative premiss with an affirmative conclusion, the major premiss must be affirmative and the conclusion negative. It follows immediately that, in order to avoid illicit major, the major premiss must be All P is M (where M is the middle term and P the major term). The conclusion, therefore, must be No S is P (S being the minor term); and this requires that, in order to avoid undistributed middle and illicit minor, the minor premiss should be No S is M or No M is S. Hence the syllogism is in Camestres or in Camenes.
351. Given a valid syllogism with two universal premisses and a particular conclusion, such that the same conclusion cannot be inferred, if for either of the premisses is substituted its subaltern, determine the mood and figure of the syllogism. [K.]
Let S, M, P be respectively the minor, middle, and major terms of the given syllogism. Then, since the conclusion is particular, it must be either Some S is P or Some S is not P. 406
First, if possible, let it be Some S is P.
The only term which need be distributed in the premisses is M. But since we have two universal premisses, two terms must be distributed in them as subjects.[438] One of these distributions must be superfluous; and it follows that for one of the premisses we may substitute its subaltern, and still get the same conclusion.
The conclusion cannot then be Some S is P.
Secondly, if possible, let the conclusion be Some S is not P.
If the subject of the minor premiss is S, we may clearly substitute its subaltern without affecting the conclusion. The subject of the minor premiss must therefore be M, which will thus be distributed in this premiss. M cannot also be distributed in the major, or else it is clear that its subaltern might be substituted for the minor and nevertheless the same conclusion inferred. The major premiss must, therefore, be affirmative with M for its predicate. This limits us to the syllogism—
| All P is M, | |
| No M is S, | |
| therefore, | Some S is not P ; |
and this syllogism, which is AEO in figure 4, does fulfil the given conditions, for it becomes invalid if either of the premisses is made particular.
The above amounts to a general proof of the proposition laid down in section [246]:—Every syllogism in which there are two universal premisses with a particular conclusion is a strengthened syllogism with the single exception of AEO in figure 4.
[438] We here include the case in which the middle term is itself twice distributed.
352. Given two valid syllogisms in the same figure in which the major, middle, and minor terms are respectively the same, shew, without reference to the mnemonic verses, that if the minor premisses are subcontraries, the conclusions will be identical. [K.]
The minor premiss of one of the syllogisms must be O, and the major premiss of this syllogism must, therefore, be A and the conclusion O. The middle and the major terms having then to be distributed in the premisses, this syllogism is determined, namely,—
| All P is M, | |
| Some S is not M, | |
| therefore, | Some S is not P. |
407 Since the other syllogism is to be in the same figure, its minor premiss must be Some S is M ; the major must therefore be universal, and in order to distribute the middle term it must be negative. This syllogism therefore is also determined, namely,—
| No P is M, | |
| Some S is M, | |
| therefore, | Some S is not P. |
The conclusions of the two syllogisms are thus shewn to be identical.
353. Find out in which of the valid syllogisms the combination of one premiss with the subcontrary of the conclusion would establish the subcontrary of the other premiss. [J.]
In the original syllogism (α) let X (universal) and Y (particular) prove Z (particular), the minor, middle, and major terms being S M, and P, respectively. Then we are to have another syllogism (β) in which X and Z1 (the sub-contrary of Z) prove Y1 (the sub-contrary of Y). In β, S or P will be the middle term.
It is clear that only one term can be distributed in α if the conclusion is affirmative, and only two if the conclusion is negative. Hence S cannot be distributed in α, and it follows that it cannot be distributed in the premisses of β. The middle term of β must therefore be P, and as X must consequently contain P it must be the major premiss of α and Y the minor premiss.
Z must be either SiP or SoP. First, let Z = SiP. Then it is clear that X = MaP, Z1 = SoP, Y1 = SoM, Y = SiM. Secondly, let Z = SoP. Then Z1 = SiP, X = PaM or MeP or PeM (since it must distribute P), Y1 = SiM (if X is affirmative) or SoM (if X is negative), Y = SoM or SiM accordingly.
Hence we have four syllogisms satisfying the required conditions as follows:—
| MaP | MeP | PeM | PaM |
| SiM | SiM | SiM | SoM |
| ⎯⎯ | ⎯⎯ | ⎯⎯ | ⎯⎯ |
| SiP | SoP | SoP | SoP |
It will be observed that these are all the moods of the first and second figures, in which one premiss is particular.
354. Is it possible that there should be a valid syllogism such that, each of the premisses being converted, a new syllogism is obtainable giving a conclusion in which the old major and minor terms have changed places? Prove the correctness of your answer by general reasoning, and if it is in the 408 affirmative, determine the syllogism or syllogisms fulfilling the given conditions. [K.]
If such a syllogism be possible, it cannot have two affirmative premisses, or (since A can only be converted per accidens) we should have two particular premisses in the new syllogism.
Therefore, the original syllogism must have one negative premiss. This cannot be O, since O is inconvertible.
Therefore, one premiss of the original syllogism must be E.
First, let this be the major premiss. Then the minor premiss must be affirmative, and its converse (being a particular affirmative), will not distribute either of its terms. But this converse will be the major premiss of the new syllogism, which also must have a negative conclusion. We should then have illicit major in the new syllogism; and hence the above supposition will not give us the desired result.
Secondly, let the minor premiss of the original syllogism be E. The major premiss in order to distribute the old major term must be A, with the major term as subject. We get then the following, satisfying the given conditions:—
| All P is M, |
| No M is S, or No S is M, |
| therefore, No S is P, or Some S is not P ; |
that is, we really have four syllogisms, such that both premisses being converted, thus,
| No S is M, or No M is S, |
| Some M is P, |
we have a new syllogism yielding a conclusion in which the old major and minor terms have changed places, namely,
Some P is not S.
Symbolically,—
| PaM, | SeM, | ⎱ | ||||
| MeS, | ⎱ | or | MeS, | ⎰ | ||
| or | SeM, | ⎰ | MiP, | |||
| ⎯⎯ | ⎯⎯ | |||||
| ∴ or | SeP SoP | ⎱ ⎰ | ∴ | PoS. | ||
If it be required to retain the quantity of the original conclusion, that conclusion must be SoP, in this case then we have only two syllogisms fulfilling the given conditions.
355. Shew that if the proportion of B’s out of the class A is greater than that out of the class not-A, then the proportion 409 of A’s out of the class B will be greater than that out of the class not-B.[439] [J.]
[439] This and the following problem cannot properly be called problems on the syllogism. They are given as examples in numerical logic.
Let the number of A’s be denoted by N(A), the number of AB’s by N(AB), &c.
Then, since Every A is AB or Ab (by the law of excluded middle) and No A is both AB and Ab (by the law of contradiction), it follows that
N(A) = N(AB) + N(Ab).
We have to shew that
| N(AB) | N(Ab) | |
| ⎯⎯ | > | ⎯⎯ |
| N(B) | N(b) |
follows from
| N(AB) | N(aB) | |
| ⎯⎯ | > | ⎯⎯ . |
| N(A) | N(a) |
This can be done by substituting
N(AB) + N(Ab) for N(A), &c.
Thus,
| N(AB) | N(aB) | ||
| ⎯⎯ | > | ⎯⎯ , | |
| N(A) | N(a) | ||
| N(a) | N(A) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(aB) | N(AB) | ||
| N(aB) + N(ab) | N(AB) + N(Ab) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(aB) | N(AB) | ||
| N(ab) | N(Ab) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(aB) | N(AB) | ||
| N(ab) | N(aB) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(Ab) | N(AB) | ||
| N(Ab) + N(ab) | N(AB) + N(aB) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(Ab) | N(AB) | ||
| N(b) | N(B) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(Ab) | N(AB) | ||
| N(AB) | N(Ab) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ . |
| N(B) | N(b) |
356. Given the number (U) of objects in the Universe, and the number of objects in each of the classes x1, x2, x3, … xn, shew that the least number of objects in the class (x1x2x3 … xn)
= U − N (x1) − N (x2) − N (x3) … − N (xn). 410
where N (x1) means the number of things which are not x1; N (x2) means the number of things which are not x2; &c. [J.]
Given N (x1), N (x2), &c., the number of objects in the class (x1 or x2 … or xn) is greatest when no object belongs to any pair of the classes x1, x2, …; and in this case it = N (x1) + N (x2) … + N (xn).
Hence the least number in the contradictory class, x1x2x3 … xn,
= U − N (x1) − N (x2) … − N (xn).
357. Prove that with three given propositions (of the forms A, E, I, O) it is never possible to construct more than one valid syllogism. [K.]
358. On the supposition that no proposition is interpreted as implying the existence either of its subject or of its predicate, find in what cases the reduction of syllogisms to figure 1 is invalid. [K.]
359. Given a valid syllogism, determine the conditions under which the contradictories of the premisses will furnish premisses for another valid syllogism containing the same terms. How will the conclusions of the two syllogisms be related to one another? [K.]
360. Shew that the number of paupers who are blind males is equal to the excess, if any, of the sum of the whole number of blind persons, added to the whole number of male persons, added to the number of those who being paupers are neither blind nor males, above the sum of the whole number of paupers, added to the number of those who not being paupers are blind, and to the number of those who not being paupers are male. [Jevons, Principles of Science.]
361. Shew that, if X and Y are any two propositions containing a common term, then (a) one of the four combinations XY, XYʹ, XʹY, XʹYʹ will always form unstrengthened premisses for a valid syllogism; (b) either only one of the four combinations will do so; or, if two, the syllogisms so formed will be of the same mood. [RR.]
362. Two arguments whose premisses are mutually consistent but which contain sub-contrary conclusions are formed in the same figure with the same middle term.
Find out directly from the general rules of syllogism what can be known with regard to the moods and figure of the two given arguments. [J.]
411 363. Some M is not P, All S is all M. What conclusion follows from the combination of these premisses?
Can you infer anything either about S in terms of P or about P in terms, of S from the knowledge that both the above propositions are false? [K.]
364. (i) Either all M is all P or Some M is not P ; (ii) Some S is not M. What is all that can be inferred (a) about S in terms of P, (b) about P in terms of S, from the knowledge that both the above statements are false? [K.]
365. (a) “A good temper is proof of a good conscience, and the combination of these is proof of a good digestion, which again always produces one or the other.” Shew that this is precisely equivalent to the following: “A good temper is proof of a good digestion, and a good digestion of a good conscience.”
(b) Examine (by diagrams or otherwise) the following argument:—“Patriotism and humanitarianism must be either incompatible or inseparable; and though family-affection and humanitarianism are compatible, yet either may exist without the other; hence, family affection may exist without patriotism.” Reduce the argument, if you can, to ordinary syllogistic form; and determine whether the premisses state anything more than is necessary to prove the conclusion. [J.]
366. “All scientific persons are willing to learn; all unscientific persons are credulous; therefore, some who are credulous are not willing to learn, and some who are willing to learn are not credulous.”
Shew that the ordinary rules of immediate and mediate inference justify this reasoning; but that a certain assumption is involved in thus avoiding the apparent illicit process. Shew also that, accepting the validity of obversion and simple conversion, we have an analogous case in any inference of a particular from a universal, [J.]
367. An invalid syllogism of the second figure with a particular premiss is found to break the general rules of the syllogism in this respect only, that the middle term is undistributed. If the particular premiss is false and the other true, what do we know about the truth or falsity of the conclusion? [K.]
368. A syllogism is found to offend against none of the syllogistic rules except that with two affirmative premisses it has a negative conclusion. Determine the mood and figure of the syllogism. [K.]
412 369. Given two valid syllogisms in the same figure in which the major, middle, and minor terms are respectively the same, shew, without reference to the mnemonic verses, that if the minor premisses are contradictories, the conclusions will not be contradictories. [K.]
370. Find two syllogisms, having neither strengthened premisses nor weakened conclusions, and having M and N respectively as their middle terms, which satisfy the following conditions: (a) their conclusions are to be subcontraries; (b) their premisses are to prove that Some M is N, and to be consistent with the fact that Some M is not N. [J.]
371. Is it possible that there should be two syllogisms having a common premiss such that their conclusions, being combined as premisses in a new syllogism, may give a universal conclusion? If so, determine what the two syllogisms must be. [N.]
372. Three given propositions form the premisses and conclusion of a valid syllogism which is neither strengthened nor weakened. Shew that if two of the propositions are replaced by their contra-complementaries, the argument will still be valid, provided that the proposition remaining unaltered is either a universal premiss or a particular conclusion. [J.]
373. A certain proposition stands as minor premiss of a syllogism in the second figure whose major term is X. The same proposition stands also as major premiss of a syllogism in the third figure whose minor term is Y. If the given syllogisms are both formally and materially correct, shew how in every case we may conclude syllogistically that “some Y is not X” [J.]
374. Find out the valid syllogisms that may be constructed without using a universal premiss of the same quality as the conclusion.
Shew how these syllogisms may be directly reduced to one another; and represent diagrammatically the combined information that they yield, on the supposition that they have the same minor, middle, and major terms respectively. [J.]
375. Express the exact information contained in the two propositions, All S is M, All M is P, by means of (1) two propositions having S and not-S respectively as subjects; (2) two propositions having M and not-M respectively as subjects; (3) two propositions having P and not-P respectively as subjects. [K.]