CHAPTER I.

THE RULES OF THE SYLLOGISM.

197. The Terms of the Syllogism.—A reasoning which consists of three propositions of the traditional categorical form, and which contains three and only three terms, is called a categorical syllogism.

Of the three terms contained in a categorical syllogism, two appear in the conclusion and also in one or other of the premisses, while the third appears in the premisses only. That which appears as the predicate of the conclusion, and in one of the premisses, is called the major term ; that which appears as the subject of the conclusion, and in one of the premisses, is called the minor term ;[306] and that which appears in both the premisses, but not in the conclusion (being that term by their relations to which the mutual relation of the two other terms is determined), is called the middle term.

[³⁰⁶] The major and minor terms are also sometimes called the extremes of the syllogism.

Thus, in the syllogism,—

	All M is P,
	All S is M,
therefore,	All S is P ;

S is the minor term, M the middle term, and P the major term.

286 These respective designations of the terms of a syllogism resulted from such a syllogism as that just given being regarded as typical. With the exception of the somewhat rare case in which the terms of a proposition are coextensive, the above syllogism may be represented by the following diagram. Here

clearly the major term is the largest in extent, and the minor the smallest, while the middle occupies an intermediate position.

But we have no guarantee that the same relation between the terms of a syllogism will hold, when one of the premisses is negative or particular. Thus, the syllogism—No M is P, All S is M, therefore, No S is P—yields as one case

where the major term may be the smallest in extent, and the middle the largest. Again, the syllogism—No M is P, Some S is M, therefore, Some S is not P—yields as one case

where the major term may be the smallest in extent and the minor the largest.

Whilst, however, the middle term is not always a middle term in extent, it is always a middle term in the sense that by its means the two other terms are connected, and their mutual relation determined.

287 198. The Propositions of the Syllogism.—Every categorical syllogism consists of three propositions. Of these one is the conclusion. The premisses are called the major premiss and the minor premiss according as they contain the major term or the minor term respectively.

Thus,	All M is P	(major premiss),
	All S is M	(minor premiss),
therefore,	All S is P	(conclusion).

It is usual (as in the above syllogism) to state the major premiss first and the conclusion last. This is, however, nothing more than a convention. The order of the premisses in no way affects the validity of a syllogism, and has indeed no logical significance, though in certain cases it may be of some rhetorical importance. Jevons (Principles of Science, 6, § 14) argues that the cogency of a syllogism is more clearly recognisable when the minor premiss is stated first. But it is doubtful whether any general rule of this kind can be laid down. In favour of the traditional order, it is to be said that in what is usually regarded as the typical syllogism (All M is P, All S is M, therefore, All S is P) there is a philosophical ground for stating the major premiss first, since that premiss gives the general rule, of which the minor premiss enables us to make a particular application.

199. The Rules of the Syllogism.—The rules of the categorical syllogism as usually stated are as follows:—
(1) Every syllogism contains three and only three terms.
(2) Every syllogism consists of three and only three propositions.

These two so-called rules are not properly speaking rules for the validity of an argument. They simply serve to define the syllogism as a particular form of argument. A reasoning which does not fulfil these conditions may be formally valid, but we do not call it a syllogism.[307] The four rules that follow 288 are really rules in the sense that if, when we have got the reasoning into the form of a syllogism, they are not fulfilled, then the reasoning is invalid.[308]

[³⁰⁷] For example, B is greater than C, A is greater than B, therefore, A is greater than C.

Here is a valid reasoning which consists of three propositions. But it contains more than three terms; for the predicate of the second premiss is “greater than B,” while the subject of the first premiss is “B.” It is, therefore, as it stands, not a syllogism. Whether reasonings of this kind admit of being reduced to syllogistic form is a problem which will be discussed [subsequently].

[³⁰⁸] Apparent exceptions to these rules will be shewn in sections [205] and [206] to result from the attempt to apply them to reasonings which have not first been reduced to syllogistic form.

(3) No one of the three terms of a syllogism may be used ambiguously; and the middle term must be distributed once at least in the premisses.

This rule is frequently given in the form: “The middle term must be distributed once at least, and must not be ambiguous.” But it is obvious that we have to guard against ambiguous major and ambiguous minor as well as against ambiguous middle. The fallacy resulting from the ambiguity of one of the terms of a syllogism is a case of quaternio terminorum, that is, a fallacy of four terms.

The necessity of distributing the middle term may be illustrated by the aid of the Eulerian diagrams. Given, for instance. All P is M and All S is M, we may have any one of the five following cases:—

Here all the five relations that are à priori possible between S and P are still possible. We have, therefore, no conclusion.

If in a syllogism the middle term is distributed in neither premiss, we are said to have a fallacy of undistributed middle.

289 (4) No term may be distributed in the conclusion which was not distributed in one of the premisses.

The breach of this rule is called illicit process of the major, or illicit process of the minor, as the case may be; or, more briefly, illicit major or illicit minor.

(5) From two negative premisses nothing can be inferred.

This rule may, like rule 3, be very well illustrated by means of the Eulerian diagrams.

(6) If one premiss is negative, the conclusion must be negative; and to prove a negative conclusion, one of the premisses must be negative.[309]

[³⁰⁹] This rule and the second corollary given in the following section are sometimes combined into the one rule, Conclusio sequitur partem deteriorem ; i.e., the conclusion follows the worse or weaker premiss both in quality and in quantity, a negative being considered weaker than an affirmative and a particular than a universal.

200. Corollaries from the Rules of the Syllogism.—From the rules given in the preceding section, three corollaries may be deduced:—[310]

[³¹⁰] The formulation of these corollaries may in some cases help towards the more immediate detection of unsound syllogisms.

(i) From two particular premisses nothing can be inferred.
Two particular premisses must be either
(α) both negative,
or (β) both affirmative,
or (γ) one negative and one affirmative.
But in case (α), no conclusion follows by rule 5.
In case (β), since no term can be distributed in two particular affirmative propositions, the middle term cannot be distributed, and therefore by rule 3 no conclusion follows.
In case (γ), if any valid conclusion is possible, it must be negative (rule 6). The major term, therefore, will be distributed in the conclusion; and hence we must have two terms distributed in the premisses, namely, the middle and the major (rules 3, 4). But a particular negative proposition and a particular affirmative proposition between them distribute only one term. Therefore, no conclusion can be obtained.

(ii) If one premiss is particular, the conclusion must be particular.
290 We must have either
(α) two negative premisses, but this case is rejected by rule 5;
or (β) two affirmative premisses;
or (γ) one affirmative and one negative.
In case (β) the premisses, being both affirmative and one of them particular, can distribute but one term between them. This must be the middle term by rule 3. The minor term is, therefore, undistributed in the premisses, and the conclusion must be particular by rule 4.
In case (γ) the premisses will between them distribute two and only two terms. These must be the middle by rule 3, and the major by rule 4 (since we have a negative premiss, necessitating by rule 6 a negative conclusion, and therefore the distribution of the major term in the conclusion). Again, therefore, the minor cannot be distributed in the premisses, and the conclusion must be particular by rule 4.

De Morgan (Formal Logic, p. 14) gives the following proof of this corollary:—“If two propositions P and Q together prove a third R, it is plain that P and the denial of R prove the denial of Q. For P and Q cannot be true together without R. Now, if possible, let P (a particular) and Q (a universal) prove R (a universal). Then P (particular) and the denial of R (particular) prove the denial of Q. But two particulars can prove nothing.”[311]

[³¹¹]Further attention will be called in a later chapter to the general principle upon which this proof is based. See section [264].

(iii) From a particular major and a negative minor nothing can be inferred.
Since the minor premiss is negative, the major premiss must by rule 5 be affirmative. But it is also particular, and it therefore follows that the major term cannot be distributed in it. Hence, by rule 4, it must be undistributed in the conclusion, i.e., the conclusion must be affirmative. But also, by rule 6, since we have a negative premiss, it must be negative. This contradiction establishes the corollary that from the given premisses no conclusion can be drawn.

The following mnemonic lines, attributed to Petrus Hispanus, 291 afterwards Pope John XXI., sum up the rules of the syllogism and the first two corollaries:

Distribuas medium: nec quartus terminus adsit:
Utraque nec praemissa negans, nec particularis:
Sectetur partem conclusio deteriorem;
Et non distribuat, nisi cum praemissa, negetve.

201. Restatement of the Rules of the Syllogism.—It has been already pointed out that the first two of the rules given in section [199] are to be regarded as a description of the syllogism rather than as rules for its validity. Again, the part of rule 3 relating to ambiguity may be regarded as contained in the proviso that there shall be only three terms; for, if one of the terms is ambiguous, there are really four terms, and hence no syllogism according to our definition of syllogism. The rules may, therefore, be reduced to four; and they may be restated as follows:—

A. Two rules of distribution:
(1) The middle term must be distributed once at least in the premisses;
(2) No term may be distributed in the conclusion which was not distributed in one of the premisses;
B. Two rules of quality:
(3) From two negative premisses no conclusion follows;
(4) If one premiss is negative, the conclusion must be negative; and to prove a negative conclusion, one of the premisses must be negative.[312]

[³¹²] The rules of quality might also be stated as follows; To prove an affirmative conclusion, both premisses must be affirmative; To prove a negative conclusion, one premiss must be affirmative and the other negative.

202. Dependence of the Rules of the Syllogism upon one another.—The four rules just given are not ultimately independent of one another. It may be shewn that a breach of the second, or of the third, or of the first part of the fourth involves indirectly a breach of the first; or, again, that a breach of the first, or of the third, or of the first part of the fourth involves indirectly a breach of the second.

292 (i) The rule that two negative premisses yield no conclusion may be deduced from the rule that the middle term must be distributed once at least in the premisses.

This is shewn by De Morgan (Formal Logic, p. 13). He takes two universal negative premisses E, E. In whatever figure they may be, they can be reduced by conversion to

No P is M,
No S is M.

Then by obversion they become (without losing any of their force),—

All P is not-M,
All S is not-M ;

and we have undistributed middle. Hence rule 3 is exhibited as a corollary from rule 1. For if any connexion between S and P can be inferred from the first pair of premisses, it must also be inferable from the second pair.

The case in which one of the premisses is particular is dealt with by De Morgan as follows;—“Again, No Y is X, Some Ys are not Zs, may be converted into

Every X is (a thing which is not Y),
Some (things which are not Zs) are Ys,

in which there is no middle term.”

This is not satisfactory, since we may often exhibit a valid syllogism in such a form that there appear to be four terms; e.g., All M is P, All S is M, may be reduced to All M is P, No S is not-M, and there is now no middle term.

The case in question may, however, be disposed of by saying that if we cannot infer anything from two negative premisses both of which are universal, à fortiori we cannot from two negative premisses one of which is particular.[313]

[³¹³] This argument holds good in the special case under consideration even if we interpret particulars, but not universals, as implying the existence of their subjects. For the validity of the above proof that two universal negatives yield no conclusion remains unaffected even if we allow to universals the maximum of existential import.

(ii) The rules that from two negative premisses nothing can be inferred and that if one premiss is negative the conclusion must be negative are mutually deducible from one another.

The following proof that the second of these rules is deducible from the first is suggested by De Morgan’s deduction of 293 the second corollary as given in section [200]. If two propositions P and Q together prove a third R, it is plain that P and the denial of R prove the denial of Q. For P and Q cannot be true together without R. Now, if possible, let P (a negative) and Q (an affirmative) prove R (an affirmative). Then P (a negative) and the denial of R (a negative) prove the denial of Q. But by hypothesis two negatives prove nothing.

It may be shewn similarly that if we start by assuming the second of the rules then the first is deducible from it.

(iii) Any syllogism involving directly an illicit process of major or minor involves indirectly a fallacy of undistributed middle, and vice versâ.[314]

[³¹⁴] For this theorem and its proof I am indebted to Mr Johnson.

Let P and Q be the premisses and R the conclusion of a syllogism involving illicit major or minor, a term X which is undistributed in P being distributed in R. Then the contradictory of R combined with P must prove the contradictory of Q. But any term distributed in a proposition is undistributed in its contradictory. X is therefore undistributed in the contradictory of R, and by hypothesis it is undistributed in P. But X is the middle term of the new syllogism, which is therefore guilty of the fallacy of undistributed middle. It is thus shewn that any syllogism involving directly a fallacy of illicit major or minor involves indirectly a fallacy of undistributed middle.

Adopting a similar line of argument, we might also proceed in the opposite direction, and exhibit the rule relating to the distribution of the middle term as a corollary from the rule relating to the distribution of the major and minor terms.

203. Statement of the independent Rules of the Syllogism.—The theorems established in the preceding section shew that the first part of rule 4 (as given in section [201]) is a corollary from rule 3, and that rule 3 is in its turn a corollary from rule 1; also that rules 1 and 2 mutually involve one another, so that either one of them may be regarded as a corollary from the other. We are, therefore, left with either rule 1 or rule 2 and also with the second part of rule 4; and the independent rules of the syllogism may accordingly be stated as follows: 294
(α) Rule of Distribution:—The middle term must be distributed once at least in the premisses [or, as alternative with this, No term may be distributed in the conclusion which was not distributed in one of the premisses];
(β) Rule of Quality:—To prove a negative conclusion one of the premisses must be negative.[315]

[³¹⁵] On examination it will be found that the only syllogism rejected by this rule and not also rejected directly or indirectly by the preceding rule is the following:—All P is M, All M is S, therefore, Some S is not P. In the technical language explained in the following chapter, this is AAO in figure 4. So far, therefore, as the first three figures are concerned, we are left with a single rule, namely, a rule of distribution, which may be stated in either of the alternative forms given above.

It should be clearly understood that it is not meant that every invalid syllogism will offend directly against one of these two rules. As a direct test for the detection of invalid syllogisms we must still fall back upon the four rules given in section [201].[316] All that we have succeeded in shewing is that ultimately these four rules are not independent of one another.

[³¹⁶] If, for example, for our rule of distribution we select the rule relating to the distribution of the middle term, then the invalid syllogism,

	All M is P,
	No S is M,
therefore,	No S is P,

does not directly involve a breach of either of our two independent rules. But if this syllogism is valid, then must also the following syllogism be valid:

	All M is P (original major),
	Some S is P (contradictory of original conclusion),
therefore	Some S is M (contradictory of original minor);

and here we have undistributed middle. Hence the rule relating to the distribution of the middle term establishes indirectly the invalidity of the syllogism in question. The principle involved is the same as that on which we shall find the process of indirect reduction to be based.

Take, again, the syllogism: PaM, SeM, ∴ SaP. This does not directly offend against the rules given above; but the reader will find that its validity involves the validity of another syllogism in which a direct transgression of these rules occurs.

204. Proof of the Rule of Quality.—For the following very interesting and ingenious proof of the Rule of Quality (as stated in the preceding section) I am indebted to Mr R. A. P. Rogers, of Trinity College, Dublin. In this proof the symbol f_n( ) is used to denote the form of a proposition, the terms which the 295 proposition contains in any given case being inserted within the brackets. Thus, if f_x(P, M) symbolises All M is P, then f_x(B, A) will symbolise All A is B: or, again, if f_y(S, M) symbolises Some S is not M, then f_y(B, A) will symbolise Some B is not A. It will be observed that the order in which the terms are given does not necessarily correspond with the order of subject and predicate.

Let f₁( ), f₂( ), f₃( ) be propositions belonging to the traditional schedule. Then “f₁(P, M), f₂(S, M), ∴ f₃(S, P)” will be the expression of a syllogism; and, since the syllogism is a process of formal reasoning, if the above syllogism is valid in any case, it will hold good if other terms are substituted for S, M, P (or any of them). Thus, substituting S for M, and S for P, if “f₁(P, M), f₂(S, M), ∴ f₃(S, P)” is a valid syllogism, then “f₁(S, S), f₂(S, S), ∴ f₃(S, S)” will be a valid syllogism.

It follows, by contraposition, that if “f₁(S, S), f₂(S, S), ∴ f₃(S, S)” is an invalid syllogism, then “f₁(P, M), f₂(S, M), ∴ f₃(S, P)” will be an invalid syllogism.

If possible, let f₁( ) and f₂( ) be affirmative, while f₃( ) is negative. Then f₁(S, S) and f₂(S, S) will be formally true propositions, while f₃(S, S) is formally false. Hence f₃(S, S) cannot be a valid inference from f₁(S, S) and f₂(S, S); in other words, “f₁(S, S), f₂(S, S), ∴ f₃(S, S)” must be an invalid syllogism. Consequently, “f₁(P. M), f₂(S, M), ∴ f₃(S, P)” cannot be a valid syllogism; that is, we cannot have a valid syllogism in which both premisses are affirmative and the conclusion negative.

205. Two negative premisses may yield a valid conclusion; but not syllogistically.—Jevons remarks: “The old rules of logic informed us that from two negative premisses no conclusion could be drawn, but it is a fact that the rule in this bare form does not hold universally true; and I am not aware that any precise explanation has been given of the conditions under which it is or is not imperative. Consider the following example,—Whatever is not metallic is not capable of powerful magnetic influence, Carbon is not metallic, therefore, Carbon is not capable of powerful magnetic influence. Here we have two distinctly negative premisses, and yet they yield a perfectly 296 valid negative conclusion. The syllogistic rule is actually falsified in its bare and general statement” (Principles of Science, 4, § 10).[317]

[³¹⁷] Lotze (Logic, § 89; Outlines of Logic, §§ 40-42) holds that two negative premisses invalidate a syllogism in figure 1 or figure 2, but not necessarily in figure 3. The example upon which he relies is this,—No M is P, No M is S, therefore, Some not-S is not P. The argument in the text may be applied to this example as well as to the one given by Jevons.

This apparent exception is, however, no real exception. The reasoning (which may be expressed symbolically in the form, No not-M is P, No S is M, therefore, No S is P) is certainly valid; but if we regard the premisses as negative it has four terms S, P, M, and not-M, and is therefore no syllogism. Reducing it to syllogistic form, the minor becomes by obversion All S is not-M, an affirmative proposition.[318] It is not the case, therefore, that we have succeeded in finding a valid syllogism with two negative premisses. In other words, while we must not say that from two negative premisses nothing follows, it remains true that if a syllogism regularly expressed has two negative premisses it is invalid.[319] It must not be considered that this is a mere technicality, and that Jevons’s example shews that the rule is at any rate of no practical value. It is not possible to formulate specific rules at all except with reference to some defined form of reasoning; and no given rule is vitiated either 297 theoretically or for practical purposes because it does not apply outside the form to which alone it professes to apply.[320]

[³¹⁸] It may be added that it is in this form that the cogency of the argument is most easily to be recognised. Of course every affirmation involves a denial and vice versâ ; but it may fairly be said that in Jevons’s example the primary force of the minor premiss, considered in connexion with the major premiss, is to affirm that carbon belongs to the class of non-metallic substances, rather than to deny that it belongs to the class of metallic substances.

[³¹⁹] By a syllogism regularly expressed we mean a reasoning consisting of three propositions, which not only contain between them three and only three terms, but which are also expressed in the traditional categorical forms. Attention must be called to this because, if we introduce additional propositional forms of the kind indicated on page [146], we may have a valid reasoning with two negative premisses, which satisfies the condition of containing only three terms; for example,

	No M is P,
	Some M is not S,
therefore,	There is something besides S and P.

It will be found that this reasoning is easily reducible to a valid syllogism in Ferison.

[³²⁰] A case similar to that adduced by Jevons is dealt with in the Port Royal Logic (Professor Baynes’s translation, p. 211) as follows:—“There are many reasonings, of which all the propositions appear negative, and which are, nevertheless, very good, because there is in them one which is negative only in appearance, and in reality affirmative, as we have already shewn, and as we may still further see by this example: That which has no parts cannot perish by the dissolution of its parts; The soul has no parts; therefore, The soul cannot perish by the dissolution of its parts. There are several who advance such syllogisms to shew that we have no right to maintain unconditionally this axiom of logic, Nothing can be inferred from pure negatives ; but they have not observed that, in sense, the minor of this and such other syllogisms is affirmative, since the middle, which is the subject of the major, is in it the attribute. Now the subject of the major is not that which has parts, but that which has not parts, and thus the sense of the minor is, The soul is a thing without parts, which is a proposition affirmative of a negative attribute.” Ueberweg also, who himself gives a clear explanation of the case, shews that it was not overlooked by the older logicians; and he thinks it not improbable that the doctrine of qualitative aequipollence between two judgments (i.e., obversion) resulted from the consideration of this very question (System of Logic, § 106). Compare, further, Whately’s treatment of the syllogism, “No man is happy who is not secure; no tyrant is secure; therefore, no tyrant is happy” (Logic, II. 4, § 7).

The truth is that by the aid of the process of obversion the premisses of every valid syllogism may be expressed as negatives, though the reasoning will then no longer be technically in the form of a syllogism; for example, the propositions which constitute the premisses of a syllogism in Barbara—All M is P, All S is M, therefore, All S is P—may be written in a negative form, thus, No M is not-P, No S is not-M, and the conclusion All S is P still follows.

206. Other apparent exceptions to the Rules of the Syllogism.—It is curious that the logicians who have laid so much stress on the case considered in the preceding section do not appear to have observed that, as soon as we admit more than three terms, other apparent breaches of the syllogistic rules may occur in what are perfectly valid reasonings. Thus, the premisses All P is M and All S is M, in which M is not distributed, yield the conclusion Some not-S is not-P;[321] and 298 hence we might argue that undistributed middle does not invalidate an argument. Again, from the premisses All M is P, All not-M is S, we may infer Some S is not P,[322] although there is apparently an illicit process of the major. It is unnecessary after what has been said in the preceding section to give examples of valid reasonings in which we have a negative premiss with an affirmative conclusion, or two affirmative premisses with a negative conclusion, or a particular major with a negative minor. Any valid syllogism which is affirmative throughout will yield the first and, if it has a particular major, also the last of these by the obversion of the minor premiss, and the second by the obversion of the conclusion. The only syllogistic rules, indeed, which still hold good when more than three terms are admitted are the rule providing against illicit minor and the first two corollaries.

[³²¹] By the contraposition of both premisses this reasoning is reduced to the valid syllogistic form, All not-M is not-P, All not-M is not-S, therefore, Some not-S is not-P.

[³²²] By the inversion of the first premiss, this reasoning is reduced to the valid syllogistic form, Some not-M is not P, All not-M is S, therefore, Some S is not P. Compare section [104].

But of course none of the above examples really invalidate the syllogistic rules; for these rules have been formulated solely with reference to reasonings of a certain form, namely, those which contain three and only three terms. In every case the reasoning inevitably conforms to the rule which it appears to violate, as soon as, by the aid of immediate inferences, the superfluous number of terms has been eliminated.

207. Syllogisms with two singular premisses.—Bain (Logic, Deduction, p. 159) argues that an apparent syllogism with two singular premisses cannot be regarded as a genuine syllogistic or deductive inference; and he illustrates his view by reference to the following syllogism:

	Socrates fought at Delium,
	Socrates was the master of Plato,
therefore,	The master of Plato fought at Delium.

The argument is that “the proposition ‘Socrates was the master of Plato and fought at Delium,’ compounded out of the two premisses, is nothing more than a grammatical abbreviation,” whilst the step hence to the conclusion is a mere omission of something that had previously been said. “Now, we never 299 consider that we have made a real inference, a step in advance, when we repeat less than we are entitled to say, or drop from a complex statement some portion not desired at the moment. Such an operation keeps strictly within the domain of Equivalence or Immediate Inference. In no way, therefore, can a syllogism with two singular premisses be viewed as a genuine syllogistic or deductive inference.”

This argument leads up to some interesting considerations, but it proves too much. In the following syllogisms the premisses may be similarly compounded together:

All men are mortal,	⎱	All men are mortal and rational ;
All men are rational,	⎰
therefore, Some rational beings are mortal.
All men are mortal,	⎱	All men including kings are mortal ;
All kings are men,	⎰
therefore, All kings are mortal.[323]

[³²³] Compare with the above the following syllogism which has two singular premisses:—The Lord Chancellor receives a higher salary than the Prime Minister; Lord Herschell is the Lord Chancellor; therefore, Lord Herschell receives a higher salary than the Prime Minister. These premisses would presumably be compounded by Bain into the single proposition, “The Lord Chancellor, Lord Herschell, receives a higher salary than the Prime Minister.”

Do not Bain’s criticisms apply to these syllogisms as much as to the syllogism with two singular premisses? The method of treatment adopted is indeed particularly applicable to syllogisms in which the middle term is subject in both premisses. But we may always combine the two premisses of a syllogism in a single statement, and it is always true that the conclusion of a syllogism contains a part of, and only a part of, the information contained in the two premisses taken together; hence we may always get Bain’s result.[324] In other words, in the conclusion of every syllogism “we repeat less than we are entitled to say,” or, if we care to put it so, “drop from a complex statement some portion not desired at the moment.”

[³²⁴] It may be pointed out that the general method adopted by Boole in his Laws of Thought is to sum up all his given propositions in a single proposition, and then eliminate the terms that are not required. Compare also the methods employed in [Appendix C] of the present work.

300 208. Charge of incompleteness brought against the ordinary syllogistic conclusion.—This charge (a consideration of which will appropriately supplement the discussion contained in the preceding section) is brought by Jevons (Principles of Science, 4, § 8) against the ordinary syllogistic conclusion. The premisses Potassium floats on water, Potassium is a metal yield, according to him, the conclusion Potassium metal is potassium floating on water. But “Aristotle would have inferred that some metals float on water. Hence Aristotle’s conclusion simply leaves out some of the information afforded in the premisses ; it even leaves us open to interpret the some metals in a wider sense than we are warranted in doing.”

In reply to this it may be remarked: first, that the Aristotelian conclusion does not profess to sum up the whole of the information contained in the premisses of the syllogism; secondly, that some must here be interpreted to mean merely “not none,” “one at least.” The conclusion of the above syllogism might perhaps better be written “some metal floats on water,” or “some metal or metals &c.” Lotze remarks in criticism of Jevons: “His whole procedure is simply a repetition or at the outside an addition of his two premisses; thus it merely adheres to the given facts, and such a process has never been taken for a Syllogism, which always means a movement of thought that uses what is given for the purpose of advancing beyond it…… The meaning of the Syllogism, as Aristotle framed it, would in this case be that the occurrence of a floating metal Potassium proves that the property of being so light is not incompatible with the character of metal in general” (Logic, II. 3, note). This criticism is perhaps pushed a little too far. It is hardly a fair description of Jevons’s conclusion to say that it is the mere sum of the premisses; for it brings out a relation between two terms which was not immediately apparent in the premisses as they originally stood. Still there can be no doubt that the elimination of the middle term is the very gist of syllogistic reasoning as ordinarily understood.

It may be added, as an argumentum ad hominem against Jevons, that his own conclusion also leaves out some of the information afforded in the premisses. For we cannot pass 301 back from the proposition Potassium metal is potassium floating on water to either of the original premisses.

209. The connexion between the Dictum de omni et nullo and the ordinary Rules of the Syllogism.—The dictum de omni et nullo was given by Aristotle as the axiom on which all syllogistic inference is based. It applies directly, however, to those syllogisms only in which the major term is predicate in the major premiss, and the minor term subject in the minor premiss (i.e., to what are called syllogisms in figure 1). The rules of the syllogism, on the other hand, apply independently of the position of the terms in the premisses. Nevertheless, it is interesting to trace the connexion between them. It will be found that all the rules are involved in the dictum, but some of them in a less general form, in consequence of the distinction just pointed out.

The dictum may be stated as follows:—“Whatever is predicated, whether affirmatively or negatively, of a term distributed may be predicated in like manner of everything contained under it.”

(1) The dictum provides for three and only three terms; namely, (i) a certain term which must be distributed, (ii) something predicated of this term, (iii) something contained under it. These terms are respectively the middle, major, and minor. We may consider the rule relating to the ambiguity of terms to be also contained here, since if any term is ambiguous we have practically more than three terms.

(2) The dictum provides for three and only three propositions; namely, (i) a proposition predicating something of a term distributed, (ii) a proposition declaring something to be contained under this term, (iii) a proposition making the original predication of the contained term. These propositions constitute respectively the major premiss, the minor premiss, and the conclusion, of the syllogism.

(3) The dictum prescribes not merely that the middle term shall be distributed once at least in the premisses, but more definitely that it shall be distributed in the major premiss,—“Whatever is predicated of a term distributed.”[325]

[³²⁵] This is another form of what will be found to be a special rule of figure 1, namely, that the major premiss must be universal. Compare section [244].

302 (4) Illicit process of the major is provided against indirectly. This fallacy can be committed only when the conclusion is negative; but the words “in like manner” declare that if there is a negative conclusion, the major premiss must also be negative; and since in any syllogism to which the dictum directly applies, the major term is predicate of this premiss, it will be distributed in its premiss as well as in the conclusion. Illicit process of the minor is provided against inasmuch as the dictum warrants us in making our predication in the conclusion only of what has been shewn in the minor premiss to be contained under the middle term.

(5) The proposition declaring that something is contained under the term distributed must necessarily be an affirmative proposition. The dictum provides, therefore, that the premisses shall not both be negative.[326]

[³²⁶] It really provides that the minor premiss shall be affirmative, which again is one of the special rules of figure 1.

(6) The words “in like manner” clearly provide against a breach of the rule that if one premiss is negative, the conclusion must be negative, and vice versâ.

EXERCISES.[327]

[³²⁷] The following exercises may be solved without any knowledge beyond what is contained in the preceding chapter, the assumption however being made that if no rule of the syllogism as given in section [199] or section [201] is broken, then the syllogism is valid.

210. If P is a mark of the presence of Q, and R of that of S, and if P and R are never found together, am I right in inferring that Q and S sometimes exist separately? [V.]

The premisses may be stated as follows:

All P is Q,
All R is S,
No P is R ;

and in order to establish the desired conclusion we must be able to infer at least one of the following,—Some Q is not S, Some S is not Q.
But neither of these propositions can be inferred; for they distribute respectively S and Q, and neither of these terms is distributed in the given premisses. The question is, therefore, to be answered in the negative.

303 211. If it be known concerning a syllogism in the Aristotelian system that the middle term is distributed in both premisses, what can we infer as to the conclusion? [C.]

If both premisses are affirmative, they can between them distribute only two terms, and by hypothesis the middle term is distributed twice in the premisses; hence the minor term cannot be distributed in the premisses, and it follows that the conclusion must be particular.
If one of the premisses is negative, there may be three distributed terms in the premisses; these must, however, be the middle term twice (by hypothesis) and the major term (since the conclusion must now be negative and will therefore distribute the major term); hence the minor term cannot be distributed in the premisses, and it again follows that the conclusion must be particular.
But either both premisses will be affirmative, or one affirmative and the other negative; in any case, therefore, we can infer that the conclusion will be particular.

212. Shew directly in how many ways it is possible to prove the conclusions SaP, SeP ; point out those that conform immediately to the Dictum de omni et nullo ; and exhibit the equivalence between these and the remainder. [W.]

(1) To prove All S is P.
Both premisses must be affirmative, and both must be universal.
S being distributed in the conclusion must be distributed in the minor premiss, which must therefore be All S is M.
M not being distributed in the minor must be distributed in the major, which must therefore be All M is P.
SaP can therefore be proved in only one way, namely,

	All M is P,
	All S is M,
therefore,	All S is P ;

and this syllogism conforms immediately to the Dictum.
(2) To prove No S is P.
Both premisses must be universal, and one must be negative while the other is affirmative; i.e., one premiss must be E and the other A.
First, let the major be E, i.e., either No M is P or No P is M. In each case the minor must be affirmative and must distribute S ; therefore, it will be All S is M.
304 Secondly, let the minor be E, i.e., either No S is M or No M is S. In each case the major must be affirmative and must distribute P ; therefore, it will be All P is M.
We can then prove SeP in four ways, thus,—

(i)	MeP,	(ii)	PeM,	(iii)	PaM,	(iv)	PaM,
	SaM,		SaM,		SeM,		MeS,
	⎯⎯		⎯⎯		⎯⎯		⎯⎯
	SeP.		SeP.		SeP.		SeP.

Of these, (i) only conforms immediately to the dictum, and we have to shew the equivalence between it and the others.
The only difference between (i) and (ii) is that the major premiss of the one is the simple converse of the major premiss of the other; they are, therefore, equivalent. Similarly the only difference between (iii) and (iv) is that the minor premiss of the one is the simple converse of the minor premiss of the other; they are, therefore, equivalent.
Finally, we may shew that (iv) is equivalent to (i) by transposing the premisses and converting the conclusion.

213. Given that the major term is distributed in the premisses and undistributed in the conclusion of a valid syllogism, determine the syllogism. [C.]

Since the major term is undistributed in the conclusion, the conclusion—and, therefore, both premisses—must be affirmative. Hence, in order to distribute P, the major premiss must be PaM ; and in order to distribute M (which is not distributed in the major premiss), the minor premiss must be MaS. It follows that the syllogism must be

	All P is M,
	All M is S,
therefore,	Some S is P.

214. Prove that if three propositions involving three terms (each of which occurs in two of the propositions) are together incompatible, then (a) each term is distributed at least once, and (b) one and only one of the propositions is negative.
Shew that these rules are equivalent to the rules of the syllogism. [J.]

No two of the propositions can be formally incompatible with one another, since they do not contain the same terms. But each pair must be incompatible with the third, i.e., the contradictory of any one must be deducible from the other two. It follows that 305 we shall have three valid syllogisms, in which the given propositions taken in pairs are the premisses, whilst the contradictory of the third proposition is in each case the conclusion.[328]

Then (a) each term must be distributed once at least. For if any one of the terms failed to be distributed at least once, we should obviously have undistributed middle in one of our syllogisms; and (since a term undistributed in a proposition is distributed in its contradictory) illicit major or minor in the two others. If, however, the above condition is fulfilled, it is clear that we cannot have either undistributed middle, or illicit major or minor. Hence rule (a) is equivalent to the syllogistic rules relating to the distribution of terms.
Again, (b) one of the propositions must be negative, but not more than one of them can be negative. For if all three were affirmative, then (since the contradictory of an affirmative is negative) we should in each of our syllogisms infer a negative from two affirmatives; and if two were negative, we should have two negative premisses in one of our syllogisms, and (since the contradictory of a negative is affirmative) an affirmative conclusion with a negative premiss in each of the others. If, however, the above condition is fulfilled, it is clear that we cannot have either two negative premisses, or two affirmative premisses with a negative conclusion, or a negative premiss with an affirmative conclusion. Hence rule (b) is equivalent to the syllogistic rules relating to quality.

[³²⁸] Every syllogism involves two others, in each of which one of the original premisses combined with the contradictory of the conclusion proves the contradictory of the other original premiss. Hence the three syllogisms referred to in the text mutually involve one another. Compare sections [264], [265].

215. Explain what is meant by a syllogism ; and put the following argument into syllogistic form:—"We have no right to treat heat as a substance, for it may be transformed into something which is not heat, and is certainly not a substance at all, namely, mechanical work.” [N.]

216. Put the following argument into syllogistic form:—How can anyone maintain that pain is always an evil, who admits that remorse involves pain, and yet may sometimes be a real good? [V.]

306 217. It has been pointed out by Ohm that reasoning to the following effect occurs in some works on mathematics:—“A magnitude required for the solution of a problem must satisfy a particular equation, and as the magnitude x satisfies this equation, it is therefore the magnitude required.” Examine the logical validity of this argument. [C.]

218. Obtain a conclusion from the two negative premisses,—No P is M, No S is M. [K.]

219. If it is false that the attribute B is ever found coexisting with A, and not less false that the attribute C is sometimes found absent from A, can you assert anything about B in terms of C? [C.]

220. Give examples (in symbols—taking S, M, P, as minor, middle, and major terms, respectively) in which, attempting to infer a universal conclusion where we have a particular premiss, we commit respectively one but one only of the following fallacies,—(a) undistributed middle, (b) illicit major, (c) illicit minor. Give also an example in which, making the same attempt, we commit none of the above fallacies. [K.]

221. Can an apparent syllogism break directly all the rules of the syllogism at once? [K.]

222. Can you give an instance of an invalid syllogism in which the major premiss is universal negative, the minor premiss affirmative, and the conclusion particular negative? If not, why not? [K.]

223. Shew that
(i) If both premisses of a syllogism are affirmative, and one but only one of them universal, they will between them distribute only one term;
(ii) If both premisses are affirmative and both universal, they will between them distribute two terms;
(iii) If one but only one premiss is negative, and one but only one premiss universal, they will between them distribute two terms;
(iv) If one but only one premiss is negative, and both premisses are universal, they will between them distribute three terms. [K.]

224. Ascertain how many distributed terms there may be in the premisses of a syllogism more than in the conclusion. [L.]

225. If the minor premiss of a syllogism is negative, what do you know about the position of the terms in the major? [O’S.]

307 226. If the major term of a syllogism is the predicate of the major premiss, what do you know about the minor premiss? [L.]

227. How much can you tell about a valid syllogism if you know (1) that only the middle term is distributed;
(2) that only the middle and minor terms are distributed;
(3) that all three terms are distributed? [W.]

228. What can be determined respecting a valid syllogism under each of the following conditions: (1) that only one term is distributed, and that only once; (2) that only one term is distributed, and that twice; (3) that two terms only are distributed, each only once; (4) that two terms only are distributed, each twice? [L.]

229. Two propositions are given having a term in common. If they are I and A, shew that either no conclusion or two can be deduced; but if I and E, always and only one. [T.]

230. Find out, from the rules of the syllogism, what are the valid forms of syllogism in which the major premiss is particular affirmative. [J.]

231. Given (a) that the major premiss, (b) that the minor premiss, of a valid syllogism is particular negative, determine in each case the syllogism. [K.]

232. Given that the major premiss of a valid syllogism is affirmative, and that the major term is distributed both in premisses and conclusion, while the minor term is undistributed in both, determine the syllogism. [N.]

233. Shew directly in how many ways it is possible to prove the conclusions SiP, SoP. [W.]

234. Shew that if the rule that a negative conclusion requires a negative premiss be omitted from the general rules of the syllogism, the only invalid syllogism thereby admitted is such that, if its conclusion be false whilst its premisses are true, the three terms of the syllogism must be absolutely coextensive. [O’S.]

235. Find, by direct application of the fundamental rules of syllogism, what are the valid forms of syllogism in which neither of the premisses is a universal proposition having the same quality as the conclusion. [J.]

308 236. In what cases will contradictory major premisses both yield conclusions when combined with the same minor?
How are the conclusions related?
Shew that in no case will contradictory minor premisses both yield conclusions when combined with the same major. [O’S.]

237. (a) All just actions are praiseworthy; (b) No unjust actions are expedient; (c) Some inexpedient actions are not praiseworthy; (d) Not all praiseworthy actions are inexpedient.
Do (c) and (d) follow from (a) and (b)? [K.]

238. Reduce the following arguments to ordinary syllogistic form:
(i) No M is S, Whatever is not M is P, therefore, All S is P ;
(ii) It cannot be that no not-S is P, for some M is P and no M is S ;
(iii) It is impossible for the three propositions, All M is P, Anything that is not M is not S, Some things that are not P are S, all to be true together;
(iv) Everything is M or P, Nothing is both S and M, therefore, All S is P. [K.]

239. Shew that the following syllogisms break directly or indirectly all the rules of the syllogism:
(1) All P is M, All S is M, therefore, Some S is not P ;
(2) All M is P, All M is S, therefore, No S is P. [K.]

[The so-called rules that every syllogism contains three and only three terms, and that every syllogism consists of three and only three propositions, are not here included under the rules of the syllogism.]

240. In a circular argument involving two valid syllogisms, Q and U are used as premisses to prove R, while R and V are used as premisses to prove Q ; shew that U and V must be a pair of complementary propositions, i.e., of the forms All M is N and All N is M respectively. [J.]

241. Shew that if two valid syllogisms have a common premiss while the other premisses are contradictories, both the conclusions must be particular. [K.]

242. Given the premisses of a valid syllogism, examine in what cases it is (a) possible, (b) impossible, to determine which is the minor term and which the major term. [J.]