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Deductive Logic
by St. George Stock
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478. A kind of generality might indeed he imparted even to a singular proposition by expressing it in the form 'A is always B.' Thus we may say, 'This man is always idle'—a proposition which admits of being contradicted under the form 'This man is sometimes not idle.'



CHAPTER IV.

Of Conversion.

479. Conversion is an immediate inference grounded On the transposition of the subject and predicate of a proposition.

480. In this form of inference the antecedent is technically known as the Convertend, i.e. the proposition to be converted, and the consequent as the Converse, i.e. the proposition which has been converted.

481. In a loose sense of the term we may be said to have converted a proposition when we have merely transposed the subject and predicate, when, for instance, we turn the proposition 'All A is B' into 'All B is A' or 'Some A is not B' into 'Some B is not A.' But these propositions plainly do not follow from the former ones, and it is only with conversion as a form of inference—with Illative Conversion as it is called—that Logic is concerned.

482. For conversion as a form of inference two rules have been laid down—

(1) No term must be distributed in the converse which was not distributed in the convertend.

(2) The quality of the converse must be the same as that of the convertend.

483. The first of these rules is founded on the nature of things. A violation of it involves the fallacy of arguing from part of a term to the whole.

484. The second rule is merely a conventional one. We may make a valid inference in defiance of it: but such an inference will be seen presently to involve something more than mere conversion.

485. There are two kinds of conversion—

(1) Simple.

(2) Per Accidens or by Limitation.

486. We are said to have simply converted a proposition when the quantity remains the same as before.

487. We are said to have converted a proposition per accidens, or by limitation, when the rules for the distribution of terms necessitate a reduction in the original quantity of the proposition.

488.

A can only be converted per accidens.

E and I can be converted simply.

O cannot be converted at all.

489. The reason why A can only be converted per accidens is that, being affirmative, its predicate is undistributed ( 293). Since 'All A is B' does not mean more than 'All A is some B,' its proper converse is 'Some B is A.' For, if we endeavoured to elicit the inference, 'All B is A,' we should be distributing the term B in the converse, which was not distributed in the convertend. Hence we should be involved in the fallacy of arguing from the part to the whole. Because 'All doctors are men' it by no means follows that 'All men are doctors.'

499. E and I admit of simple conversion, because the quantity of the subject and predicate is alike in each, both subject and predicate being distributed in E and undistributed in I.

/ No A is B. E < .'. No B is A.

/ Some A is B. I < .'. Some B is A.

491. The reason why O cannot be converted at all is that its subject is undistributed and that the proposition is negative. Now, when the proposition is converted, what was the subject becomes the predicate, and, as the proposition must still be negative, the former subject would now be distributed, since every negative proposition distributes its predicate. Hence we should necessarily have a term distributed in the converse which was not distributed in the convertend. From 'Some men are not doctors,' it plainly does not follow that 'Some doctors are not men'; and, generally from 'Some A is not B' it cannot be inferred that 'Some B is not A,' since the proposition 'Some A is not B' admits of the interpretation that B is wholly contained in A.



492. It may often happen as a matter of fact that in some given matter a proposition of the form 'All B is A' is true simultaneously with 'All A is B.' Thus it is as true to say that 'All equiangular triangles are equilateral' as that 'All equilateral triangles are equiangular.' Nevertheless we are not logically warranted in inferring the one from the other. Each has to be established on its separate evidence.

493. On the theory of the quantified predicate the difference between simple conversion and conversion by limitation disappears. For the quantity of a proposition is then no longer determined solely by reference to the quantity of its subject. 'All A is some B' is of no greater quantity than 'Some B is all A,' if both subject and predicate have an equal claim to be considered.

494. Some propositions occur in ordinary language in which the quantity of the predicate is determined. This is especially the case when the subject is a singular term. Such propositions admit of conversion by a mere transposition of their subject and predicate, even though they fall under the form of the A proposition, e.g.

Virtue is the condition of happiness. .'. The condition of happiness is virtue.

And again,

Virtue is a condition of happiness. .'. A condition of happiness is virtue.

In the one case the quantity of the predicate is determined by the form of the expression as distributed, in the other as undistributed.

495. Conversion offers a good illustration of the principle on which we have before insisted, namely, that in the ordinary form of proposition the subject is used in extension and the predicate in intension. For when by conversion we change the predicate into the subject, we are often obliged to attach a noun substantive to the predicate, in order that it may be taken in extension, instead of, as before, in intension, e.g.

Some mothers are unkind. .'. Some unkind persons are mothers.

Again,

Virtue is conducive to happiness. .'. One of the things which are conducive to happiness is virtue.



CHAPTER V.

Of Permutation.

496. Permutation [Footnote: Called by some writers Obversion.] is an immediate inference grounded on a change of quality in a proposition and a change of the predicate into its contradictory-term.

497. In less technical language we may say that permutation is expressing negatively what was expressed affirmatively and vice vers.

498. Permutation is equally applicable to all the four forms of proposition.

(A) All A is B. .'. No A is not-B (E).

(E) No A is B. .'. All A is not-B (A).

(I) Some A is B. .'. Some A is not not-B (O).

(O) Some A is not B. .'. Some A is not-B (I).

499, Or, to take concrete examples—

(A) All men are fallible. .'. No men are not-fallible (E).

(E) No men are perfect. .'. All men are not-perfect (A).

(I) Some poets are logical. .'. Some poets are not not-logical (O).

(O) Some islands are not inhabited. .'. Some islands are not-inhabited (I).

500. The validity of permutation rests on the principle of excluded middle, namely—That one or other of a pair of contradictory terms must be applicable to a given subject, so that, when one may be predicated affirmatively, the other may be predicated negatively, and vice vers ( 31).

501. Merely to alter the quality of a proposition would of course affect its meaning; but when the predicate is at the same time changed into its contradictory term, the original meaning of the proposition is retained, whilst the form alone is altered. Hence we may lay down the following practical rule for permutation—

Change the quality of the proposition and change the predicate into its contradictory term.

502. The law of excluded middle holds only with regard to contradictories. It is not true of a pair of positive and privative terms, that one or other of them must be applicable to any given subject. For the subject may happen to fall wholly outside the sphere to which such a pair of terms is limited. But since the fact of a term being applied is a sufficient indication of its applicability, and since within a given sphere positive and privative terms are as mutually destructive as contradictories, we may in all cases substitute the privative for the negative term in immediate inference by permutation, which will bring the inferred proposition more into conformity with the ordinary usage of language. Thus the concrete instances given above will appear as follows—

(A) All men are fallible. .'. No men are infallible (E).

(E) No men are perfect. .'. All men are imperfect (A).

(I) Some poets are logical. .'. Some poets are not illogical (O).

(O) Some islands are not inhabited. .'. Some islands are uninhabited (I).



CHAPTER VI.

Of Compound Forms of Immediate Inference.

503. Having now treated of the three simple forms of immediate inference, we go on to speak of the compound forms, and first of



Conversion by Negation.

504. When A and O have been permuted, they become respectively E and I, and, in this form, admit of simple conversion. We have here two steps of inference: but the process may be performed at a single stroke, and is then known as Conversion by Negation. Thus from 'All A is B' we may infer 'No not-B is A,' and again from 'Some A is not B' we may infer 'Some not-B is A.' The nature of these inferences will be seen better in concrete examples.

505.

(A) All poets are imaginative. .'. No unimaginative persons are poets (E).

(O) Some parsons are not clerical. .'. Some unclerical persons are parsons (I).

506. The above inferences, when analysed, will be found to resolve themselves into two steps, namely,

(1) Permutation.

(2) Simple Conversion.

(A) All A is B. .'. No A is not-B (by permutation). .'. No not-B is A (by simple conversion).

(O) Some A is not B. .'. Some A is not-B (by permutation). .'. Some not-B is A (by simple conversion).

507. The term conversion by negation has been arbitrarily limited to the exact inferential procedure of permutation followed by simple conversion. Hence it necessarily applies only to A and 0 propositions, since these when permuted become E and 1, which admit of simple conversion; whereas E and 1 themselves are permuted into A and 0, which do not. There seems to be no good reason, however, why the term 'conversion by negation' should be thus restricted in its meaning; instead of being extended to the combination of permutation with conversion, no matter in what order the two processes may be performed. If this is not done, inferences quite as legitimate as those which pass under the title of conversion by negation are left without a name.

508. From E and 1 inferences may be elicited as follows—

(E) No A is B. .'. All B is not-A (A).

(I) Some A is B. .'. Some B is not not-A (O).

(E) No good actions are unbecoming. .'. All unbecoming actions are not-good (A).

(I) Some poetical persons are logicians. .'. Some logicians are not unpoetical (O).

Or, taking a privative term for our subject,

Some unpractical persons are statesmen. .'. Some statesmen are not practical.

509. When the inferences just given are analysed, it will be found that the process of simple conversion precedes that of permutation.

510. In the case of the E proposition a compound inference can be drawn even in the original order of the processes,

No A is B. .'. Some not-B is A.

No one who employs bribery is honest. .'. Some dishonest men employ bribery.

The inference here, it must be remembered, does not refer to matter of fact, but means that one of the possible forms of dishonesty among men is that of employing bribery.

511. If we analyse the preceding, we find that the second step is conversion by limitation.

No A is B. .'. All A is not-B (by permutation). .'. Some not-B is A (by conversion per accidens).

512. From A again an inference can be drawn in the reverse order of conversion per accidens followed by permutation—

All A is B. .'. Some B is not not-A.

All ingenuous persons are agreeable. .'. Some agreeable persons are not disingenuous.

513. The intermediate link between the above two propositions is the converse per accidens of the first—'Some B is A.' This inference, however, coincides with that from 1 ( 508), as the similar inference from E ( 510) coincides with that from 0 ( 506).

514. All these inferences agree in the essential feature of combining permutation with conversion, and should therefore be classed under a common name.

515. Adopting then this slight extension of the term, we define conversion by negation as—A form of conversion in which the converse differs in quality from the convertend, and has the contradictory of one of the original terms.

516. A still more complex form of immediate inference is known as



Conversion by Contraposition.

This mode of inference assumes the following form—

All A is B. .'. All not-B is not-A.

All human beings are fallible. .'. All infallible beings are not-human.

517. This will be found to resolve itself on analysis into three steps of inference in the following order—

(1) Permutation.

(2) Simple Conversion.

(3) Permutation.

518. Let us verify this statement by performing the three steps.

All A is B. .'. No A is not-B (by permutation). .'. No not-B is A (by simple conversion). .'. All not-B is not-A (by permutation).

All Englishmen are Aryans. .'. No Englishmen are non-Aryans. .'. No non-Aryans are Englishmen. .'. All non-Aryans are non-Englishmen.

519. Conversion by contraposition may be complicated in appearance by the occurrence of a negative term in the subject or predicate or both, e.g.

All not-A is B. .'. All not-B is A.

Again,

All A is not-B. .'. All B is not-A.

Lastly,

All not-A is not-B. .'. All B is A.

520. The following practical rule will be found of use for the right performing of the process—

Transpose the subject and predicate, and substitute for each its contradictory term.

521. As concrete illustrations of the above forms of inference we may take the following—

All the men on this board that are not white are red. .'. All the men On this board that are not red are white.

Again,

All compulsory labour is inefficient. .'. All efficient labour is free (=non-compulsory).

Lastly,

All inexpedient acts are unjust. .'. All just acts are expedient.

522. Conversion by contraposition may be said to rest on the following principle—

If one class be wholly contained in another, whatever is external to the containing class is external also to the class contained.



523. The same principle may be expressed intensively as follows:—

If an attribute belongs to the whole of a subject, whatever fails to exhibit that attribute does not come under the subject.

524. This statement contemplates conversion by contraposition only in reference to the A proposition, to which the process has hitherto been confined. Logicians seem to have overlooked the fact that conversion by contraposition is as applicable to the O as to the A proposition, though, when expressed in symbols, it presents a more clumsy appearance.

Some A is not B. .'. Some not-B is not not-A.

Some wholesome things are not pleasant. .'. Some unpleasant things are not unwholesome.

525. The above admits of analysis in exactly the same way as the same process when applied to the A proposition.

Some A is not B. .'. Some A is not-B (by permutation). .'. Some not-B is A (by simple conversion). .'. Some not-B is not not-A (by permutation).

The result, as in the case of the A proposition, is the converse by negation of the original proposition permuted.

526. Contraposition may also be applied to the E proposition by the use of conversion per accidens in the place of simple conversion. But, owing to the limitation of quantity thus effected, the result arrived at is the same as in the case of the O proposition. Thus from 'No wholesome things are pleasant' we could draw the same inference as before. Here is the process in symbols, when expanded.

No A is B. .'. All A is not-B (by permutation). .'. Some not-B is A (by conversion per accidens). .'. Some not-B is not not-A (by permutation).

527. In its unanalysed form conversion by contraposition may be defined generally as—A form of conversion in which both subject and predicate are replaced by their contradictories.

528. Conversion by contraposition differs in several respects from conversion by negation.

(1) In conversion by negation the converse differs in quality from the convertend: whereas in conversion by contraposition the quality of the two is the same.

(2) In conversion by negation we employ the contradictory either of the subject or predicate, but in conversion by contraposition we employ the contradictory of both.

(3) Conversion by negation involves only two steps of immediate inference: conversion by contraposition three.

529. Conversion by contraposition cannot be applied to the ordinary E proposition except by limitation ( 526).

From 'No A is B' we cannot infer 'No not-B is not-A.' For, if we could, the contradictory of the latter, namely, 'Some not-B is not-A' would be false. But it is manifest that this is not necessarily false. For when one term is excluded from another, there must be numerous individuals which fall under neither of them, unless it should so happen that one of the terms is the direct contradictory of the other, which is clearly not conveyed by the form of the expression 'No A is B. 'No A is not-A' stands alone among E propositions in admitting of full conversion by contraposition, and the form of that is the same after it as before.

530. Nor can conversion by contraposition be applied at all to I.



From 'Some A is B' we cannot infer that 'Some not-B is not-A.' For though the proposition holds true as a matter of fact, when A and B are in part mutually exclusive, yet this is not conveyed by the form of the expression. It may so happen that B is wholly contained under A, while A itself contains everything. In this case it will be true that 'No not-B is not-A,' which contradicts the attempted inference. Thus from the proposition 'Some things are substances' it cannot be inferred that 'Some not-substances are not-things,' for in this case the contradictory is true that 'No not-substances are not-things'; and unless an inference is valid in every case, it is not formally valid at all.

531. It should be noticed that in the case of the [nu] proposition immediate inferences are possible by mere contraposition without conversion.

All A is all B. .'. All not-A is not-B.

For example, if all the equilateral triangles are all the equiangular, we know at once that all non-equilateral triangles are also non-equiangular.

532. The principle upon which this last kind of inference rests is that when two terms are co-extensive, whatever is excluded from the one is excluded also from the other.



CHAPTER VII.

Of other Forms of Immediate Inference.

533. Having treated of the main forms of immediate inference, whether simple or compound, we will now close this subject with a brief allusion to some other forms which have been recognised by logicians.

534. Every statement of a relation may furnish us with ail immediate inference in which the same fact is presented from the opposite side. Thus from 'John hit James' we infer 'James was hit by John'; from 'Dick is the grandson of Tom' we infer 'Tom is the grandfather of Dick'; from 'Bicester is north-east of Oxford' we infer 'Oxford is south-west of Bicester'; from 'So and so visited the Academy the day after he arrived in London' we infer 'So and so arrived in London the day before he visited the Academy'; from 'A is greater than B' we infer 'B is less than A'; and so on without limit. Such inferences as these are material, not formal. No law can be laid down for them except the universal postulate, that

'Whatever is true in one form of words is true in every other form of words which conveys the same meaning.'

535. There is a sort of inference which goes under the title of Immediate Inference by Added Determinants, in which from some proposition already made another is inferred, in which the same attribute is attached both to the subject and the predicate, e.g.,

A horse is a quadruped. .'. A white horse is a white quadruped.

536. Such inferences are very deceptive. The attributes added must be definite qualities, like whiteness, and must in no way involve a comparison. From 'A horse is a quadruped' it may seem at first sight to follow that 'A swift horse is a swift quadruped.' But we need not go far to discover how little formal validity there is about such an inference. From 'A horse is a quadruped' it by no means follows that 'A slow horse is a slow quadruped'; for even a slow horse is swift compared with most quadrupeds. All that really follows here is that 'A slow horse is a quadruped which is slow for a horse.' Similarly, from 'A Bushman is a man' it does not follow that 'A tall Bushman is a tall man,' but only that 'A tall Bushman is a man who is tall for a Bushman'; and so on generally.

537. Very similar to the preceding is the process known as Immediate Inference by Complex Conception, e.g.

A horse is a quadruped. .'. The head of a horse is the head of a quadruped.

538. This inference, like that by added determinants, from which it differs in name rather than in nature, may be explained on the principle of Substitution. Starting from the identical proposition, 'The head of a quadruped is the head of a quadruped,' and being given that 'A horse is a quadruped,' so that whatever is true of 'quadruped' generally we know to be true of 'horse,' we are entitled to substitute the narrower for the wider term, and in this manner we arrive at the proposition,

The head of a horse is the head of a quadruped.

539. Such an inference is valid enough, if the same caution be observed as in the case of added determinants, that is, if no difference be allowed to intervene in the relation of the fresh conception to the generic and the specific terms.



CHAPTER VIII.

Of Mediate Inferences or Syllogisms.

540. A Mediate Inference, or Syllogism, consists of two propositions, which are called the Premisses, and a third proposition known as the Conclusion, which flows from the two conjointly.

541. In every syllogism two terms are compared with one another by means of a third, which is called the Middle Term. In the premisses each of the two terms is compared separately with the middle term; and in the conclusion they are compared with one another.

542. Hence every syllogism consists of three terms, one of which occurs twice in the premisses and does not appear at all in the conclusion. This term is called the Middle Term. The predicate of the conclusion is called the Major Term and its subject the Minor Term.

543. The major and minor terms are called the Extremes, as opposed to the Mean or Middle Term.

544. The premiss in which the major term is compared with the middle is called the Major Premiss.

545. The other premiss, in which the minor term is compared with the middle, is called the Minor Premiss.

546. The order in which the premisses occur in a syllogism is indifferent, but it is usual, for convenience, to place the major premiss first.

547. The following will serve as a typical instance of a syllogism—

Middle term Major term Major Premiss. All mammals are warm-blooded Antecedent > or Minor term Middle term Premisses Minor Premiss. All whales are mammals /

Minor term Major term Consequent or .'. All whales are warm-blooded > Conclusion.

548. The reason why the names 'major, 'middle' and 'minor' terms were originally employed is that in an affirmative syllogism such as the above, which was regarded as the perfect type of syllogism, these names express the relative quantity in extension of the three terms.



549. It must be noticed however that, though the middle term cannot be of larger extent than the major nor of smaller extent than the minor, if the latter be distributed, there is nothing to prevent all three, or any two of them, from being coextensive.

550. Further, when the minor term is undistributed, we either have a case of the intersection of two classes, from which it cannot be told which of them is the larger, or the minor term is actually larger than the middle, when it stands to it in the relation of genus to species, as in the following syllogism—

All Negroes have woolly hair. Some Africans are Negroes. .'. Some Africans have woolly hair.



551. Hence the names are not applied with strict accuracy even in the case of the affirmative syllogism; and when the syllogism is negative, they are not applicable at all: since in negative propositions we have no means of comparing the relative extension of the terms employed. Had we said in the major premiss of our typical syllogism, 'No mammals are cold-blooded,' and drawn the conclusion 'No whales are cold-blooded,' we could not have compared the relative extent of the terms 'mammal' and 'cold-blooded,' since one has been simply excluded from the other.



552. So far we have rather described than defined the syllogism. All the products of thought, it will be remembered, are the results of comparison. The syllogism, which is one of them, may be so regarded in two ways—

(1) As the comparison of two propositions by means of a third.

(2) As the comparison of two terms by means of a third or middle term.

553. The two propositions which are compared with one another are the major premiss and the conclusion, which are brought into connection by means of the minor premiss. Thus in the syllogism above given we compare the conclusion 'All whales are warm-blooded' with the major premiss 'All mammals are warm-blooded,' and find that the former is contained under the latter, as soon as we become acquainted with the intermediate proposition 'All whales are mammals.'

554. The two terms which are compared with one another are of course the major and minor.

555. The syllogism is merely a form into which our deductive inferences may be thrown for the sake of exhibiting their conclusiveness. It is not the form which they naturally assume in speech or writing. Practically the conclusion is generally stated first and the premisses introduced by some causative particle as 'because,' 'since,' 'for,' &c. We start with our conclusion, and then give the reason for it by supplying the premisses.

556. The conclusion, as thus stated first, was called by logicians the Problema or Quaestio, being regarded as a problem or question, to which a solution or answer was to be found by supplying the premisses.

557. In common discourse and writing the syllogism is usually stated defectively, one of the premisses or, in some cases, the conclusion itself being omitted. Thus instead of arguing at full length

All men are fallible, The Pope is a man, .'. The Pope is fallible,

we content ourselves with saying 'The Pope is fallible, for he is a man,' or 'The Pope is fallible, because all men are so'; or perhaps we should merely say 'All men are fallible, and the Pope is a man,' leaving it to the sagacity of our hearers to supply the desired conclusion. A syllogism, as thus elliptically stated, is commonly, though incorrectly, called an Enthymeme. When the major premiss is omitted, it is called an Enthymeme of the First Order; when the minor is omitted, an Enthymeme of the Second Order; and when the conclusion is omitted an Enthymeme of the Third Order.



CHAPTER IX.

Of Mood and Figure.

558. Syllogisms may differ in two ways—

(1) in Mood;

(2) in Figure.

559. Mood depends upon the kind of propositions employed. Thus a syllogism consisting of three universal affirmatives, AAA, would be said to differ in mood from one consisting of such propositions as EIO or any other combination that might be made. The syllogism previously given to prove the fallibility of the Pope belongs to the mood AAA. Had we drawn only a particular conclusion, 'Some Popes are fallible,' it would have fallen into the mood AAI.

560. Figure depends upon the arrangement of the terms in the propositions. Thus a difference of figure is internal to a difference of mood, that is to say, the same mood can be in any figure.

561. We will now show how many possible varieties there are of mood and figure, irrespective of their logical validity.

562. And first as to mood.

Since every syllogism consists of three propositions, and each of these propositions may be either A, E, I, or O, it is clear that there will be as many possible moods as there can be combinations of four things, taken three together, with no restrictions as to repetition. It will be seen that there are just sixty-four of such combinations. For A may be followed either by itself or by E, I, or O. Let us suppose it to be followed by itself. Then this pair of premisses, AA, may have for its conclusion either A, E, I, or O, thus giving four combinations which commence with AA. In like manner there will be four commencing with AE, four with AI, and four with AO, giving a total of sixteen combinations which commence with A. Similarly there will be sixteen commencing with E, sixteen with I, sixteen with O—in all sixty-four. It is very few, however, of these possible combinations that will be found legitimate, when tested by the rules of syllogism.

563. Next as to figure.

There are four possible varieties of figure in a syllogism, as may be seen by considering the positions that can be occupied by the middle term in the premisses. For as there are only two terms in each premiss, the position occupied by the middle term necessarily determines that of the others. It is clear that the middle term must either occupy the same position in both premisses or not, that is, it must either be subject in both or predicate in both, or else subject in one and predicate in the other. Now, if we are not acquainted with the conclusion of our syllogism, we do not know which is the major and which the minor term, and have therefore no means of distinguishing between one premiss and another; consequently we must Stop here, and say that there are only three different arrangements possible. But, if the Conclusion also be assumed as known, then we are able to distinguish one premiss as the major and the other as the minor; and so we can go further, and lay down that, if the middle term does not hold the same position in both premisses, it must either be subject in the major and predicate in the minor, or else predicate in the major and subject in the minor.

564. Hence there result

The Four Figures.

When the middle term is subject in the major and predicate in the minor, we are said to have the First Figure.

When the middle term is predicate in both premisses, we are said to have the Second Figure.

When the middle term is subject in both premisses, we are said to have the Third Figure.

When the middle term is predicate in the major premiss and subject in the minor, we are said to have the Fourth Figure.

565. Let A be the major term; B the middle. C the minor.

Figure I. Figure II. Figure III. Figure IV. B—A A—B B—A A—B C—B C—B B—C B—C C—A C—A C—A C—A

All these figures are legitimate, though the fourth is comparatively valueless.

566. It will be well to explain by an instance the meaning of the assertion previously made, that a difference of figure is internal to a difference of mood. We will take the mood EIO, and by varying the position of the terms, construct a syllogism in it in each of the four figures.

I. E No wicked man is happy. I Some prosperous men are wicked. O .'. Some prosperous men are not happy.

II. E No happy man is wicked. I Some prosperous men are wicked. O .'. Some prosperous men are not happy.

III. E No wicked man is happy. I Some wicked men are prosperous. O .'. Some prosperous men are not happy.

IV. E No happy man is wicked. I Some wicked men are prosperous. O .'. Some prosperous men are not happy.

567. In the mood we have selected, owing to the peculiar nature of the premisses, both of which admit of simple conversion, it happens that the resulting syllogisms are all valid. But in the great majority of moods no syllogism would be valid at all, and in many moods a syllogism would be valid in one figure and invalid in another. As yet however we are only concerned with the conceivable combinations, apart from the question of their legitimacy.

568. Now since there are four different figures and sixty-four different moods, we obtain in all 256 possible ways of arranging three terms in three propositions, that is, 256 possible forms of syllogism.



CHAPTER X.

Of the Canon of Reasoning.

& 569. The first figure was regarded by logicians as the only perfect type of syllogism, because the validity of moods in this figure may be tested directly by their complying, or failing to comply, with a certain axiom, the truth of which is self-evident. This axiom is known as the Dictum de Omni et Nullo. It may be expressed as follows—

Whatever may be affirmed or denied of a whole class may be affirmed or denied of everything contained in that class.

570. This mode of stating the axiom contemplates predication as being made in extension, whereas it is more naturally to be regarded as being made in intension.

571. The same principle may be expressed intensively as follows—

Whatever has certain attributes has also the attributes which invariably accompany them .[Footnote: Nota notae est nota rei ipsius. 'Whatever has any mark has that which it is a mark of.' Mill, vol. i, p. 201,]

572. By Aristotle himself the principle was expressed in a neutral form thus—

'Whatever is stated of the predicate will be stated also of the subject [Footnote: [Greek: osa kat to kategoroumnou lgetai pnta ka kat to hypokeimnou rhaetsetai]. Cat. 3, I].'

This way of putting it, however, is too loose.

573. The principle precisely stated is as follows—

Whatever may be affirmed or denied universally of the predicate of an affirmative proposition, may be affirmed or denied also of the subject.

574. Thus, given an affirmative proposition 'Whales are mammals,' if we can affirm anything universally of the predicate 'mammals,' as, for instance, that 'All mammals are warm-blooded,' we shall be able to affirm the same of the subject 'whales'; and, if we can deny anything universally of the predicate, as that 'No mammals are oviparous,' we shall be able to deny the same of the subject.

575. In whatever way the supposed canon of reasoning may be stated, it has the defect of applying only to a single figure, namely, the first. The characteristic of the reasoning in that figure is that some general rule is maintained to hold good in a particular case. The major premiss lays down some general principle, whether affirmative or negative; the minor premiss asserts that a particular case falls under this principle; and the conclusion applies the general principle to the particular case. But though all syllogistic reasoning may be tortured into conformity with this type, some of it finds expression more naturally in other ways.

576. Modern logicians therefore prefer to abandon the Dictum de Omni et Nullo in any shape, and to substitute for it the following three axioms, which apply to all figures alike.



Three Axioms of Mediale Inference.

(1) If two terms agree with the same third term, they agree with one another.

(2) If one term agrees, and another disagrees, with the same third term, they disagree with one another.

(3) If two terms disagree with the same third term, they may or may not agree with one another.

577. The first of these axioms is the principle of all affirmative, the second of all negative, syllogisms; the third points out the conditions under which no conclusion can be drawn. If there is any agreement at all between the two terms and the third, as in the cases contemplated in the first and second axioms, then we have a conclusion of some kind: if it is otherwise, we have none.

578. It must be understood with regard to these axioms that, when we speak of terms agreeing or disagreeing with the same third term, we mean that they agree or disagree with the same part of it.

579. Hence in applying these axioms it is necessary to bear in mind the rules for the distinction of terms. Thus from

All B is A, No C is B,

the only inference which can be drawn is that Some A is not C (which alters the figure from the first to the fourth). For it was only part of A which was known to agree with B. On the theory of the quantified predicate we could draw the inference No C is some A.

580. It is of course possible for terms to agree with different parts of the same third term, and yet to have no connection with one another. Thus

All birds fly. All bats fly.

But we do not infer therefrom that bats are birds or vice vers.

581. On the other hand, had we said,—

All birds lay eggs, No bats lay eggs,

we might confidently have drawn the conclusion

No bats are birds

For the term 'bats,' being excluded from the whole of the term 'lay eggs,' is thereby necessarily excluded from that part of it which coincides with 'birds.'



CHAPTER XI.

Of the Generad Rules of Syllogism.

582. We now proceed to lay down certain general rules to which all valid syllogisms must conform. These are divided into primary and derivative.

I. Primary.

(1) A syllogism must consist of three propositions only.

(2) A syllogism must consist of three terms only.

(3) The middle term must be distributed at least once in the premisses.

(4) No term must be distributed in the conclusion which was not distributed in the premisses.

(5) Two negative premisses prove nothing.

(6) If one premiss be negative, the conclusion must be negative.

(7) If the conclusion be negative, one of the premisses must be negative: but if the conclusion be affirmative, both premisses must be affirmative.

II. Derivative.

(8) Two particular premisses prove nothing.

(9) If one premiss be particular, the conclusion must be particular.

583. The first two of these rules are involved in the definition of the syllogism with which we started. We said it might be regarded either as the comparison of two propositions by means of a third or as the comparison of two terms by means of a third. To violate either of these rules therefore would be inconsistent with the fundamental conception of the syllogism. The first of our two definitions indeed ( 552) applies directly only to the syllogisms in the first figure; but since all syllogisms may be expressed, as we shall presently see, in the first figure, it applies indirectly to all. When any process of mediate inference appears to have more than two premisses, it will always be found that there is more than one syllogism. If there are less than three propositions, as in the fallacy of 'begging the question,' in which the conclusion simply reiterates one of the premisses, there is no syllogism at all.

With regard to the second rule, it is plain that any attempted syllogism which has more than three terms cannot conform to the conditions of any of the axioms of mediate inference.

584. The next two rules guard against the two fallacies which are fatal to most syllogisms whose constitution is unsound.

585. The violation of Rule 3 is known as the Fallacy of Undistributed Middle. The reason for this rule is not far to seek. For if the middle term is not used in either premiss in its whole extent, we may be referring to one part of it in one premiss and to quite another part of it in another, so that there will be really no middle term at all. From such premisses as these—

All pigs are omnivorous, All men are omnivorous,

it is plain that nothing follows. Or again, take these premisses—

Some men are fallible, All Popes are men.

Here it is possible that 'All Popes' may agree with precisely that part of the term 'man,' of which it is not known whether it agrees with 'fallible' or not.

586. The violation of Rule 4 is known as the Fallacy of Illicit Process. If the major term is distributed in the conclusion, not having been distributed in the premiss, we have what is called Illicit Process of the Major; if the same is the case with the minor term, we have Illicit Process of the Minor.

587. The reason for this rule is that if a term be used in its whole extent in the conclusion, which was not so used in the premiss in which it occurred, we would be arguing from the part to the whole. It is the same sort of fallacy which we found to underlie the simple conversion of an A proposition.

588. Take for instance the following—

All learned men go mad. John is not a learned man. .'. John will not go mad.

In the conclusion 'John' is excluded from the whole class of persons who go mad, whereas in the premisses, granting that all learned men go mad, it has not been said that they are all the men who do so. We have here an illicit process of the major term.

589. Or again take the following—

All Radicals are covetous. All Radicals are poor. .'. All poor men are covetous.

The conclusion here is certainly not warranted by our premisses. For in them we spoke only of some poor men, since the predicate of an affirmative proposition is undistributed.

590. Rule 5 is simply another way of stating the third axiom of mediate inference. To know that two terms disagree with the same third term gives us no ground for any inference as to whether they agree or disagree with one another, e.g.

Ruminants are not oviparous. Sheep are not oviparous.

For ought that can be inferred from the premisses, sheep may or may not be ruminants.

591. This rule may sometimes be violated in appearance, though not in reality. For instance, the following is perfectly legitimate reasoning.

No remedy for corruption is effectual that does not render it useless. Nothing but the ballot renders corruption useless. .'. Nothing but the ballot is an effectual remedy for corruption.

But on looking into this we find that there are four terms—

No not-A is B. No not-C is A. .'. No not-C is B.

The violation of Rule 5 is here rendered possible by the additional violation of Rule 2. In order to have the middle term the same in both premisses we are obliged to make the minor affirmative, thus

No not-A is B. All not-C is not-A. .'. No not-C is B.

No remedy that fails to render corruption useless is effectual. All but the ballot fails to render corruption useless. .'. Nothing but the ballot is effectual.

592. Rule 6 declares that, if one premiss be negative, the conclusion must be negative. Now in compliance with Rule 5, if one premiss be negative, the other must be affirmative. We have therefore the case contemplated in the second axiom, namely, of one term agreeing and the other disagreeing with the same third term; and we know that this can only give ground for a judgement of disagreement between the two terms themselves—in other words, to a negative conclusion.

593. Rule 7 declares that, if the conclusion be negative, one of the premisses must be negative; but, if the conclusion be affirmative, both premisses must be affirmative. It is plain from the axioms that a judgement of disagreement can only be elicited from a judgement of agreement combined with a judgement of disagreement, and that a judgement of agreement can result only from two prior judgements of agreement.

594. The seven rules already treated of are evident by their own light, being of the nature of definitions and axioms: but the two remaining rules, which deal with particular premisses, admit of being proved from their predecessors.

595. Proof of Rule 8.—That two particular premisses prove nothing.

We know by Rule 5 that both premisses cannot be negative. Hence they must be either both affirmative, II, or one affirmative and one negative, IO or OI.

Now II premisses do not distribute any term at all, and therefore the middle term cannot be distributed, which would violate Rule 3.

Again in IO or OI premisses there is only one term distributed, namely, the predicate of the O proposition. But Rule 3 requires that this one term should be the middle term. Therefore the major term must be undistributed in the major premiss. But since one of the premisses is negative, the conclusion must be negative, by Rule 6. And every negative proposition distributes its predicate. Therefore the major term must be distributed where it occurs as predicate of the conclusion. But it was not distributed in the major premiss. Therefore in drawing any conclusion we violate Rule 4 by an illicit process of the major term.

596. Proof of Rule 9.—That, if one premiss be particular, the conclusion must be particular.

Two negative premisses being excluded by Rule 5, and two particular by Rule 8, the only pairs of premisses we can have are—

AI, AO, EI.

Of course the particular premiss may precede the universal, but the order of the premisses will not affect the reasoning.

AI premisses between them distribute one term only. This must be the middle term by Rule 3. Therefore the conclusion must be particular, as its subject cannot be distributed,

AO and EI premisses each distribute two terms, one of which must be the middle term by Rule 3: so that there is only one term left which may be distributed in the conclusion. But the conclusion must be negative by Rule 4. Therefore its predicate must be distributed. Hence its subject cannot be so. Therefore the conclusion must be particular.

597. Rules 6 and 9 are often lumped together in a single expression—'The conclusion must follow the weaker part,' negative being considered weaker than affirmative, and particular than universal.

598. The most important rules of syllogism are summed up in the following mnemonic lines, which appear to have been perfected, though not invented, by a medival logician known as Petrus Hispanus, who was afterwards raised to the Papal Chair under the title of Pope John XXI, and who died in 1277—

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



CHAPTER XII.

Of the Determination of the Legitimate Moods of Syllogism.

599. It will be remembered that there were found to be 64 possible moods, each of which might occur in any of the four figures, giving us altogether 256 possible varieties of syllogism. The task now before us is to determine how many of these combinations of mood and figure are legitimate.

600. By the application of the preceding rules we are enabled to reduce the 64 possible moods to 11 valid ones. This may be done by a longer or a shorter method. The longer method, which is perhaps easier of comprehension, is to write down the 64 possible moods, and then strike out such as violate any of the rules of syllogism.

AAA -AEA- -AIA- -AOA- -AAE- AEE -AIE- -AOE- AAI -AEI- AII -AOI- -AAO- AEO -AIO- AOO

-EAA- -EEA- -EIA- -EOA- EAE -EEE- -EIE- -EOE- -EAI- -EEI- -EII- -EOI- EAO -EEO- EIO -EOO-



601. The batches which are crossed are those in which the premisses can yield no conclusion at all, owing to their violating Rule 6 or 9; in the rest the premises are legitimate, but a wrong conclusion is drawn from each of them as are translineated.

602. IEO stands alone, as violating Rule 4. This may require a little explanation.

Since the conclusion is negative, the major term, which is its predicate, must be distributed. But the major premiss, being 1, does not distribute either subject or predicate. Hence IEO must always involve an illicit process of the major.

603. The II moods which have been left valid, after being tested by the syllogistic rules, are as follows—

AAA. AAI. AEE. AEO. AII. AOO. EAE. EAO. EIO. IAI. OAO.

604. We will now arrive at the same result by a shorter and more scientific method. This method consists in first determining what pairs of premisses are valid in accordance with Rules 6 and g, and then examining what conclusions may be legitimately inferred from them in accordance with the other rules of syllogism.

605. The major premiss may be either A, E, I or O. If it is A, the minor also may be either A, E, I or O. If it is E, the minor can only be A or I. If it is I, the minor can only be A or E. If it is O, the minor can only be A. Hence there result 9 valid pairs of premisses.

AA. AE. AI. AO. EA. EI. IA. IE. OA.

Three of these pairs, namely AA, AE, EA, yield two conclusions apiece, one universal and one particular, which do not violate any of the rules of syllogism; one of them, IE, yields no conclusion at all; the remaining five have their conclusion limited to a single proposition, on the principle that the conclusion must follow the weaker part. Hence we arrive at the same result as before, of II legitimate moods—

AAA. AAI. AEE. AEO. EAE. EAO. AII. AOO. EIO. IAI. OAO.



CHAPTER XIII.

Of the Special Rules of the Four Figures.

606. Our next task must be to determine how far the 11 moods which we arrived at in the last chapter are valid in the four figures. But before this can be done, we must lay down the



Special Rules of the Four Figures.

FIGURE 1.

Rule 1, The minor premiss must be affirmative.

Rule 2. The major premiss must be universal.

FIGURE II.

Rule 1. One or other premiss must be negative.

Rule 2. The conclusion must be negative.

Rule 3. The major premiss must be universal.

FIGURE III.

Rule 1. The minor premiss must be affirmative.

Rule 2. The conclusion must be particular.

FIGURE IV.

Rule 1. When the major premiss is affirmative, the minor must be universal.

Rule 2. When the minor premiss is particular, the major must be negative.

Rule 3, When the minor premiss is affirmative, the conclusion must be particular.

Rule 4. When the conclusion is negative, the major premiss must be universal.

Rule 5. The conclusion cannot be a universal affirmative.

Rule 6. Neither of the premisses can be a particular negative.

607. The special rules of the first figure are merely a reassertion in another form of the Dictum de Omni et Nullo. For if the major premiss were particular, we should not have anything affirmed or denied of a whole class; and if the minor premiss were negative, we should not have anything declared to be contained in that class. Nevertheless these rules, like the rest, admit of being proved from the position of the terms in the figure, combined with the rules for the distribution of terms ( 293).



Proof of the Special Rules of the Four Figures.

FIGURE 1.

608. Proof of Rule 1.—The minor premiss must be affirmative.

B—A C—B C—A

If possible, let the minor premiss be negative. Then the major must be affirmative (by Rule 5), [Footnote: This refers to the General Rules of Syllogism.] and the conclusion must be negative (by Rule 6). But the major being affirmative, its predicate is undistributed; and the conclusion being negative, its predicate is distributed. Now the major term is in this figure predicate both in the major premiss and in the conclusion. Hence there results illicit process of the major term. Therefore the minor premiss must be affirmative.

609. Proof of Rule 2.—The major premiss must be universal.

Since the minor premiss is affirmative, the middle term, which is its predicate, is undistributed there. Therefore it must be distributed in the major premiss, where it is subject. Therefore the major premiss must be universal.

FIGURE II.

610. Proof of Rule 1,—One or other premiss must be negative.

A—B C—B C—A

The middle term being predicate in both premisses, one or other must be negative; else there would be undistributed middle.

611. Proof of Rule 2.—The conclusion must be negative.

Since one of the premisses is negative, it follows that the conclusion also must be so (by Rule 6).

612. Proof of Rule 3.—The major premiss must be universal.

The conclusion being negative, the major term will there be distributed. But the major term is subject in the major premiss. Therefore the major premiss must be universal (by Rule 4).

FIGURE III.

613. Proof of Rule 1.—The minor premiss must be affirmative.

B—A B—C C—A

The proof of this rule is the same as in the first figure, the two figures being alike so far as the major term is concerned.

614. Proof of Rule 2.—The conclusion must be particular.

The minor premiss being affirmative, the minor term, which is its predicate, will be undistributed there. Hence it must be undistributed in the conclusion (by Rule 4). Therefore the conclusion must be particular.

FIGURE IV.

615. Proof of Rule I.—When the major premiss is affirmative, the minor must be universal.

If the minor were particular, there would be undistributed middle. [Footnote: Shorter proofs are employed in this figure, as the student is by this time familiar with the method of procedure.]

616. Proof of Rule 2.—When the minor premiss is particular, the major must be negative.

A—B B—C C—A

This rule is the converse of the preceding, and depends upon the same principle.

617. Proof of Rule 3.—When the minor premiss is affirmative, the conclusion must be particular.

If the conclusion were universal, there would be illicit process of the minor.

618. Proof of Rule 4.—When the conclusion is negative, the major premiss must be universal.

If the major premiss were particular, there would be illicit process of the major.

619. Proof of Rule 5.—The conclusion CANNOT be A UNIVERSAL affirmative.

The conclusion being affirmative, the premisses must be so too (by Rule 7). Therefore the minor term is undistributed in the minor premiss, where it is predicate. Hence it cannot be distributed in the conclusion (by Rule 4). Therefore the affirmative conclusion must be particular.

620. Proof of Rule 6.—Neither of the premisses can lie a, PARTICULAR NEGATIVE.

If the major premiss were a particular negative, the conclusion would be negative. Therefore the major term would be distributed in the conclusion. But the major premiss being particular, the major term could not be distributed there. Therefore we should have an illicit process of the major term.

If the minor premiss were a particular negative, then, since the major must be affirmative (by Rule 5), we should have undistributed middle.



CHAPTER XIV

Of the Determination of the Moods that are valid in the Four Figures.

621. By applying the special rules just given we shall be able to determine how many of the eleven legitimate moods are valid in the four figures.

$622. These eleven legitimate moods were found to be

AAA. AAI. AEE. AEO. AII. AOO. EAE. EAO. EIO. IAI. OAO.

FIGURE 1.

623. The rule that the major premiss must be universal excludes the last two moods, IAI, OAO. The rule that the minor premiss must be affirmative excludes three more, namely, AEE, AEO, AOO.

Thus we are left with six moods which are valid in the first figure, namely,

AAA. EAE. AII. EIO. AAI. EAO.

FIGURE II.

624. The rule that one premiss must be negative excludes four moods, namely, AAA, AAI, AII, IAI. The rule that the major must be universal excludes OAO. Thus we are left with six moods which are valid in the second figure, namely,

EAE. AEE. EIO. AOO. EAO. AEO.

FIGURE III.

625. The rule that the conclusion must be particular confines us to eight moods, two of which, namely AEE and AOO, are excluded by the rule that the minor premiss must be affirmative.

Thus we are left with six moods which are valid in the third figure, namely,

AAI. IAI. AII. EAO. OAO. EIO.

FIGURE IV.

626. The first of the eleven moods, AAA, is excluded by the rule that the conclusion cannot be a universal affirmative.

Two more moods, namely AOO and OAO, are excluded by the rule that neither of the premisses can be a particular negative.

AII violates the rule that when the major premiss is affirmative, the minor must be universal.

EAE violates the rule that, when the minor premiss is affirmative, the conclusion must be particular. Thus we are left with six moods which are valid in the fourth figure, namely,

AAI. AEE. IAI. EAO. EIO. AEO.

627. Thus the 256 possible forms of syllogism have been reduced to two dozen legitimate combinations of mood and figure, six moods being valid in each of the four figures.

FIGURE I. AAA. EAE. AII. EIO. (AAI. EAO.)

FIGURE II. EAE. AEE. EIO. AGO. (EAO. AEO.)

FIGURE III. AAI. IAI. AII. EAO. OAO. EIO.

FIGURE IV. AAI. AEE. IAI. EAO. EIO. (AEO.)

628. The five moods enclosed in brackets, though valid, are useless. For the conclusion drawn is less than is warranted by the premisses. These are called Subaltern Moods, because their conclusions might be inferred by subalternation from the universal conclusions which can justly be drawn from the same premisses. Thus AAI is subaltern to AAA, EAO to EAE, and so on with the rest.

629. The remaining 19 combinations of mood and figure, which are loosely called 'moods,' though in strictness they should be called 'figured moods,' are generally spoken of under the names supplied by the following mnemonics—

Barbara, Celarent, Darii, Ferioque prioris; Cesare, Camestres, Festino, Baroko secund; Tertia Darapti, Disamis, Datisi, Felapton, Bokardo, Ferison habet; Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison: Quinque Subalterni, totidem Generalibus orti, Nomen habent nullum, nee, si bene colligis, usum.

630. The vowels in these lines indicate the letters of the mood. All the special rules of the four figures can be gathered from an inspection of them. The following points should be specially noted.

The first figure proves any kind of conclusion, and is the only one which can prove A.

The second figure proves only negatives.

The third figure proves only particulars.

The fourth figure proves any conclusion except A.

631. The first figure is called the Perfect, and the rest the Imperfect figures. The claim of the first to be regarded as the perfect figure may be rested on these grounds—

1. It alone conforms directly to the Dictum de Omni et Nullo.

2. It suffices to prove every kind of conclusion, and is the only figure in which a universal affirmative proposition can be established.

3. It is only in a mood of this figure that the major, middle and minor terms are to be found standing in their relative order of extension.

632. The reason why a universal affirmative, which is of course infinitely the most important form of proposition, can only be proved in the first figure may be seen as follows.

Proof that A can only be established in figure I.

An A conclusion necessitates both premisses being A propositions (by Rule 7). But the minor term is distributed in the conclusion, as being the subject of an A proposition, and must therefore be distributed in the minor premiss, in order to which it must be the subject. Therefore the middle term must be the predicate and is consequently undistributed. In order therefore that the middle term may be distributed, it must be subject in the major premiss, since that also is an A proposition. But when the middle term is subject in the major and predicate in the minor premiss, we have what is called the first figure.



CHAPTER XV.

Of the Special Canons of the Four Figures.

633. So far we have given only a negative test of legitimacy, having shown what moods are not invalidated by running counter to any of the special rules of the four figures. We will now lay down special canons for the four figures, conformity to which will serve as a positive test of the validity of a given mood in a given figure. The special canon of the first figure—will of course be practically equivalent to the Dictum de Omni et Nullo. All of them will be expressed in terms of extension, for the sake of perspicuity.



Special Canons of the Four Figures.

FIGURE 1.

634. CANON. If one term wholly includes or excludes another, which wholly or partly includes a third, the first term wholly or partly includes or excludes the third.

Here four cases arise—



(1) Total inclusion (Barbara).

All B is A. All C is B. .'. All C is A.



(2) Partial inclusion (Darii).

All B is A. Some C is B. .'. Some C is A.



(3) Total exclusion (Celarent).

No B is A. All C is B. .'. No C is A.



(4) Partial exclusion (Ferio).

No B is A. Some C is B. .'. Some C is not A.

FIGURE II.

635. CANON. If one term is excluded from another, which wholly or partly includes a third, or is included in another from which a third is wholly or partly excluded, the first is excluded from the whole or part of the third.

Here we have four cases, all of exclusion—

(1) Total exclusion on the ground of inclusion in an excluded term (Cesare).



No A is B. All C is B. .'. No C is A.

(2) Partial exclusion on the ground of a similar partial inclusion (Festino).



No A is B. Some C is B. .'. Some C is not A.

(3) Total exclusion on the ground of exclusion from an including term (Camestres).



All A is B. No C is B. .'. No C is A.

(4) Partial exclusion on the ground of a similar partial exclusion (Baroko).



All A is B. Some C is not B. .'. Some C is not A.

FIGURE III.

636. CANON. If two terms include another term in common, or if the first includes the whole and the second a part of the same term, or vice vers, the first of these two terms partly includes the second; and if the first is excluded from the whole of a term which is wholly or in part included in the second, or is excluded from part of a term which is wholly included in the second, the first is excluded from part of the second.

Here it is evident from the statement that six cases arise—

(1) Total inclusion of the same term in two others (Darapti).



All B is A. All B is C. .'. some C is A.

(2) Total inclusion in the first and partial inclusion in the second (Datisi).



All B is A. Some B is C. .'. some C is A.

(3) Partial inclusion in the first and total inclusion in the second (Disamis).



Some B is A. All B is C. .'. some C is A.

(4) Total exclusion of the first from a term which is wholly included in the second (Felapton).



No B is A. All B is C. .'. some C is not A.

(5) Total exclusion of the first from a term which is partly included in the second (Ferison).



No B is A. Some B is C. .'. some C is not A.

(6) Exclusion of the first from part of a term which is wholly included in the second (Bokardo).



Some B is not A. All B is C. .'. Some C is not A.

FIGURE IV.

637. CANON. If one term is wholly or partly included in another which is wholly included in or excluded from a third, the third term wholly or partly includes the first, or, in the case of total inclusion, is wholly excluded from it; and if a term is excluded from another which is wholly or partly included in a third, the third is partly excluded from the first.

Here we have five cases—

(1) Of the inclusion of a whole term (Bramsntip).

All A is B. All B is C. .'. Some C is (all) A.

(2) Of the inclusion of part of a term (DIMARIS).



Some A is B. All B is C. .'. Some C is (some) A,

(3) Of the exclusion of a whole term (Camenes).



All A is B. No B is C. .'. No C is A.

(4) Partial exclusion on the ground of including the whole of an excluded term (Fesapo).



No A is B. All B is C. .'. Some C is not A.

(5) Partial exclusion on the ground of including part of an excluded term (Fresison).



No A is B. Some B is C. .'. Some C is not A.

638. It is evident from the diagrams that in the subaltern moods the conclusion is not drawn directly from the premisses, but is an immediate inference from the natural conclusion. Take for instance AAI in the first figure. The natural conclusion from these premisses is that the minor term C is wholly contained in the major term A. But instead of drawing this conclusion we go on to infer that something which is contained in C, namely some C, is contained in A.



All B is A. All C is B. .'. all C is A. .'. some C is A.

Similarly in EAO in figure 1, instead of arguing that the whole of C is excluded from A, we draw a conclusion which really involves a further inference, namely that part of C is excluded from A.



No B is A. All C is B. .'. no C is A. .'. some C is not A.

639. The reason why the canons have been expressed in so cumbrous a form is to render the validity of all the moods in each figure at once apparent from the statement. For purposes of general convenience they admit of a much more compendious mode of expression.

640. The canon of the first figure is known as the Dictum de Omni et Nullo—

What is true (distributively) of a whole term is true of all that it includes.

641. The canon of the second figure is known as the Dictum de Diverse—

If one term is contained in, and another excluded from a third term, they are mutually excluded.

642. The canon of the third figure is known as the Dictum de Exemplo et de Excepto—

Two terms which contain a common part partly agree, or, if one contains a part which the other does not, they partly differ.

643. The canon of the fourth figure has had no name assigned to it, and does not seem to admit of any simple expression. Another mode of formulating it is as follows:—

Whatever is affirmed of a whole term may have partially affirmed of it whatever is included in that term (Bramantip, Dimaris), and partially denied of it whatever is excluded (Fesapo); whatever is affirmed of part of a term may have partially denied of it whatever is wholly excluded from that term (Fresison); and whatever is denied of a whole term may have wholly denied of it whatever is wholly included in that term (Camenes).

644. From the point of view of intension the canons of the first three figures may be expressed as follows.

645. Canon of the first figure. Dictum de Omni et Nullo—

An attribute of an attribute of anything is an attribute of the thing itself.

646. Canon of the second figure. Dictum de Diverso—

If a subject has an attribute which a class has not, or vice versa, the subject does not belong to the class.

647. Canon of the third figure.

1. Dictum de Exemplo—

If a certain attribute can be affirmed of any portion of the members of a class, it is not incompatible with the distinctive attributes of that class.

2. Dictum de Excepto—

If a certain attribute can be denied of any portion of the members of a class, it is not inseparable from the distinctive attributes of that class.



CHAPTER XVI.

Of the Special Uses of the Four Figures.

648. The first figure is useful for proving the properties of a thing.

649. The second figure is useful for proving distinctions between things.

650. The third figure is useful for proving instances or exceptions.

651. The fourth figure is useful for proving the species of a genus.

FIGURE 1.

652.

B is or is not A. C is B. .'. C is or is not A.

We prove that C has or has not the property A by predicating of it B, which we know to possess or not to possess that property.

Luminous objects are material. Comets are luminous. .'. Comets are material.

No moths are butterflies. The Death's head is a moth. .'. The Death's head is not a butterfly.

FIGURE II.

653.

A is B. A is not B. C is not B. C is B. .'. C is not A. .'. C is not A.

We establish the distinction between C and A by showing that A has an attribute which C is devoid of, or is devoid of an attribute which C has.

All fishes are cold-blooded. A whale is not cold-blooded. .'. A whale is not a fish.

No fishes give milk. A whale gives milk. .'. A whale is not a fish.

FIGURE III.

654.

B is A. B is not A. B is C. B is C. .'. Some C is A. .'. Some C is not A.

We produce instances of C being A by showing that C and A meet, at all events partially, in B. Thus if we wish to produce an instance of the compatibility of great learning with original powers of thought, we might say

Sir William Hamilton was an original thinker. Sir William Hamilton was a man of great learning. .'. Some men of great learning are original thinkers.

Or we might urge an exception to the supposed rule about Scotchmen being deficient in humour under the same figure, thus—

Sir Walter Scott was not deficient in humour. Sir Walter Scott was a Scotchman. .'. Some Scotchmen are not deficient in humour.

FIGURE IV.

655.

All A is B, No A is B. All B is C. All B is C. .'. Some C is A .'.Some C is not A.

We show here that A is or is not a species of C by showing that A falls, or does not fall, under the class B, which itself falls under C. Thus—

All whales are mammals. All mammals are warm-blooded. .'. Some warm-blooded animals are whales. No whales are fishes. All fishes are cold-blooded. .'. Some cold-blooded animals are not whales.



CHAPTER XVII.

Of the Syllogism with three figures.

656. It will be remembered that in beginning to treat of figure ( 565) we pointed out that there were either four or three ligures possible according as the conclusion was assumed to be known or not. For, if the conclusion be not known, we cannot distinguish between the major and the minor term, nor, consequently, between one premiss and another. On this view the first and the fourth figures are the same, being that arrangement of the syllogism in which the middle term occupies a different position in one premiss from what it does in the other. We will now proceed to constitute the legitimate moods and figures of the syllogism irrespective of the conclusion.

657. When the conclusion is set out of sight, the number of possible moods is the same as the number of combinations that can be made of the four things, A, E, I, O, taken two together, without restriction as to repetition. These are the following 16:—

AA EA IA OA AE -EE- IE -OE- AI EI -II- -OI- AO -EO- -IO- -OO-

of which seven may be neglected as violating the general rules of the syllogism, thus leaving us with nine valid moods—

AA. AE. AI. AO. EA. EI. IA. IE. OA.

658. We will now put these nine moods successively into the three figures. By so doing it will become apparent how far they are valid in each.

659. Let it be premised that

when the extreme in the premiss that stands first is predicate in the conclusion, we are said to have a Direct Mood;

when the extreme in the premiss that stands second is predicate in the conclusion, we are said to have an Indirect Mood.

660. FIGURE 1.

Mood AA. All B is A. All C is B. .'. All C is A, or Some A is C, (Barbara & Bramantip).

Mood AE. All B is A. No C is B. .'. Illicit Process, or Some A is not C, (Fesapo).

Mood AI. All B is A. Some C is B. .'. Some C is A, or Some A is C. (Darii & Disamis).

Mood AO. All B is A. Some C is not B. .'. Illicit Process, (Ferio).

Mood EA. No B is A. All C is B. .'. No C is A, or No A is C, (Celarent & Camenes).

Mood EI. No B is A. Some C is B. .'. Some C is not A, or Illicit Process.

Mood IA. Some B is A. All C is B. .'. Undistributed Middle.

Mood IE. Some B is C. Some B is not A. No A is B. All C is B. .'. Illicit Process, or Some C is not A, (Fresison).

Mood OA. Some B is not A. All C is B. .'. Undistributed Middle.

661. Thus we are left with six valid moods, which yield four direct conclusions and five indirect ones, corresponding to the four moods of the original first figure and the five moods of the original fourth, which appear now as indirect moods of the first figure.

662. But why, it maybe asked, should not the moods of the first figure equally well be regarded as indirect moods of the fourth? For this reason-that all the moods of the fourth figure can be elicited out of premisses in which the terms stand in the order of the first, whereas the converse is not the case. If, while retaining the quantity and quality of the above premisses, i. e. the mood, we were in each case to transpose the terms, we should find that we were left with five valid moods instead of six, since AI in the reverse order of the terms involves undistributed middle; and, though we should have Celarent indirect to Camenes, and Darii to Dimaris, we should never arrive at the conclusion of Barbara or have anything exactly equivalent to Ferio. In place of Barbara, Bramantip would yield as an indirect mood only the subaltern AAI in the first figure. Both Fesapo and Fresison would result in an illicit process, if we attempted to extract the conclusion of Ferio from them as an indirect mood. The nearest approach we could make to Ferio would be the mood EAO in the first figure, which may be elicited indirectly from the premisses of CAMENES, being subaltern to CELARENT. For these reasons the moods of the fourth figure are rightly to be regarded as indirect moods of the first, and not vice vers.

$663. FIGURE II.

Mood AA. All A is B. All C is B. .'. Undistributed Middle.

Mood AE. All A is B. No C is B. .'. No C is A, or No A is C, (Camestres & Cesare).

Mood AI. All A is B. Some C is B. .'. Undistributed Middle.

Mood AO. All A is B. Some C is not B. .'. Some C is not A, (Baroko), or Illicit Process.

Mood EA. No A is B. All C is B. .'. No C is A, or No A is C, (Cesare & Carnestres).

Mood EI No A is B. Some C is B. .'. Some C is not A, (Festino), or Illicit Process.

Mood IA. Some A is B. All C is B. .'. Undistributed Middle.

Mood IE. Some A is B. No C is B. .'. Illicit Process, or Some A is not C, (Festino).

Mood OA. Some A is not B. All C is B. .'. Illicit Process, or Some A is not C, (Baroko).

664. Here again we have six valid moods, which yield four direct conclusions corresponding to Cesare, CARNESTRES, FESTINO and BAROKO. The same four are repeated in the indirect moods.

665. FIGURE III.

Mood AA. All B is A. All B is C. .'. Some C is A, or Some A is C, (Darapti).

Mood AE. All B is A. No B is C. .'. Illicit Process, or Some A is not C, (Felapton).

Mood AI. All B is A, Some B is C. .'. Some C is A, or Some A is C, (Datisi & Disamis).

Mood AO. All B is A. Some B is not C. .'. Illicit Process, Or Some A is not C, (Bokardo).

Mood EA. No B is A. All B is C. .'. Some C is not A, (Felapton), or Illicit Process.

Mood EI. No B is A. Some B is C. .'. Some C is not A, (Ferison), or Illicit Process.

Mood IA. Some B is A. All B is C. .'. Some C is A, Or Some A is C, (Disamis & Datisi).

Mood IE. Some B is A. No B is C. .'. Illicit Process, or Some A is not C, (Ferison).

Mood QA. Some B is not A. All B is C. .'. Some C is not A, (Bokardo), or Illicit Process.

666. In this figure every mood is valid, either directly or indirectly. We have six direct moods, answering to Darapti, Disamis, Datisi, Felapton, Bokardo and Ferison, which are simply repeated by the indirect moods, except in the case of Darapti, which yields a conclusion not provided for in the mnemonic lines. Darapti, though going under one name, has as much right to be considered two moods as Disamis and Datisi.



CHAPTER XVIII.

Of Reduction.

667. We revert now to the standpoint of the old logicians, who regarded the Dictum de Omni et Nullo as the principle of all syllogistic reasoning. From this point of view the essence of mediate inference consists in showing that a special case, or class of cases, comes under a general rule. But a great deal of our ordinary reasoning does not conform to this type. It was therefore judged necessary to show that it might by a little manipulation be brought into conformity with it. This process is called Reduction.

668. Reduction is of two kinds—

(1) Direct or Ostensive.

(2) Indirect or Ad Impossibile.

669. The problem of direct, or ostensive, reduction is this—

Given any mood in one of the imperfect figures (II, III and IV) how to alter the form of the premisses so as to arrive at the same conclusion in the perfect figure, or at one from which it can be immediately inferred. The alteration of the premisses is effected by means of immediate inference and, where necessary, of transposition.

670. The problem of indirect reduction, or reductio (per deductionem) ad impossibile, is this—Given any mood in one of the imperfect figures, to show by means of a syllogism in the perfect figure that its conclusion cannot be false.

671. The object of reduction is to extend the sanction of the Dictum de Omni et Nullo to the imperfect figures, which do not obviously conform to it.

672. The mood required to be reduced is called the Reducend; that to which it conforms, when reduced, is called the Reduct.



Direct or Ostensive Reduction.

673. In the ordinary form of direct reduction, the only kind of immediate inference employed is conversion, either simple or by limitation; but the aid of permutation and of conversion by negation and by contraposition may also be resorted to.

674. There are two moods, Baroko and Bokardo, which cannot be reduced ostensively except by the employment of some of the means last mentioned. Accordingly, before the introduction of permutation into the scheme of logic, it was necessary to have recourse to some other expedient, in order to demonstrate the validity of these two moods. Indirect reduction was therefore devised with a special view to the requirements of Baroko and Bokardo: but the method, as will be seen, is equally applicable to all the moods of the imperfect figures.

675. The mnemonic lines, 'Barbara, Celarent, etc., provide complete directions for the ostensive reduction of all the moods of the second, third, and fourth figures to the first, with the exception of Baroko and Bokardo. The application of them is a mere mechanical trick, which will best be learned by seeing the process performed.

676. Let it be understood that the initial consonant of each name of a figured mood indicates that the reduct will be that mood which begins with the same letter. Thus the B of Bramantip indicates that Bramantip, when reduced, will become Barbara.

677. Where m appears in the name of a reducend, me shall have to take as major that premiss which before was minor, and vice versa-in other words, to transpose the premisses, m stands for mutatio or metathesis.

678. s, when it follows one of the premisses of a reducend, indicates that the premiss in question must be simply converted; when it follows the conclusion, as in Disamis, it indicates that the conclusion arrived at in the first figure is not identical in form with the original conclusion, but capable of being inferred from it by simple conversion. Hence s in the middle of a name indicates something to be done to the original premiss, while s at the end indicates something to be done to the new conclusion.

679. P indicates conversion per accidens, and what has just been said of s applies, mutatis mutandis, to p.

680. k may be taken for the present to indicate that Baroko and Bokardo cannot be reduced ostensively.

681. FIGURE II.

Cesare. / Celarent. No A is B. = / No B is A. All C is B. / All C is B. .'. No C is A. / .'. No C is A.

Camestres. / Celarent. All A is B. = / No B is C. No C is B. / All A is B. .'. No C is A. / .'. No A is C. .'. No C is A.

Festino. Ferio. No A is B. / No B is A. Some C is B. = Some C is B. .'. Some C is not A./ .'. Some C is not A. [Baroko]

682. FIGURE III.

Darapti. / Darii. All B is A. = / All B is A. All B is C. / Some C is B. .'. Some C is A. / Some C is A.

Disamis. / Darii. Some B is A. = / All B is C. All B is C. / Some A is B. .'. Some C is A. / .'. Some A is C. .'. Some C is A.

Datisi. / Darii. All B is A. = / All B is A. Some B is C. / Some C is B. .'. Some C is A. / .'. Some C is A.

Felapton. / Ferio. No B is A. = / No B is A. All B is C. / Some C is B. .'. Some C is not-A. / .'. Some C is not-A.

[Bokardo].

Ferison. / Ferio. No B is A. = / No B is A. Some B is C. / Some C is B .'. Some C is not A. / .'. Some C is not A.

683. FIGURE IV.

Bramantip. / Barbara. All A is B. = / All B is C. All B is C. / All A is B. .. Some C is A. / .. All A is C. .'. Some C is A.

Camenes Celarent All A is B / No B is C. No B is C. = All A is B. .. No C is A./ .'. No A is C. .'. No C is A.

Dimaris. Darii. Some A is B. / All B is C. All B is C. = Some A is B. .'. Some C is A./ .'. Some A is C. .'. Some C is A.

Fesapo. Ferio. No A is B. / No B is A. All B is C. = Some C is B. .'. Some C is not A./ .'. Some C is not A.

Fresison. Ferio. No A is B. / No B is A. Some B is C. = Some C is B. .'. Some C is not A./ .'. Some C is not A.

684. The reason why Baroko and Bokardo cannot be reduced ostensively by the aid of mere conversion becomes plain on an inspection of them. In both it is necessary, if we are to obtain the first figure, that the position of the middle term should be changed in one premiss. But the premisses of both consist of A and 0 propositions, of which A admits only of conversion by limitation, the effect of which would be to produce two particular premisses, while 0 does not admit of conversion at all,

It is clear then that the 0 proposition must cease to be 0 before we can get any further. Here permutation comes to our aid; while conversion by negation enables us to convert the A proposition, without loss of quantity, and to elicit the precise conclusion we require out of the reduct of Boltardo.

(Baroko) Fanoao. Ferio. All A is B. / No not-B is A. Some C is not-B. = Some C is not-B. .'. Some C is not-A./ .'. Some C is not-A.

(Bokardo) Donamon. Darii. Some B is not-A. / All B is C. All B is C. = Some not-A is B .'. Some C is not-A./ .'. Some not-A is C. .'. Some C is not-A.

685. In the new symbols, Fanoao and Donamon, [pi] has been adopted as a symbol for permutation; n signifies conversion by negation. In Donamon the first n stands for a process which resolves itself into permutation followed by simple conversion, the second for one which resolves itself into simple conversion followed by permutation, according to the extended meaning which we have given to the term 'conversion by negation.' If it be thought desirable to distinguish these two processes, the ugly symbol Do[pi]samos[pi] may be adopted in place of Donamon.

686. The foregoing method, which may be called Reduction by Negation, is no less applicable to the other moods of the second figure than to Baroko. The symbols which result from providing for its application would make the second of the mnemonic lines run thus—

Benare[pi], Cane[pi]e, Denilo[pi], Fano[pi]o secundae.

687. The only other combination of mood and figure in which it will be found available is Camenes, whose name it changes to Canene.

688.

(Cesare) Benarea. Barbara. No A is B. / All B is not-A. All C is B. = All C is B. .'. No C is A. / .'. All C is not-A. .'. No C is A.

(Camestres) Cane[pi]e. Celarent. All A is B. / No not-B is A. No C is B. = All C is not-B. .'. No C is A. / .'. No C is A.

(Festino) Denilo[pi]. Darii. No A is B. / All B is not-A. Some C is B. = Some C is B. .'. Some C is not A./ .'. Some C is not-A. .'. Some C is not A.

(Camenes) Canene. Celarent. All A is B. / No not-B is A. No B is C. = All C is not-B. .'. No C is A. / .'. No C is A.

689. The following will serve as a concrete instance of Cane[pi]e reduced to the first figure.

All things of which we have a perfect idea are perceptions. A substance is not a perception. .'. A substance is not a thing of which we have a perfect idea.

When brought into Celarent this becomes—

No not-perception is a thing of which we have a perfect idea. A substance is a not-perception. .'. No substance is a thing of which we have a perfect idea.

690. We may also bring it, if we please, into Barbara, by permuting the major premiss once more, so as to obtain the contrapositive of the original—

All not-perceptions are things of which we have an imperfect idea. All substances are not-perceptions. .'. All substances are things of which we have an imperfect idea.



Indirect Reduction.

691. We will apply this method to Baroko.

All A is B. All fishes are oviparous. Some C is not B. Some marine animals are not oviparous. .'. Some C is not A. .'. Some marine animals are not fishes.

692. The reasoning in such a syllogism is evidently conclusive: but it does not conform, as it stands, to the first figure, nor (permutation apart) can its premisses be twisted into conformity with it. But though we cannot prove the conclusion true in the first figure, we can employ that figure to prove that it cannot be false, by showing that the supposition of its falsity would involve a contradiction of one of the original premisses, which are true ex hypothesi.

693. If possible, let the conclusion 'Some C is not A' be false. Then its contradictory 'All C is A' must be true. Combining this as minor with the original major, we obtain premisses in the first figure,

All A is B, All fishes are oviparous, All C is A, All marine animals are fishes,

which lead to the conclusion

All C is B, All marine animals are oviparous.

But this conclusion conflicts with the original minor, 'Some C is not B,' being its contradictory. But the original minor is ex hypothesi true. Therefore the new conclusion is false. Therefore it must either be wrongly drawn or else one or both of its premisses must be false. But it is not wrongly drawn; since it is drawn in the first figure, to which the Dictum de Omni et Nullo applies. Therefore the fault must lie in the premisses. But the major premiss, being the same with that of the original syllogism, is ex hypothesi true. Therefore the minor premiss, 'All C is A,' is false. But this being false, its contradictory must be true. Now its contradictory is the original conclusion, 'Some C is not A,' which is therefore proved to be true, since it cannot be false.

694. It is convenient to represent the two syllogisms in juxtaposition thus—

Baroko. Barbara. All A is B. All A is B. Some C is not B. / All C is A. .'. Some C is not A. / All C is B.

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