A is followed by B .'. Something followed by B is A—

would be clumsy formalism. We usually say, and we ought to say—

A is followed by B .'. B follows A (or is preceded by A).

Now, any relation between two terms may be viewed from either side—A: B or B: A. It is in both cases the same fact; but, with the altered point of view, it may present a different character. For example, in the Immediate Inference—A > B .'. B < A—a diminishing turns into an increasing ratio, whilst the fact predicated remains the same. Given, then, a relation between two terms as viewed from one to the other, the same relation viewed from the other to the one may be called the Reciprocal. In the cases of Equality, Co-existence and Simultaneity, the given relation and its reciprocal are not only the same fact, but they also have the same character: in the cases of Greater and Less and Sequence, the character alters.

We may, then, state the following rule for the conversion of propositions in which the whole relation explicitly stated is taken as the copula: Transpose the terms, and for the given relation substitute its reciprocal. Thus—

A is the cause of B .'. B is the effect of A.

The rule assumes that the reciprocal of a given relation is definitely known; and so far as this is true it may be extended to more concrete relations—

A is a genus of B .'. B is a species of A A is the father of B .'. B is a child of A.

But not every relational expression has only one definite reciprocal. If we are told that A is the brother of B, we can only infer that B is either the brother or the sister of A. A list of all reciprocal relations is a desideratum of Logic.

Sec. 5. Obversion (otherwise called Permutation or AEquipollence) is Immediate Inference by changing the quality of the given proposition and substituting for its predicate the contradictory term. The given proposition is called the 'obvertend,' and the inference from it the 'obverse.' Thus the obvertend being—Some philosophers are consistent reasoners, the obverse will be—Some philosophers are not inconsistent reasoners.

The legitimacy of this mode of reasoning follows, in the case of affirmative propositions, from the principle of Contradiction, that if any term be affirmed of a subject, the contradictory term may be denied (chap. vi. Sec. 3). To obvert affirmative propositions, then, the rule is—Insert the negative sign, and for the predicate substitute its contradictory term.

A. All S is P .'. No S is not-P All men are fallible .'. No men are infallible.

I. Some S is P .'. some S is not-P Some philosophers are consistent .'. Some philosophers are not inconsistent.

In agreement with this mode of inference, we have the rule of modern English grammar, that 'two negatives make an affirmative.'

Again, by the principle of Excluded Middle, if any term be denied of a subject, its contradictory may be affirmed: to obvert negative propositions, then, the rule is—Remove the negative sign, and for the predicate substitute its contradictory term.

E. No S is P .'. All S is not-P No matter is destructible .'. All matter is indestructible.

O. Some S is not P .'. Some S is not-P Some ideals are not attainable .'. Some ideals are unattainable.

Thus, by obversion, each of the four propositions retains its quantity but changes its quality: A. to E., I. to O., E. to A., O. to I. And all the obverses are infinite propositions, the affirmative infinites having the sense of negatives, and the negative infinites having the sense of affirmatives.

Again, having obtained the obverse of a given proposition, it may be desirable to recover the obvertend; or it may at any time be requisite to change a given infinite proposition into the corresponding direct affirmative or negative; and in such cases the process is still obversion. Thus, if No S is not-P be given us to recover the obvertend or to find the corresponding affirmative; the proposition being formally negative, we apply the rule for obverting negatives: 'Remove the negative sign, and for the predicate substitute its contradictory.' This yields the affirmative All S is P. Similarly, to obtain the obvertend of All S is not-P, apply the rule for obverting Affirmatives; and this yields No S is P.

Sec. 6. Contrariety.—We have seen in chap. iv. Sec. 8, that contrary terms are such that no two of them are predicable in the same way of the same subject, whilst perhaps neither may be predicable of it. Similarly, Contrary Propositions may be defined as those of which no two are ever both true together, whilst perhaps neither may be true; or, in other words, both may be false. This is the relation between A. and E. when concerned with the same matter: as A.—All men are wise; E.—No men are wise. Such propositions cannot both be true; but they may both be false, for some men may be wise and some not. They cannot both be true; for, by the principle of Contradiction, if wise may be affirmed of All men, not-wise must be denied; but All men are not-wise is the obverse of No men are wise, which therefore may also be denied.

At the same time we cannot apply to A. and E. the principle of Excluded Middle, so as to show that one of them must be true of the same matter. For if we deny that All men are wise, we do not necessarily deny the attribute 'wise' of each and every man: to say that Not all are wise may mean no more than that Some are not. This gives a proposition in the form of O.; which, as we have seen, does not imply its subalternans, E.

If, however, two Singular Propositions, having the same matter, but differing in quality, are to be treated as universals, and therefore as A. and E., they are, nevertheless, contradictory and not merely contrary; for one of them must be false and the other true.

Sec. 7. Contradiction is a relation between two propositions analogous to that between contradictory terms (one of which being affirmed of a subject the other is denied)—such, namely, that one of them is false and the other true. This is the case with the forms A. and O., and E. and I., in the same matter. If it be true that All men are wise, it is false that Some men are not wise (equivalent by obversion to Some men are not-wise); or else, since the 'Some men' are included in the 'All men,' we should be predicating of the same men that they are both 'wise' and 'not-wise'; which would violate the principle of Contradiction. Similarly, No men are wise, being by obversion equivalent to All men are not-wise, is incompatible with Some men are wise, by the same principle of Contradiction.

But, again, if it be false that All men are wise, it is always true that Some are not wise; for though in denying that 'wise' is a predicate of 'All men' we do not deny it of each and every man, yet we deny it of 'Some men.' Of 'Some men,' therefore, by the principle of Excluded Middle, 'not-wise' is to be affirmed; and Some men are not-wise, is by obversion equivalent to Some men are not wise. Similarly, if it be false that No men are wise, which by obversion is equivalent to All men are not-wise, then it is true at least that Some men are wise.

By extending and enforcing the doctrine of relative terms, certain other inferences are implied in the contrary and contradictory relations of propositions. We have seen in chap. iv. that the contradictory of a given term includes all its contraries: 'not-blue,' for example, includes red and yellow. Hence, since The sky is blue becomes by obversion, The sky is not not-blue, we may also infer The sky is not red, etc. From the truth, then, of any proposition predicating a given term, we may infer the falsity of all propositions predicating the contrary terms in the same relation. But, on the other hand, from the falsity of a proposition predicating a given term, we cannot infer the truth of the predication of any particular contrary term. If it be false that The sky is red, we cannot formally infer, that The sky is blue (cf. chap. iv. Sec. 8).

Sec. 8. Sub-contrariety is the relation of two propositions, concerning the same matter that may both be true but are never both false. This is the case with I. and O. If it be true that Some men are wise, it may also be true that Some (other) men are not wise. This follows from the maxim in chap. vi. Sec. 6, not to go beyond the evidence.

For if it be true that Some men are wise, it may indeed be true that All are (this being the subalternans): and if All are, it is (by contradiction) false that Some are not; but as we are only told that Some men are, it is illicit to infer the falsity of Some are not, which could only be justified by evidence concerning All men.

But if it be false that Some men are wise, it is true that Some men are not wise; for, by contradiction, if Some men are wise is false, No men are wise is true; and, therefore, by subalternation, Some men are not wise is true.

Sec. 9. The Square of Opposition.—By their relations of Subalternation, Contrariety, Contradiction, and Sub-contrariety, the forms A. I. E. O. (having the same matter) are said to stand in Opposition: and Logicians represent these relations by a square having A. I. E. O. at its corners:

A. Contraries E.

S Co s S u nt e u b ra i b a di r a l ct o l t ct o t e di r e r ra i r n nt e n s Co s s

I. Sub-contraries O.

As an aid to the memory, this diagram is useful; but as an attempt to represent the logical relations of propositions, it is misleading. For, standing at corners of the same square, A. and E., A. and I., E. and O., and I. and O., seem to be couples bearing the same relation to one another; whereas we have seen that their relations are entirely different. The following traditional summary of their relations in respect of truth and falsity is much more to the purpose:

(1) If A. is true, I. is true, E. is false, O. is false. (2) If A. is false, I. is unknown, E. is unknown, O. is true. (3) If I. is true, A. is unknown, E. is false, O. is unknown. (4) If I. is false, A. is false, E. is true, O. is true. (5) If E. is true, A. is false, I. is false, O. is true. (6) If E. is false, A. is unknown, I. is true, O. is unknown. (7) If O. is true, A. is false, I. is unknown, E. is unknown. (8) If O. is false, A. is true, I. is true, E. is false.

Where, however, as in cases 2, 3, 6, 7, alleging either the falsity of universals or the truth of particulars, it follows that two of the three Opposites are unknown, we may conclude further that one of them must be true and the other false, because the two unknown are always Contradictories.

Sec. 10. Secondary modes of Immediate Inference are obtained by applying the process of Conversion or Obversion to the results already obtained by the other process. The best known secondary form of Immediate Inference is the Contrapositive, and this is the converse of the obverse of a given proposition. Thus:

DATUM. OBVERSE. CONTRAPOSITIVE.

A. All S is P .'. No S is not-P .'. No not-P is S I. Some S is P .'. Some S is not not-P .'. (none) E. No S is P .'. All S is not-P .'. Some not-P is S O. Some S is not P .'. Some S is not-P .'. Some not-P is S

There is no contrapositive of I., because the obverse of I. is in the form of O., and we have seen that O. cannot be converted. O., however, has a contrapositive (Some not-P is S); and this is sometimes given instead of the converse, and called the 'converse by negation.'

Contraposition needs no justification by the Laws of Thought, as it is nothing but a compounding of conversion with obversion, both of which processes have already been justified. I give a table opposite of the other ways of compounding these primary modes of Immediate Inference.

A I E O ————————————————————————————————————————

1 All A is B Some A is B No A is B Some A is not B

————————————————————————————————————————

Obverse 2 No A is b Some A is not b All A is b Some A is b

————————————————————————————————————————

Converse 3 Some B is A Some B is A No B is A

————————————————————————————————————————

Obverse of 4 Some B is not a Some B is not a All B is a Converse

————————————————————————————————————————

Contra- positive 5 No b is A Some b is A Some b is A

————————————————————————————————————————

Obverse of 6 All b is a Some b is not a Some b is not a Contrapos

————————————————————————————————————————

Converse of Obverse 7 Some a is B of Converse

————————————————————————————————————————

Obverse of Converse of 8 Some a is not b Obverse of Converse

————————————————————————————————————————

Converse of Obverse 9 Some a is b of Contrapos

————————————————————————————————————————

Obverse of Converse of 10 Some a is not B Obverse of Contrapos

————————————————————————————————————————

In this table a and b stand for not-A and not-B and had better be read thus: for No A is b, No A is not-B; for All b is a (col. 6), All not-B is not-A; and so on.

It may not, at first, be obvious why the process of alternately obverting and converting any proposition should ever come to an end; though it will, no doubt, be considered a very fortunate circumstance that it always does end. On examining the results, it will be found that the cause of its ending is the inconvertibility of O. For E., when obverted, becomes A.; every A, when converted, degenerates into I.; every I., when obverted, becomes O.; O cannot be converted, and to obvert it again is merely to restore the former proposition: so that the whole process moves on to inevitable dissolution. I. and O. are exhausted by three transformations, whilst A. and E. will each endure seven.

Except Obversion, Conversion and Contraposition, it has not been usual to bestow special names on these processes or their results. But the form in columns 7 and 10 (Some a is B—Some a is not B), where the original predicate is affirmed or denied of the contradictory of the original subject, has been thought by Dr. Keynes to deserve a distinctive title, and he has called it the 'Inverse.' Whilst the Inverse is one form, however, Inversion is not one process, but is obtained by different processes from E. and A. respectively. In this it differs from Obversion, Conversion, and Contraposition, each of which stands for one process.

The Inverse form has been objected to on the ground that the inference All A is B .'. Some not-A is not B, distributes B (as predicate of a negative proposition), though it was given as undistributed (as predicate of an affirmative proposition). But Dr. Keynes defends it on the ground that (1) it is obtained by obversions and conversions which are all legitimate and (2) that although All A is B does not distribute B in relation to A, it does distribute B in relation to some not-A (namely, in relation to whatever not-A is not-B). This is one reason why, in stating the rule in chap. vi. Sec. 6, I have written: "an immediate inference ought to contain nothing that is not contained, or formally implied, in the proposition from which it is inferred"; and have maintained that every term formally implies its contradictory within the suppositio.

Sec. 11. Immediate Inferences from Conditionals are those which consist—(1) in changing a Disjunctive into a Hypothetical, or a Hypothetical into a Disjunctive, or either into a Categorical; and (2) in the relations of Opposition and the equivalences of Obversion, Conversion, and secondary or compound processes, which we have already examined in respect of Categoricals. As no new principles are involved, it may suffice to exhibit some of the results.

We have already seen (chap. v. Sec. 4) how Disjunctives may be read as Hypotheticals and Hypotheticals as Categoricals. And, as to Opposition, if we recognise four forms of Hypothetical A. I. E. O., these plainly stand to one another in a Square of Opposition, just as Categoricals do. Thus A. and E. (If A is B, C is D, and If A is B, C is not D) are contraries, but not contradictories; since both may be false (C may sometimes be D, and sometimes not), though they cannot both be true. And if they are both false, their subalternates are both true, being respectively the contradictories of the universals of opposite quality, namely, I. of E., and O. of A. But in the case of Disjunctives, we cannot set out a satisfactory Square of Opposition; because, as we saw (chap. v. Sec. 4), the forms required for E. and O. are not true Disjunctives, but Exponibles.

The Obverse, Converse, and Contrapositive, of Hypotheticals (admitting the distinction of quality) may be exhibited thus:

DATUM. OBVERSE.

A. If A is B, C is D If A is B, C is not d I. Sometimes when A is B, C is D Sometimes when A is B, C is not d E. If A is B, C is not D If A is B, C is d O. Sometimes when A is B, C is not D Sometimes when A is B, C is d

CONVERSE. CONTRAPOSITIVE.

Sometimes when C is D, A is B If C is d, A is not B Sometimes when C is D, A is B (none) If C is D, A is not B Sometimes when C is d, A is B (none) Sometimes when C is d, A is B

As to Disjunctives, the attempt to put them through these different forms immediately destroys their disjunctive character. Still, given any proposition in the form A is either B or C, we can state the propositions that give the sense of obversion, conversion, etc., thus:

DATUM.—A is either B or C; OBVERSE.—A is not both b and c; CONVERSE.—Something, either B or C, is A; CONTRAPOSITIVE.—Nothing that is both b and c is A.

For a Disjunctive in I., of course, there is no Contrapositive. Given a Disjunctive in the form Either A is B or C is D, we may write for its Obverse—In no case is A b, and C at the same time d. But no Converse or Contrapositive of such a Disjunctive can be obtained, except by first casting it into the hypothetical or categorical form.

The reader who wishes to pursue this subject further, will find it elaborately treated in Dr. Keynes' Formal Logic, Part II.; to which work the above chapter is indebted.



CHAPTER VIII

ORDER OF TERMS, EULER'S DIAGRAMS, LOGICAL EQUATIONS, EXISTENTIAL IMPORT OF PROPOSITIONS

Sec. 1. Of the terms of a proposition which is the Subject and which the Predicate? In most of the exemplary propositions cited by Logicians it will be found that the subject is a substantive and the predicate an adjective, as in Men are mortal. This is the relation of Substance and Attribute which we saw (chap. i. Sec. 5) to be the central type of relations of coinherence; and on this model other predications may be formed in which the subject is not a substance, but is treated as if it were, and could therefore be the ground of attributes; as Fame is treacherous, The weather is changeable. But, in literature, sentences in which the adjective comes first are not uncommon, as Loud was the applause, Dark is the fate of man, Blessed are the peacemakers, and so on. Here, then, 'loud,' 'dark' and 'blessed' occupy the place of the logical subject. Are they really the subject, or must we alter the order of such sentences into The applause was loud, etc.? If we do, and then proceed to convert, we get Loud was the applause, or (more scrupulously) Some loud noise was the applause. The last form, it is true, gives the subject a substantive word, but 'applause' has become the predicate; and if the substantive 'noise' was not implied in the first form, Loud is the applause, by what right is it now inserted? The recognition of Conversion, in fact, requires us to admit that, formally, in a logical proposition, the term preceding the copula is subject and the one following is predicate. And, of course, materially considered, the mere order of terms in a proposition can make no difference in the method of proving it, nor in the inferences that can be drawn from it.

Still, if the question is, how we may best cast a literary sentence into logical form, good grounds for a definite answer may perhaps be found. We must not try to stand upon the naturalness of expression, for Dark is the fate of man is quite as natural as Man is mortal. When the purpose is not merely to state a fact, but also to express our feelings about it, to place the grammatical predicate first may be perfectly natural and most effective. But the grounds of a logical order of statement must be found in its adaptation to the purposes of proof and inference. Now general propositions are those from which most inferences can be drawn, which, therefore, it is most important to establish, if true; and they are also the easiest to disprove, if false; since a single negative instance suffices to establish the contradictory. It follows that, in re-casting a literary or colloquial sentence for logical purposes, we should try to obtain a form in which the subject is distributed—is either a singular term or a general term predesignate as 'All' or 'No.' Seeing, then, that most adjectives connote a single attribute, whilst most substantives connote more than one attribute; and that therefore the denotation of adjectives is usually wider than that of substantives; in any proposition, one term of which is an adjective and the other a substantive, if either can be distributed in relation to the other, it is nearly sure to be the substantive; so that to take the substantive term for subject is our best chance of obtaining an universal proposition. These considerations seem to justify the practice of Logicians in selecting their examples.

For similar reasons, if both terms of a proposition are substantive, the one with the lesser denotation is (at least in affirmative propositions) the more suitable subject, as Cats are carnivores. And if one term is abstract, that is the more suitable subject; for, as we have seen, an abstract term may be interpreted by a corresponding concrete one distributed, as Kindness is infectious; that is, All kind actions suggest imitation.

If, however, a controvertist has no other object in view than to refute some general proposition laid down by an opponent, a particular proposition is all that he need disentangle from any statement that serves his purpose.

Sec. 2. Toward understanding clearly the relations of the terms of a proposition, it is often found useful to employ diagrams; and the diagrams most in use are the circles of Euler.

These circles represent the denotation of the terms. Suppose the proposition to be All hollow-horned animals ruminate: then, if we could collect all ruminants upon a prairie, and enclose them with a circular palisade; and segregate from amongst them all the hollow-horned beasts, and enclose them with another ring-fence inside the other; one way of interpreting the proposition (namely, in denotation) would be figured to us thus:



An Universal Affirmative may also state a relation between two terms whose denotation is co-extensive. A definition always does this, as Man is a rational animal; and this, of course, we cannot represent by two distinct circles, but at best by one with a thick circumference, to suggest that two coincide, thus:



The Particular Affirmative Proposition may be represented in several ways. In the first place, bearing in mind that 'Some' means 'some at least, it may be all,' an I. proposition may be represented by Figs. 1 and 2; for it is true that Some horned animals ruminate, and that Some men are rational. Secondly, there is the case in which the 'Some things' of which a predication is made are, in fact, not all; whilst the predicate, though not given as distributed, yet might be so given if we wished to state the whole truth; as if we say Some men are Chinese. This case is also represented by Fig. 1, the outside circle representing 'Men,' and the inside one 'Chinese.' Thirdly, the predicate may appertain to some only of the subject, but to a great many other things, as in Some horned beasts are domestic; for it is true that some are not, and that certain other kinds of animals are, domestic. This case, therefore, must be illustrated by overlapping circles, thus:



The Universal Negative is sufficiently represented by a single Fig. (4): two circles mutually exclusive, thus:



That is, No horned beasts are carnivorous.

Lastly, the Particular Negative may be represented by any of the Figs. 1, 3, and 4; for it is true that Some ruminants are not hollow-horned, that Some horned animals are not domestic, and that Some horned beasts are not carnivorous.

Besides their use in illustrating the denotative force of propositions, these circles may be employed to verify the results of Obversion, Conversion, and the secondary modes of Immediate Inference. Thus the Obverse of A. is clear enough on glancing at Figs. 1 and 2; for if we agree that whatever term's denotation is represented by a given circle, the denotation of the contradictory term shall be represented by the space outside that circle; then if it is true that All hollow horned animals are ruminants, it is at the same time true that No hollow-horned animals are not-ruminants; since none of the hollow-horned are found outside the palisade that encloses the ruminants. The Obverse of I., E. or O. may be verified in a similar manner.

As to the Converse, a Definition is of course susceptible of Simple Conversion, and this is shown by Fig. 2: 'Men are rational animals' and 'Rational animals are men.' But any other A. proposition is presumably convertible only by limitation, and this is shown by Fig. 1; where All hollow-horned animals are ruminants, but we can only say that Some ruminants are hollow-horned.

That I. may be simply converted may be seen in Fig. 3, which represents the least that an I. proposition can mean; and that E. may be simply converted is manifest in Fig. 4.

As for O., we know that it cannot be converted, and this is made plain enough by glancing at Fig. 1; for that represents the O., Some ruminants are not hollow-horned, but also shows this to be compatible with All hollow-horned animals are ruminants (A.). Now in conversion there is (by definition) no change of quality. The Converse, then, of Some ruminants are not hollow-horned must be a negative proposition, having 'hollow-horned' for its subject, either in E. or O.; but these would be respectively the contrary and contradictory of All hollow-horned animals are ruminants; and, therefore, if this be true, they must both be false.

But (referring still to Fig. 1) the legitimacy of contrapositing O. is equally clear; for if Some ruminants are not hollow-horned, Some animals that are not hollow-horned are ruminants, namely, all the animals between the two ring-fences. Similar inferences may be illustrated from Figs. 3 and 4. And the Contraposition of A. may be verified by Figs. 1 and 2, and the Contraposition of E. by Fig. 4.

Lastly, the Inverse of A. is plain from Fig. 1—Some things that are not hollow-horned are not ruminants, namely, things that lie outside the outer circle and are neither 'ruminants' nor 'hollow-horned.' And the Inverse of E may be studied in Fig. 4—Some things that are not-horned beasts are carnivorous.

Notwithstanding the facility and clearness of the demonstrations thus obtained, it may be said that a diagrammatic method, representing denotations, is not properly logical. Fundamentally, the relation asserted (or denied) to exist between the terms of a proposition, is a relation between the terms as determined by their attributes or connotation; whether we take Mill's view, that a proposition asserts that the connotation of the subject is a mark of the connotation of the predicate; or Dr. Venn's view, that things denoted by the subject (as having its connotation) have (or have not) the attribute connoted by the predicate; or, the Conceptualist view, that a judgment is a relation of concepts (that is, of connotations). With a few exceptions artificially framed (such as 'kings now reigning in Europe'), the denotation of a term is never directly and exhaustively known, but consists merely in 'all things that have the connotation.' If the value of logical training depends very much upon our habituating ourselves to construe propositions, and to realise the force of inferences from them, according to the connotation of their terms, we shall do well not to turn too hastily to the circles, but rather to regard them as means of verifying in denotation the conclusions that we have already learnt to recognise as necessary in connotation.

Sec. 3. The equational treatment of propositions is closely connected with the diagrammatic. Hamilton thought it a great merit of his plan of quantifying the predicate, that thereby every proposition is reduced to its true form—an equation. According to this doctrine, the proposition All X is all Y (U.) equates X and Y; the proposition All X is some Y (A.) equates X with some part of Y; and similarly with the other affirmatives (Y. and I.). And so far it is easy to follow his meaning: the Xs are identical with some or all the Ys. But, coming to the negatives, the equational interpretation is certainly less obvious. The proposition No X is Y (E.) cannot be said in any sense to equate X and Y; though, if we obvert it into All X is some not-Y, we have (in the same sense, of course, as in the above affirmative forms) X equated with part at least of 'not-Y.'

But what is that sense? Clearly not the same as that in which mathematical terms are equated, namely, in respect of some mode of quantity. For if we may say Some X is some Y, these Xs that are also Ys are not merely the same in number, or mass, or figure; they are the same in every respect, both quantitative and qualitative, have the same positions in time and place, are in fact identical. The proposition 2+2=4 means that any two things added to any other two are, in respect of number, equal to any three things added to one other thing; and this is true of all things that can be counted, however much they may differ in other ways. But All X is all Y means that Xs and Ys are the same things, although they have different names when viewed in different aspects or relations. Thus all equilateral triangles are equiangular triangles; but in one case they are named from the equality of their angles, and in the other from the equality of their sides. Similarly, 'British subjects' and 'subjects of King George V' are the same people, named in one case from the person of the Crown, and in the other from the Imperial Government. These logical equations, then, are in truth identities of denotation; and they are fully illustrated by the relations of circles described in the previous section.

When we are told that logical propositions are to be considered as equations, we naturally expect to be shown some interesting developments of method in analogy with the equations of Mathematics; but from Hamilton's innovations no such thing results. This cannot be said, however, of the equations of Symbolic Logic; which are the starting-point of very remarkable processes of ratiocination. As the subject of Symbolic Logic, as a whole, lies beyond the compass of this work, it will be enough to give Dr. Venn's equations corresponding with the four propositional forms of common Logic.

According to this system, universal propositions are to be regarded as not necessarily implying the existence of their terms; and therefore, instead of giving them a positive form, they are translated into symbols that express what they deny. For example, the proposition All devils are ugly need not imply that any such things as 'devils' really exist; but it certainly does imply that Devils that are not ugly do not exist. Similarly, the proposition No angels are ugly implies that Angels that are ugly do not exist. Therefore, writing x for 'devils,' y for 'ugly,' and [y] for 'not-ugly,' we may express A., the universal affirmative, thus:

A. x[y] = 0.

That is, x that is not y is nothing; or, Devils that are not-ugly do not exist. And, similarly, writing x for 'angels' and y for 'ugly,' we may express E., the universal negative, thus:

E. xy = 0.

That is, x that is y is nothing; or, Angels that are ugly do not exist.

On the other hand, particular propositions are regarded as implying the existence of their terms, and the corresponding equations are so framed as to express existence. With this end in view, the symbol v is adopted to represent 'something,' or indeterminate reality, or more than nothing. Then, taking any particular affirmative, such as Some metaphysicians are obscure, and writing x for 'metaphysicians,' and y for 'obscure,' we may express it thus:

I. xy = v.

That is, x that is y is something; or, Metaphysicians that are obscure do occur in experience (however few they may be, or whether they all be obscure). And, similarly, taking any particular negative, such as Some giants are not cruel, and writing x for 'giants' and y for 'not-cruel,' we may express it thus:

O. x[y] = v.

That is, x that is not y is something; or, giants that are not-cruel do occur—in romances, if nowhere else.

Clearly, these equations are, like Hamilton's, concerned with denotation. A. and E. affirm that the compound terms x[y] and xy have no denotation; and I. and O. declare that x[y] and xy have denotation, or stand for something. Here, however, the resemblance to Hamilton's system ceases; for the Symbolic Logic, by operating upon more than two terms simultaneously, by adopting the algebraic signs of operations, +,-, x, / (with a special signification), and manipulating the symbols by quasi-algebraic processes, obtains results which the common Logic reaches (if at all) with much greater difficulty. If, indeed, the value of logical systems were to be judged of by the results obtainable, formal deductive Logic would probably be superseded. And, as a mental discipline, there is much to be said in favour of the symbolic method. But, as an introduction to philosophy, the common Logic must hold its ground. (Venn: Symbolic Logic, c. 7.)

Sec. 4. Does Formal Logic involve any general assumption as to the real existence of the terms of propositions?

In the first place, Logic treats primarily of the relations implied in propositions. This follows from its being the science of proof for all sorts of (qualitative) propositions; since all sorts of propositions have nothing in common except the relations they express.

But, secondly, relations without terms of some sort are not to be thought of; and, hence, even the most formal illustrations of logical doctrines comprise such terms as S and P, X and Y, or x and y, in a symbolic or representative character. Terms, therefore, of some sort are assumed to exist (together with their negatives or contradictories) for the purposes of logical manipulation.

Thirdly, however, that Formal Logic cannot as such directly involve the existence of any particular concrete terms, such as 'man' or 'mountain,' used by way of illustration, is implied in the word 'formal,' that is, 'confined to what is common or abstract'; since the only thing common to all terms is to be related in some way to other terms. The actual existence of any concrete thing can only be known by experience, as with 'man' or 'mountain'; or by methodically justifiable inference from experience, as with 'atom' or 'ether.' If 'man' or 'mountain,' or 'Cuzco' be used to illustrate logical forms, they bring with them an existential import derived from experience; but this is the import of language, not of the logical forms. 'Centaur' and 'El Dorado' signify to us the non-existent; but they serve as well as 'man' and 'London' to illustrate Formal Logic.

Nevertheless, fourthly, the existence or non-existence of particular terms may come to be implied: namely, wherever the very fact of existence, or of some condition of existence, is an hypothesis or datum. Thus, given the proposition All S is P, to be P is made a condition of the existence of S: whence it follows that an S that is not P does not exist (x[y] = 0). On the further hypothesis that S exists, it follows that P exists. On the hypothesis that S does not exist, the existence of P is problematic; but, then, if P does exist we cannot convert the proposition; since Some P is S (P existing) would involve the existence of S; which is contrary to the hypothesis.

Assuming that Universals do not, whilst Particulars do, imply the existence of their subjects, we cannot infer the subalternate (I. or O.) from the subalternans (A. or E.), for that is to ground the actual on the problematic; and for the same reason we cannot convert A. per accidens.

Assuming, again, a certain suppositio or universe, to which in a given discussion every argument shall refer, then, any propositions whose terms lie outside that suppositio are irrelevant, and for the purposes of that discussion are sometimes called "false"; though it seems better to call them irrelevant or meaningless, seeing that to call them false implies that they might in the same case be true. Thus propositions which, according to the doctrine of Opposition, appear to be Contradictories, may then cease to be so; for of Contradictories one is true and the other false; but, in the case supposed, both are meaningless. If the subject of discussion be Zoology, all propositions about centaurs or unicorns are absurd; and such specious Contradictories as No centaurs play the lyre—Some centaurs do play the lyre; or All unicorns fight with lions—Some unicorns do not fight with lions, are both meaningless, because in Zoology there are no centaurs nor unicorns; and, therefore, in this reference, the propositions are not really contradictory. But if the subject of discussion or suppositio be Mythology or Heraldry, such propositions as the above are to the purpose, and form legitimate pairs of Contradictories.

In Formal Logic, in short, we may make at discretion any assumption whatever as to the existence, or as to any condition of the existence of any particular term or terms; and then certain implications and conclusions follow in consistency with that hypothesis or datum. Still, our conclusions will themselves be only hypothetical, depending on the truth of the datum; and, of course, until this is empirically ascertained, we are as far as ever from empirical reality. (Venn: Symbolic Logic, c. 6; Keynes: Formal Logic, Part II. c. 7: cf. Wolf: Studies in Logic.)



CHAPTER IX

FORMAL CONDITIONS OF MEDIATE INFERENCE

Sec. 1. A Mediate Inference is a proposition that depends for proof upon two or more other propositions, so connected together by one or more terms (which the evidentiary propositions, or each pair of them, have in common) as to justify a certain conclusion, namely, the proposition in question. The type or (more properly) the unit of all such modes of proof, when of a strictly logical kind, is the Syllogism, to which we shall see that all other modes are reducible. It may be exhibited symbolically thus:

M is P; S is M: .'. S is P.

Syllogisms may be classified, as to quantity, into Universal or Particular, according to the quantity of the conclusion; as to quality, into Affirmative or Negative, according to the quality of the conclusion; and, as to relation, into Categorical, Hypothetical and Disjunctive, according as all their propositions are categorical, or one (at least) of their evidentiary propositions is a hypothetical or a disjunctive.

To begin with Categorical Syllogisms, of which the following is an example:

All authors are vain; Cicero is an author: .'. Cicero is vain.

Here we may suppose that there are no direct means of knowing that Cicero is vain; but we happen to know that all authors are vain and that he is an author; and these two propositions, put together, unmistakably imply that he is vain. In other words, we do not at first know any relation between 'Cicero' and 'vanity'; but we know that these two terms are severally related to a third term, 'author,' hence called a Middle Term; and thus we perceive, by mediate evidence, that they are related to one another. This sort of proof bears an obvious resemblance (though the relations involved are not the same) to the mathematical proof of equality between two quantities, that cannot be directly compared, by showing the equality of each of them to some third quantity: A = B = C .'. A = C. Here B is a middle term.

We have to inquire, then, what conditions must be satisfied in order that a Syllogism may be formally conclusive or valid. A specious Syllogism that is not really valid is called a Parasyllogism.

Sec. 2. General Canons of the Syllogism.

(1) A Syllogism contains three, and no more, distinct propositions.

(2) A Syllogism contains three, and no more, distinct univocal terms.

These two Canons imply one another. Three propositions with less than three terms can only be connected in some of the modes of Immediate Inference. Three propositions with more than three terms do not show that connection of two terms by means of a third, which is requisite for proving a Mediate Inference. If we write—

All authors are vain; Cicero is a statesman—

there are four terms and no middle term, and therefore there is no proof. Or if we write—

All authors are vain; Cicero is an author: .'. Cicero is a statesman—

here the term 'statesman' occurs without any voucher; it appears in the inference but not in the evidence, and therefore violates the maxim of all formal proof, 'not to go beyond the evidence.' It is true that if any one argued—

All authors are vain; Cicero wrote on philosophy: .'. Cicero is vain—

this could not be called a bad argument or a material fallacy; but it would be a needless departure from the form of expression in which the connection between the evidence and the inference is most easily seen.

Still, a mere adherence to the same form of words in the expression of terms is not enough: we must also attend to their meaning. For if the same word be used ambiguously (as 'author' now for 'father' and anon for 'man of letters'), it becomes as to its meaning two terms; so that we have four in all. Then, if the ambiguous term be the Middle, no connection is shown between the other two; if either of the others be ambiguous, something seems to be inferred which has never been really given in evidence.

The above two Canons are, indeed, involved in the definition of a categorical syllogism, which may be thus stated: A Categorical Syllogism is a form of proof or reasoning (way of giving reasons) in which one categorical proposition is established by comparing two others that contain together only three terms, or that have one and only one term in common.

The proposition established, derived, or inferred, is called the Conclusion: the evidentiary propositions by which it is proved are called the Premises.

The term common to the premises, by means of which the other terms are compared, is called the Middle Term; the subject of the conclusion is called the Minor Term; the predicate of the conclusion, the Major Term.

The premise in which the minor term occurs is called the Minor Premise; that in which the major term occurs is called the Major Premise. And a Syllogism is usually written thus:

Major Premise—All authors (Middle) are vain (Major);

Minor Premise—Cicero (Minor) is an author (Middle):

Conclusion—.'. Cicero (Minor) is vain (Major).

Here we have three propositions with three terms, each term occurring twice. The minor and major terms are so called, because, when the conclusion is an universal affirmative (which only occurs in Barbara; see chap. x. Sec. 6), its subject and predicate are respectively the less and the greater in extent or denotation; and the premises are called after the peculiar terms they contain: the expressions 'major premise' and 'minor premise' have nothing to do with the order in which the premises are presented; though it is usual to place the major premise first.

(3) No term must be distributed in the conclusion unless it is distributed in the premises.

It is usual to give this as one of the General Canons of the Syllogism; but we have seen (chap. vi. Sec. 6) that it is of wider application. Indeed, 'not to go beyond the evidence' belongs to the definition of formal proof. A breech of this rule in a syllogism is the fallacy of Illicit Process of the Minor, or of the Major, according to which term has been unwarrantably distributed. The following parasyllogism illicitly distributes both terms of the conclusion:

All poets are pathetic; Some orators are not poets: .'. No orators are pathetic.

(4) The Middle Term must be distributed at least once in the premises (in order to prove a conclusion in the given terms).

For the use of mediate evidence is to show the relation of terms that cannot be directly compared; this is only possible if the middle term furnishes the ground of comparison; and this (in Logic) requires that the whole denotation of the middle should be either included or excluded by one of the other terms; since if we only know that the other terms are related to some of the middle, their respective relations may not be with the same part of it.

It is true that in what has been called the "numerically definite syllogism," an inference may be drawn, though our canon seems to be violated. Thus:

60 sheep in 100 are horned; 60 sheep in 100 are blackfaced: .'. at least 20 blackfaced sheep in 100 are horned.

But such an argument, though it may be correct Arithmetic, is not Logic at all; and when such numerical evidence is obtainable the comparatively indefinite arguments of Logic are needless. Another apparent exception is the following:

Most men are 5 feet high; Most men are semi-rational: .'. Some semi-rational things are 5 feet high.

Here the Middle Term (men) is distributed in neither premise, yet the indisputable conclusion is a logical proposition. The premises, however, are really arithmetical; for 'most' means 'more than half,' or more than 50 per cent.

Still, another apparent exception is entirely logical. Suppose we are given, the premises—All P is M, and All S is M—the middle term is undistributed. But take the obverse of the contrapositive of both premises:

All m is p; All m is s: .'. Some s is p.

Here we have a conclusion legitimately obtained; but it is not in the terms originally given.

For Mediate Inference depending on truly logical premises, then, it is necessary that one premise should distribute the middle term; and the reason of this may be illustrated even by the above supposed numerical exceptions. For in them the premises are such that, though neither of the two premises by itself distributes the Middle, yet they always overlap upon it. If each premise dealt with exactly half the Middle, thus barely distributing it between them, there would be no logical proposition inferrible. We require that the middle term, as used in one premise, should necessarily overlap the same term as used in the other, so as to furnish common ground for comparing the other terms. Hence I have defined the middle term as 'that term common to both premises by means of which the other terms are compared.'

(5) One at least of the premises must be affirmative; or, from two negative premises nothing can be inferred (in the given terms).

The fourth Canon required that the middle term should be given distributed, or in its whole extent, at least once, in order to afford sure ground of comparison for the others. But that such comparison may be effected, something more is requisite; the relation of the other terms to the Middle must be of a certain character. One at least of them must be, as to its extent or denotation, partially or wholly identified with the Middle; so that to that extent it may be known to bear to the other term, whatever relation we are told that so much of the Middle bears to that other term. Now, identity of denotation can only be predicated in an affirmative proposition: one premise, then, must be affirmative.

If both premises are negative, we only know that both the other terms are partly or wholly excluded from the Middle, or are not identical with it in denotation: where they lie, then, in relation to one another we have no means of knowing. Similarly, in the mediate comparison of quantities, if we are told that A and C are both of them unequal to B, we can infer nothing as to the relation of C to A. Hence the premises—

No electors are sober; No electors are independent—

however suggestive, do not formally justify us in inferring any connection between sobriety and independence. Formally to draw a conclusion, we must have affirmative grounds, such as in this case we may obtain by obverting both premises:

All electors are not-sober; All electors are not-independent: .'. Some who are not-independent are not-sober.

But this conclusion is not in the given terms.

(6) (a) If one premise be negative, the conclusion must be negative: and (b) to prove a negative conclusion, one premise must be negative.

(a) For we have seen that one premise must be affirmative, and that thus one term must be partly (at least) identified with the Middle. If, then, the other premise, being negative, predicates the exclusion of the remaining term from the Middle, this remaining term must be excluded from the first term, so far as we know the first to be identical with the Middle: and this exclusion will be expressed by a negative conclusion. The analogy of the mediate comparison of quantities may here again be noticed: if A is equal to B, and B is unequal to C, A is unequal to C.

(b) If both premises be affirmative, the relations to the Middle of both the other terms are more or less inclusive, and therefore furnish no ground for an exclusive inference. This also follows from the function of the middle term.

For the more convenient application of these canons to the testing of syllogisms, it is usual to derive from them three Corollaries:

(i) Two particular premises yield no conclusion.

For if both premises be affirmative, all their terms are undistributed, the subjects by predesignation, the predicates by position; and therefore the middle term must be undistributed, and there can be no conclusion.

If one premise be negative, its predicate is distributed by position: the other terms remaining undistributed. But, by Canon 6, the conclusion (if any be possible) must be negative; and therefore its predicate, the major term, will be distributed. In the premises, therefore, both the middle and the major terms should be distributed, which is impossible: e.g.,

Some M is not P; Some S is M: .'. Some S is not P.

Here, indeed, the major term is legitimately distributed (though the negative premise might have been the minor); but M, the middle term, is distributed in neither premise, and therefore there can be no conclusion.

Still, an exception may be made by admitting a bi-designate conclusion:

Some P is M; Some S is not M: .'. Some S is not some P.

(ii) If one premise be particular, so is the conclusion.

For, again, if both premises be affirmative, they only distribute one term, the subject of the universal premise, and this must be the middle term. The minor term, therefore, is undistributed, and the conclusion must be particular.

If one premise be negative, the two premises together can distribute only two terms, the subject of the universal and the predicate of the negative (which may be the same premise). One of these terms must be the middle; the other (since the conclusion is negative) must be the major. The minor term, therefore, is undistributed, and the conclusion must be particular.

(iii) From a particular major and a negative minor premise nothing can be inferred.

For the minor premise being negative, the major premise must be affirmative (5th Canon); and therefore, being particular, distributes the major term neither in its subject nor in its predicate. But since the conclusion must be negative (6th Canon), a distributed major term is demanded, e.g.,

Some M is P; No S is M: .'. ———

Here the minor and the middle terms are both distributed, but not the major (P); and, therefore, a negative conclusion is impossible.

Sec. 3. First Principle or Axiom of the Syllogism.—Hitherto in this chapter we have been analysing the conditions of valid mediate inference. We have seen that a single step of such inference, a Syllogism, contains, when fully expressed in language, three propositions and three terms, and that these terms must stand to one another in the relations required by the fourth, fifth, and sixth Canons. We now come to a principle which conveniently sums up these conditions; it is called the Dictum de omni et nullo, and may be stated thus:

Whatever is predicated (affirmatively or negatively) of a term distributed,

With which term another term can be (partly or wholly) identified,

May be predicated in like manner (affirmatively or negatively) of the latter term (or part of it).

Thus stated (nearly as by Whately in the introduction to his Logic) the Dictum follows line by line the course of a Syllogism in the First Figure (see chap. X. Sec. 2). To return to our former example: All authors are vain is the same as—Vanity is predicated of all authors; Cicero is an author is the same as—Cicero is identified as an author; therefore Cicero is vain, or—Vanity may be predicated of Cicero. The Dictum then requires: (1) three propositions; (2) three terms; (3) that the middle term be distributed; (4) that one premise be affirmative, since only by an affirmative proposition can one term be identified with another; (5) that if one premise be negative the conclusion shall be so too, since whatever is predicated of the middle term is predicated in like manner of the minor.

Thus far, then, the Dictum is wholly analytic or verbal, expressing no more than is implied in the definitions of 'Syllogism' and 'Middle Term'; since (as we have seen) all the General Canons (except the third, which is a still more general condition of formal proof) are derivable from those definitions. However, the Dictum makes a further statement of a synthetic or real character, namely, that when these conditions are fulfilled an inference is justified; that then the major and minor terms are brought into comparison through the middle, and that the major term may be predicated affirmatively or negatively of all or part of the minor. It is this real assertion that justifies us in calling the Dictum an Axiom.

Sec. 4. Whether the Laws of Thought may not fully explain the Syllogism without the need of any synthetic principle has, however, been made a question. Take such a syllogism as the following:

All domestic animals are useful; All pugs are domestic animals: .'. All pugs are useful.

Here (an ingenious man might urge), having once identified pugs with domestic animals, that they are useful follows from the Law of Identity. If we attend to the meaning, and remember that what is true in one form of words is true in any other form, then, all domestic animals being useful, of course pugs are. It is merely a case of subalternation: we may put it in this way:

All domestic animals are useful: .'. Some domestic animals (e.g., pugs) are useful.

The derivation of negative syllogisms from the Law of Contradiction (he might add) may be shown in a similar manner.

But the force of this ingenious argument depends on the participial clause—'having once identified pugs with domestic animals.' If this is a distinct step of the reasoning, the above syllogism cannot be reduced to one step, cannot be exhibited as mere subalternation, nor be brought directly under the law of Identity. If 'pug,' 'domestic,' and 'useful' are distinct terms; and if 'pug' and 'useful' are only known to be connected because of their relations to 'domestic': this is something more than the Laws of Thought provide for: it is not Immediate Inference, but Mediate; and to justify it, scientific method requires that its conditions be generalised. The Dictum, then, as we have seen, does generalise these conditions, and declares that when such conditions are satisfied a Mediate Inference is valid.

But, after all (to go back a little), consider again that proposition All pugs are domestic animals: is it a distinct step of the reasoning; that is to say, is it a Real Proposition? If, indeed, 'domestic' is no part of the definition of 'pug,' the proposition is real, and is a distinct part of the argument. But take such a case as this:

All dogs are useful; All pugs are dogs.

Here we clearly have, in the minor premise, only a verbal proposition; to be a dog is certainly part of the definition of 'pug.' But, if so, the inference 'All pugs are useful' involves no real mediation, and the argument is no more than this:

All dogs are useful; .'. Some dogs (e.g., pugs) are useful.

Similarly, if the major premise be verbal, thus:

All men are rational; Socrates is a man—

to conclude that 'Socrates is rational' is no Mediate Inference; for so much was implied in the minor premise, 'Socrates is a man,' and the major premise adds nothing to this.

Hence we may conclude (as anticipated in chap. vii. Sec. 3) that 'any apparent syllogism, having one premise a verbal proposition, is really an Immediate Inference'; but that, if both premises are real propositions, the Inference is Mediate, and demands for its explanation something more than the Laws of Thought.

The fact is that to prove the minor to be a case of the middle term may be an exceedingly difficult operation (chap. xiii. Sec. 7). The difficulty is disguised by ordinary examples, used for the sake of convenience.

Sec. 5. Other kinds of Mediate Inference exist, yielding valid conclusions, without being truly syllogistic. Such are mathematical inferences of Equality, as—

A = B = C .'. A = C.

Here, according to the usual logical analysis, there are strictly four terms—(1) A, (2) equal to B, (3) B, (4) equal to C.

Similarly with the argument a fortiori,

A > B > C .'. (much more) A > C.

This also is said to contain four terms: (1) A, (2) greater than B, (3) B, (4) greater than C. Such inferences are nevertheless intuitively sound, may be verified by trial (within the limits of sense-perception), and are generalised in appropriate axioms of their own, corresponding to the Dictum of the syllogism; as 'Things equal to the same thing are equal to one another,' etc.

Now, surely, this is an erroneous application of the usual logical analysis of propositions. Both Logic and Mathematics treat of the relations of terms; but whilst Mathematics employs the sign = for only one kind of relation, and for that relation exclusive of the terms; Logic employs the same signs (is or is not) for all relations, recognising only a difference of quality in predication, and treating every other difference of relation as belonging to one of the terms related. Thus Logicians read A—is—equal to B: as if equal to B could possibly be a term co-relative with A. Whence it follows that the argument A = B = C .'. A = C contains four terms; though everybody sees that there are only three.

In fact (as observed in chap. ii. Sec. 2) the sign of logical relation (is or is not), whilst usually adequate for class-reasoning (coinherence) and sometimes extensible to causation (because a cause implies a class of events), should never be stretched to include other relations in such a way as to sacrifice intelligence to formalism. And, besides mathematical or quantitative relations, there are others (usually considered qualitative because indefinite) which cannot be justly expressed by the logical copula. We ought to read propositions expressing time-relations (and inferences drawn accordingly) thus:

B—is before—C; A—is before—B: .'. A—is before—C.

And in like manner A—is simultaneous with—B; etc. Such arguments (as well as the mathematical) are intuitively sound and verifiable, and might be generalised in axioms if it were worth while: but it is not, because no method could be founded on such axioms.

The customary use of relative terms justifies some Mediate Inferences, as, The father of a father is a grand-father.

Some cases, however, that at first seem obvious, are really delusive unless further data be supplied. Thus A co-exists with B, B with C; .'. A with C—is not sound unless B is an instantaneous event; for where B is perdurable, A may co-exist with it at one time and C at another.

Again: A is to the left of B, B of C; .'. A of C. This may pass; but it is not a parallel argument that if A is north of B and B west of C, then A is north-west of C: for suppose that A is a mile to the north of B, and B a yard to the west of C, then A is practically north of C; at least, its westward position cannot be expressed in terms of the mariner's compass. In such a case we require to know not only the directions but the distances of A and C from B; and then the exact direction of A from C is an affair of mathematical calculation.

Qualitative reasoning concerning position is only applicable to things in one dimension of space, or in time considered as having one dimension. Under these conditions we may frame the following generalisation concerning all Mediate Inferences: Two terms definitely related to a third, and one of them positively, are related to one another as the other term is related to the third (that is, positively or negatively); provided that the relations given are of the same kind (that is, of Time, or Coinherence, or Likeness, or Equality).

Thus, to illustrate by relations of Time—

B is simultaneous with C; A is not simultaneous with B: .'. A is not simultaneous with C.

Here the relations are of the same kind but of different logical quality, and (as in the syllogism) a negative copula in the premises leads to a negative conclusion.

An examination in detail of particular cases would show that the above generalisation concerning all Mediate Inferences is subject to too many qualifications to be called an Axiom; it stands to the real Axioms (the Dictum, etc.) as the notion of the Uniformity of Nature does to the definite principles of natural order (cf. chap. xiii. Sec. 9).



CHAPTER X

CATEGORICAL SYLLOGISMS

Sec. 1. The type of logical, deductive, mediate, categorical Inference is a Syllogism directly conformable with the Dictum: as—

All carnivores (M) are excitable (P); Cats (S) are carnivores (M): .'. Cats (S) are excitable (P).

In this example P is predicated of M, a term distributed; in which term, M, S is given as included; so that P may be predicated of S.

Many arguments, however, are of a type superficially different from the above: as—

No wise man (P) fears death (M); Balbus (S) fears death (M): .'. Balbus (S) is not a wise man (P).

In this example, instead of P being predicated of M, M is predicated of P, and yet S is given as included not in P, but in M. The divergence of such a syllogism from the Dictum may, however, be easily shown to be superficial by writing, instead of No wise man fears death, the simple, converse, No man who fears death is wise.

Again:

Some dogs (M) are friendly to man (P); All dogs (M) are carnivores (S): .'. Some carnivores (S) are friendly to man (P).

Here P is predicated of M undistributed; and instead of S being included in M, M is included in S: so that the divergence from the type of syllogism to which the Dictum directly applies is still greater than in the former case. But if we transpose the premises, taking first

All dogs (M) are carnivores (P),

then P is predicated of M distributed; and, simply converting the other premise, we get—

Some things friendly to man (S) are dogs (M):

whence it follows that—

Some things friendly to man (S) are carnivores (P);

and this is the simple converse of the original conclusion.

Once more:

No pigs (P) are philosophers (M); Some philosophers (M) are hedonists (S): .'. Some hedonists (S) are not pigs (P).

In this case, instead of P being predicated of M distributed, M is predicated of P distributed; and instead of S (or part of it) being included in M, we are told that some M is included in S. Still there is no real difficulty. Simply convert both the premises, and we have:

No philosophers (M) are pigs (P); Some hedonists (S) are philosophers (M).

Whence the same conclusion follows; and the whole syllogism plainly conforms directly to the Dictum.

Such departures as these from the normal syllogistic form are said to constitute differences of Figure (see Sec. 2); and the processes by which they are shown to be unessential differences are called Reduction (see Sec. 6).

Sec. 2. Figure is determined by the position of the Middle Term in the premises; of which position there are four possible variations. The middle term may be subject of the major premise, and predicate of the minor, as in the first example above; and this position, being directly conformable to the requirements of the Dictum, is called the First Figure. Or the middle term may be predicate of both premises, as in the second of the above examples; and this is called the Second Figure. Or the middle term may be subject of both premises, as in the third of the above examples; and this is called the Third Figure. Or, finally, the middle term may be predicate of the major premise, and subject of the minor, as in the fourth example given above; and this is the Fourth Figure.

It may facilitate the recollection of this most important point if we schematise the figures thus:

I. II. III. IV.

M -P P -M M -P P -M / / / S -M S -M M -S M -S

The horizontal lines represent the premises, and at the angles formed with them by the slanting or by the perpendicular lines the middle term occurs. The schema of Figure IV. resembles Z, the last letter of the alphabet: this helps one to remember it in contrast with Figure I., which is thereby also remembered. Figures II. and III. seem to stand back to back.

Sec. 3. The Moods of each Figure are the modifications of it which arise from different combinations of propositions according to quantity and quality. In Figure I., for example, four Moods are recognised: A.A.A., E.A.E., A.I.I., E.I.O.

A. All M is P; A. All S is M: A. .'. All S is P.

E. No M is P; A. All S is M: E. .'. No S is P.

A. All M is P; I. Some S is M: I. .'. Some S is P.

E. No M is P; I. Some S is M: O. .'. Some S is not P.

Now, remembering that there are four Figures, and four kinds of propositions (A. I. E. O.), each of which propositions may be major premise, minor premise, or conclusion of a syllogism, it appears that in each Figure there may be 64 Moods, and therefore 256 in all. On examining these 256 Moods, however, we find that only 24 of them are valid (i.e., of such a character that the conclusion strictly follows from the premises), whilst 5 of these 24 are needless, because their conclusions are 'weaker' or less extensive than the premises warrant; that is to say, they are particular when they might be universal. Thus, in Figure I., besides the above 4 Moods, A.A.I. and E.A.O. are valid in the sense of being conclusive; but they are superfluous, because included in A.A.A. and E.A.E. Omitting, then, these 5 needless Moods, which are called 'Subalterns' because their conclusions are subaltern (chap. vii. Sec. 2) to those of other Moods, there remain 19 Moods that are valid and generally recognised.

Sec. 4. How these 19 Moods are determined must be our next inquiry. There are several ways more or less ingenious and interesting; but all depend on the application, directly or indirectly, of the Six Canons, which were shown in the last chapter to be the conditions of Mediate Inference.

(1) One way is to begin by finding what Moods of Figure I. conform to the Dictum. Now, the Dictum requires that, in the major premise, P be predicated of a term distributed, from which it follows that no Mood can be valid whose major premise is particular, as in I.A.I. or O.A.O. Again, the Dictum requires that the minor premise be affirmative ("with which term another is identified"); so that no Mood can be valid whose minor premise is negative, as in A.E.E. or A.O.O. By such considerations we find that in Figure I., out of 64 Moods possible, only six are valid, namely, those above-mentioned in Sec. 3, including the two subalterns. The second step of this method is to test the Moods of the Second, Third, and Fourth Figures, by trying whether they can be reduced to one or other of the four Moods of the First (as briefly illustrated in Sec. 1, and to be further explained in Sec. 6).

(2) Another way is to take the above six General or Common Canons, and to deduce from them Special Canons for testing each Figure: an interesting method, which, on account of its length, will be treated of separately in the next section.

(3) Direct application of the Common Canons is, perhaps, the simplest plan. First write out the 64 Moods that are possible without regard to Figure, and then cross out those which violate any of the Canons or Corollaries, thus:

[Transcriber's Note: Moods surrounded with square brackets were crossed out in the original text.]

AAA, [AAE] (6th Can. b). AAI. [AAO] (6th Can. b). [AEA] (6th Can. a) AEE, [AEI] (6th Can. a) AEO, [AIA] (Cor. ii.) [AIE] (6th Can. b) AII, [AIO] (6th Can. b) [AOA] (6th Can. a) [AOE] (Cor. ii.) [AOI] (6th Can. a) AOO.

Whoever has the patience to go through the remaining 48 Moods will discover that of the whole 64 only 11 are valid, namely:

A.A.A., A.A.I., A.E.E., A.E.O., A.I.I., A.O.O., E.A.E., E.A.O., E.I.O., I.A.I., O.A.O.

These 11 Moods have next to be examined in each Figure, and if valid in every Figure there will still be 44 moods in all. We find, however, that in the First Figure, A.E.E., A.E.O., A.O.O. involve illicit process of the major term (3rd Can.); I.A.I., O.A.O. involve undistributed Middle (4th Can.); and A.A.I., E.A.O. are subalterns. In the Second Figure all the affirmative Moods, A.A.A., A.A.I., A.I.I., I.A.I., involve undistributed Middle; O.A.O. gives illicit process of the major term; and A.E.O., E.A.O. are subalterns. In the Third Figure, A.A.A., E.A.E., involve illicit process of the minor term (3rd Can.); A.E.E., A.E.O., A.O.O., illicit process of the major term. In the Fourth Figure, A.A.A. and E.A.E. involve illicit process of the minor term; A.I.I., A.O.O., undistributed Middle; O.A.O. involves illicit process of the major term; and A.E.O. is subaltern.

Those moods of each Figure which, when tried by these tests, are not rejected, are valid, namely:

Fig. I.—A.A.A., E.A.E., A.I.I., E.I.O. (A.A.I., E.A.O., Subaltern);

Fig. II.—E.A.E., A.E.E., E.I.O., A.O.O. (E.A.O., A.E.O., Subaltern);

Fig. III.—A.A.I., I.A.I., A.I.I., E.A.O., O.A.O., E.I.O.;

Fig. IV.—A.A.I., A.E.E., I.A.I., E.A.O., E.I.O. (A.E.O., Subaltern).

Thus, including subaltern Moods, there are six valid in each Figure. In Fig. III. alone there is no subaltern Mood, because in that Figure there can be no universal conclusion.

Sec. 5. Special Canons of the several Figures, deduced from the Common Canons, enable us to arrive at the same result by a somewhat different course. They are not, perhaps, necessary to the Science, but afford a very useful means of enabling one to thoroughly appreciate the character of formal syllogistic reasoning. Accordingly, the proof of each rule will be indicated, and its elaboration left to the reader. There is no difficulty, if one bears in mind that Figure is determined by the position of the middle term.

Fig. I., Rule (a): The minor premise must be affirmative.

For, if not, in negative Moods there will be illicit process of the major term. Applying this rule to the eleven possible Moods given in Sec. 4, as remaining after application of the Common Canons, it eliminates A.E.E., A.E.O., A.O.O.

(b) The major premise must be universal.

For, if not, the minor premise being affirmative, the middle term will be undistributed. This rule eliminates I.A.I., O.A.O.; leaving six Moods, including two subalterns.

Fig. II. (a) One premise must be negative.

For else neither premise will distribute the middle term. This rule eliminates A.A.A., A.A.I., A.I.I., I.A.I.

(b) The major premise must be universal.

For else, the conclusion being negative, there will be illicit process of the major term. This eliminates I.A.I., O.A.O.; leaving six Moods, including two subalterns.

Fig. III. (a) The minor premise must be affirmative.

For else, in negative moods there will be illicit process of the major term. This rule eliminates A.E.E., A.E.O., A.O.O.

(b) The conclusion must be particular.

For, if not, the minor premise being affirmative, there will be illicit process of the minor term. This eliminates A.A.A., A.E.E., E.A.E.; leaving six Moods.

Fig. IV. (a) When the major premise is affirmative, the minor must be universal.

For else the middle term is undistributed. This eliminates A.I.I., A.O.O.

(b) When the minor premise is affirmative the conclusion must be particular.

Otherwise there will be illicit process of the minor term. This eliminates A.A.A., E.A.E.

(c) When either premise is negative, the major must be universal.

For else, the conclusion being negative, there will be illicit process of the major term. This eliminates O.A.O.; leaving six Moods, including one subaltern.

Sec. 6. Reduction is either—(1) Ostensive or (2) Indirect. Ostensive Reduction consists in showing that an argument given in one Mood can also be stated in another; the process is especially used to show that the Moods of the second, third, and fourth Figures are equivalent to one or another Mood of the first Figure. It thus proves the validity of the former Moods by showing that they also essentially conform to the Dictum, and that all Categorical Syllogisms are only superficial varieties of one type of proof.

To facilitate Reduction, the recognised Moods have all had names given them; which names, again, have been strung together into mnemonic verses of great force and pregnancy:

Barbara, Celarent, Darii, Ferioque prioris: Cesare, Camestres, Festino, Baroco, secundae: Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet: Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison.

In the above verses the names of the Moods of Fig. I. begin with the first four consonants B, C, D, F, in alphabetical order; and the names of all other Moods likewise begin with these letters, thus signifying (except in Baroco and Bocardo) the mood of Fig. I., to which each is equivalent, and to which it is to be reduced: as Bramantip to Barbara, Camestres to Celarent, and so forth.

The vowels A, E, I, O, occurring in the several names, give the quantity and quality of major premise, minor premise, and conclusion in the usual order.

The consonants s and p, occurring after a vowel, show that the proposition which the vowel stands for is to be converted either (s) simply or (p) per accidens; except where s or p occurs after the third vowel of a name, the conclusion: then it refers not to the conclusion of the given Mood (say Disamis), but to the conclusion of that Mood of the first Figure to which the given Mood is reduced (Darii).

M (mutare, metathesis) means 'transpose the premises' (as of Camestres).

C means 'substitute the contradictory of the conclusion for the foregoing premise,' a process of the Indirect Reduction to be presently explained (see Baroco, Sec. 8).

The other consonants, r, n, t (with b and d, when not initial), occurring here and there, have no mnemonic significance.

What now is the problem of Reduction? The difference of Figures depends upon the position of the Middle Term. To reduce a Mood of any other Figure to the form of the First, then, we must so manipulate its premises that the Middle Term shall be subject of the major premise and predicate of the minor premise.

Now in Fig. II. the Middle Term is predicate of both premises; so that the minor premise may need no alteration, and to convert the major premise may suffice. This is the case with Cesare, which reduces to Celarent by simply converting the major premise; and with Festino, which by the same process becomes Ferio. In Camestres, however, the minor premise is negative; and, as this is impossible in Fig. I., the premises must be transposed, and the new major premise must be simply converted: then, since the transposition of the premises will have transposed the terms of the conclusion (according to the usual reading of syllogisms), the new conclusion must be simply converted in order to prove the validity of the original conclusion. The process may be thus represented (s.c. meaning 'simply convert')

Camestres. Celarent.

All P is M; —— /—-> No M is S; c/ / s/ / No S is M: ——/ —-> All P is M:

s.c. .'. No S is P. <—————- No P is S.

The Ostensive Reduction of Baroco also needs special explanation; for as it used to be reduced indirectly, its name gives no indication of the ostensive process. To reduce it ostensively let us call it Faksnoko, where k means 'obvert the foregoing premise.' By thus obverting (k) and simply converting (s) (in sum, contrapositing) the major premise, and obverting the minor premise, we get a syllogism in Ferio, thus:

Baroco or Faksnoko. Ferio. contrap All P is M; ———————————-> No m (not-M) is P;

obv Some S is not M: ———————————-> Some S is m (not-M): .'. Some S is not P. .'. Some S is not P.

In Fig. III. the middle term is subject of both premises; so that, to reduce its Moods to the First Figure, it may be enough to convert the minor premise. This is the case with Darapti, Datisi, Felapton, and Ferison. But, with Disamis, since the major premise must in the First Figure be universal, we must transpose the premises, and then simply convert the new minor premise; and, lastly, since the major and minor terms have now changed places, we must simply convert the new conclusion in order to verify the old one. Thus:

Disamis. Darii.

Some M is P; —— /—-> All M is S; s./ / /c. / All M is S: ——/ —-> Some P is M:

s.c. .'. Some S is P. <——————- .'. Some P is S.

Bocardo, like Baroco, indicates by its name the indirect process. To reduce it ostensively let its name be Doksamrosk, and proceed thus:

Bocardo or Doksamrosk. Darii.

Some M is not P; ————— /————-> All M is S; / / / contrap / All M is S: —————/ ————-> Some p (not-P) is M:

convert & obvert .'. Some S is not P. <————————————- .'. Some p (not-P) is S.

In Fig. IV. the position of the middle term is, in both premises, the reverse of what it is in the First Figure; we may therefore reduce its Moods either by transposing the premises, as with Bramantip, Camenes, and Dimaris; or by converting both premises, the course pursued with Fesapo and Fresison. It may suffice to illustrate by the case of Bramantip:

Bramantip. Barbara.

All P is M; ————— /———> All M is S; / / All M is S: —————/ ———> All P is M:

convert per acc. .'. Some S is P. <———————————- .'. All P is S.

This case shows that a final significant consonant (s, p, or sk) in the name of any Mood refers to the conclusion of the new syllogism in the First Figure; since p in Bramantip cannot refer to that Mood's own conclusion in I.; which, being already particular, cannot be converted per accidens.

Finally, in Fig. I., Darii and Ferio differ respectively from Barbara and Celarent only in this, that their minor premises, and consequently their conclusions, are subaltern to the corresponding propositions of the universal Moods; a difference which seems insufficient to give them rank as distinct forms of demonstration. And as for Barbara and Celarent, they are easily reducible to one another by obverting their major premises and the new conclusions, thus:

Barbara. Celarent. obv. All M is P; ———————————-> No M is p (not-P);

All S is M: ———————————-> All S is M:

obv. .'. All S is P. <—————————- .'. No S is p (not-P).

There is, then, only one fundamental syllogism.

Sec. 7. A new version of the mnemonic lines was suggested in Mind No. 27, with the object of (1) freeing them from all meaningless letters, (2) showing by the name of each Mood the Figure to which it belongs, (3) giving names to indicate the ostensive reduction of Baroco and Bocardo. To obtain the first two objects, l is used as the mark of Fig. I., n of Fig II., r of Fig. III., t of Fig. IV. The verses (to be scanned discreetly) are as follows:

Balala, Celalel, Dalii, Felioque prioris:

{Faksnoko} Cesane, Camenes, Fesinon, { } secundae: { Banoco,}

Tertia, Darapri, Drisamis, Darisi, Ferapro,

Doksamrosk} }, Ferisor habet: Quarta insuper addit. Bocaro }

Bamatip, Cametes, Dimatis, Fesapto, Fesistot.

De Morgan praised the old verses as "more full of meaning than any others that ever were made"; and in defence of the above alteration it may be said that they now deserve that praise still more.

Sec. 8. Indirect reduction is the process of proving a Mood to be valid by showing that the supposition of its invalidity involves a contradiction. Take Baroco, and (since the doubt as to its validity is concerned not with the truth of the premises, but with their relation to the conclusion) assume the premises to be true. Then, if the conclusion be false, its contradictory is true. The conclusion being in O., its contradictory will be in A. Substituting this A. for the minor premise of Baroco, we have the premises of a syllogism in Barbara, which will be found to give a conclusion in A., contradictory of the original minor premise; thus: