Some S is P; ∴ Some P is S.Some poets are business-like; ∴ Some business-like men are poets.
Here the convertend and the converse say the same thing, and this is true if that is.
We have, then, two cases of simple conversion: of I. (as above) and of E. For E.:
No S is P; ∴ No P is S.No ruminants are carnivores; ∴ No carnivores are ruminants.
In converting I., the predicate (P) when taken as the new subject, being preindesignate, is treated as particular; and in converting E., the predicate (P), when taken as the new subject, is treated as universal, according to the rule inchap. v. § 1.
A. is the one case of conversion by limitation:
All S is P; ∴ Some P is S.All cats are grey in the dark; ∴ Some things grey in the dark are cats.
The predicate is treated as particular, when taking it for the new subject, according to the rule not to go beyond the evidence. To infer thatAll things grey in the dark are catswould be palpably absurd; yet no error of reasoning is commoner than the simple conversion of A. The validity of conversion by limitation may be shown thus: if,All S is P, then, by subalternation,Some S is P, and therefore, by simple conversion,Some P is S.
O. cannot be truly converted. If we take the proposition:Some S is not P, to convert this intoNo P is S, orSome P is not S, would break the rule inchap. vi. § 6; sinceS,undistributed in the convertend, would be distributed in the converse. If we are told thatSome men are not cooks, we cannot infer thatSome cooks are not men. This would be to assume that 'Some men' are identical with 'All men.'
By quantifying the predicate, indeed, we may convert O. simply, thus:
Some men are not cooks∴No cooks are some men.
And the same plan has some advantage in converting A.; for by the usual methodper accidens, the converse of A. being I., if we convert this again it is still I., and therefore means less than our original convertend. Thus:
All S is P ∴ Some P is S ∴ Some S is P.
Such knowledge, as thatAll S(the whole of it)is P, is too precious a thing to be squandered in pure Logic; and it may be preserved by quantifying the predicate; for if we convert A. to Y., thus—
All S is P ∴ Some P is all S—
we may reconvert Y. to A. without any loss of meaning. It is the chief use of quantifying the predicate that, thereby, every proposition is capable of simple conversion.
The conversion of propositions in which the relation of terms is inadequately expressed (seechap. ii., § 2) by the ordinary copula (isoris not) needs a special rule. To argue thus—
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(oris preceded by A).
Now, any relation between two terms may be viewed from either side—A: BorB: 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 turnsinto 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 AA 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 thatA is the brother of B, we can only infer thatB is either the brother or the sister of A. A list of all reciprocal relations is a desideratum of Logic.
§ 5. Obversion (otherwise called Permutation or Æquipollence) 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. § 3). To obvert affirmative propositions, then, the rule is—Insert thenegative sign, and for the predicate substitute its contradictory term.
A.All S is P ∴ No S is not-PAll men are fallible ∴ No men are infallible.I.Some S is P ∴ some S is not-PSome 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-PNo matter is destructible ∴ All matter is indestructible.O.Some S is not P ∴ Some S is not-PSome 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, ifNo S is not-Pbe 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 affirmativeAll S is P.Similarly, to obtain the obvertend ofAll S is not-P, apply the rule for obverting Affirmatives; and this yieldsNo S is P.
§ 6. Contrariety.—We have seen inchap. iv. § 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, ifwisemay be affirmed ofAll men, not-wisemust be denied; butAll men are not-wiseis the obverse ofNo 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 thatAll men are wise, we do not necessarily deny the attribute 'wise' of each and every man: to say thatNot all are wisemay mean no more than thatSome 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.
§ 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 thatAll men are wise, it is false thatSome men are not wise(equivalent by obversion toSome 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 toAll men are not-wise, is incompatible withSome men are wise, by the same principle of Contradiction.
But, again, if it be false thatAll men are wise, it is always true thatSome 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; andSome men are not-wise, is by obversion equivalent toSome men are not wise. Similarly, if it be false thatNo men are wise, which by obversion is equivalent toAll men are not-wise, then it is true at least thatSome 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 inchap. iv.that the contradictory of a given term includes all its contraries: 'not-blue,' for example, includes red and yellow. Hence, sinceThe sky is bluebecomes by obversion,The sky is not not-blue, we may also inferThe 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 thatThe sky is red, we cannot formally infer, thatThe sky is blue(cf.chap. iv. § 8).
§ 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 itbe true thatSome men are wise, it may also be true thatSome (other) men are not wise. This follows from the maxim inchap. vi. § 6, not to go beyond the evidence.
For if it be true thatSome men are wise, it may indeed be true thatAll are(this being the subalternans): and ifAll are, it is (by contradiction) false thatSome are not; but as we are only told thatSome men are, it is illicit to infer the falsity ofSome are not, which could only be justified by evidence concerningAll men.
But if it be false thatSome men are wise, it is true thatSome men are not wise; for, by contradiction, ifSome men are wiseis false,No men are wiseis true; and, therefore, by subalternation,Some men are not wiseis true.
§ 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:
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.
§ 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 SI.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 SO.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.1All A is B.Some A is B.No A is B.Some A is not B.Obverse.2No A is b.Some A is not b.All A is b.Some A is b.Converse.3Some B is A.Some B is A.No B is A.—Obverse of Converse.4Some B is not a.Some B is not a.All B is a.—Contrapositive.5No b is A.—Some b is A.Some b is A.Obverse of Contrapositive.6All b is a.—Some b is not a.Some b is not a.Converse of Obverse of Converse.7——Some a is B.—Obverse of Converse of Obverse of Converse.8——Some a is not b.—Converse of Obverse of Contrapositive.9Some a is b.———Obverse of Converse of Obverse of Contrapositive.10Some a is not B.———
In this tableaandbstand fornot-Aandnot-Band had better be read thus: forNo A is b, No A is not-B; forAll 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 inferenceAll A is B ∴ Some not-A is not B, distributesB(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 althoughAll A is Bdoes not distributeBin relation toA, it does distributeBin relation to somenot-A(namely, in relation to whatevernot-Aisnot-B). This is one reason why, in stating the rule inchap. vi. § 6, Ihave 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 thesuppositio.
§ 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. § 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, andIf A is B, C is not D) are contraries, but not contradictories; since both may be false (Cmay sometimes beD, 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. § 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 DIf A is B, C is not dI. Sometimeswhen A is B, C is DSometimeswhen A is B, C is not dE.If A is B, C is not DIf A is B, C is dO. Sometimeswhen A is B, C is not DSometimeswhen A is B, C is dConverse.Contrapositive.Sometimeswhen C is D, A is BIf C is d, A is not BSometimeswhen C is D, A is B(none)If C is D, A is not BSometimeswhen C is d, A is B(none)Sometimeswhen 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 formA 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 formEither 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.
§ 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 inMen are mortal. This is the relation of Substance and Attribute which we saw (chap. i. § 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; asFame is treacherous, The weather is changeable. But, in literature, sentences in which the adjective comes first are not uncommon, asLoud 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 intoThe applause was loud, etc.? If we do, and then proceed to convert, we getLoud 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 logicalproposition, 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, forDark is the fate of manis quite as natural asMan 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 leastin affirmative propositions) the more suitable subject, asCats 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, asKindness 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.
§ 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 beAll 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:
Fig. 1.Fig. 1.
An Universal Affirmative may also state a relation between two terms whose denotation is co-extensive. A definition always does this, asMan is a rational animal; and this, of course, we cannot represent by two distinctcircles, but at best by one with a thick circumference, to suggest that two coincide, thus:
Fig. 2.Fig. 2.
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 thatSome horned animals ruminate, and thatSome 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 saySome 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 inSome 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:
Fig. 3.Fig. 3.
The Universal Negative is sufficiently represented by a single Fig. (4): two circles mutually exclusive, thus:
Fig. 4.Fig. 4.
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 thatSome ruminants are not hollow-horned, thatSome horned animals are not domestic, and thatSome 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 thatAll hollow horned animals are ruminants, it is at the same time true thatNo 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; whereAll hollow-horned animals are ruminants, but we can only say thatSome 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 withAll hollow-horned animals are ruminants(A.). Now in conversion there is (by definition) no change of quality. The Converse, then, ofSome ruminants are not hollow-hornedmust be a negative proposition, having 'hollow-horned' for its subject, either in E. or O.; but these would be respectively the contrary and contradictory ofAll 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 ifSome 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.
§ 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 propositionAll X is all Y(U.) equates X and Y; the propositionAll 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 propositionNo X is Y(E.) cannot be said in any sense to equate X and Y; though, if we obvert it intoAll 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 saySome 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 everyrespect, 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. ButAll X is all Ymeans 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 propositionAll devils are uglyneed not imply that any such things as 'devils' really exist; but it certainly does imply thatDevils that are not ugly donot exist. Similarly, the propositionNo angels are uglyimplies thatAngels that are ugly do not exist. Therefore, writingxfor 'devils,'yfor 'ugly,' andȳfor 'not-ugly,' we may express A., the universal affirmative, thus:
A.xȳ= 0.
That is,x that is not y is nothing; or,Devils that are not-ugly do not exist. And, similarly, writingxfor 'angels' andyfor '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 asSome metaphysicians are obscure, and writingxfor 'metaphysicians,' andyfor '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 asSome giants are not cruel, and writingxfor 'giants' andyfor 'not-cruel,' we may express it thus:
O.xȳ= 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 xȳ and xy have no denotation; and I. and O. declare that xȳ and xy have denotation, or stand for something. Here, however, the resemblance to Hamilton's systemceases; for the Symbolic Logic, by operating upon more than two terms simultaneously, by adopting the algebraic signs of operations, +,-, ×, ÷ (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.)
§ 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 therelationsimplied 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 propositionAll 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 (xȳ= 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; sinceSome P is S(P existing) would involve the existence of S; which is contrary to the hypothesis.
Assuming that Universalsdo not, whilst Particularsdo, 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 certainsuppositioor universe, to which in a given discussion every argument shall refer, then, any propositions whose terms lie outside thatsuppositioare 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 speciousContradictories asNo centaurs play the lyre—Some centaurs do play the lyre; orAll 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 orsuppositiobe 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.)
§ 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: