CHAPTER III.
THE OPPOSITION OF PROPOSITIONS.[107]
[107] This chapter will be mainly concerned with the opposition of categorical propositions; and, as regards categoricals, complications arising in connexion with their existential interpretation will for the present be postponed.
79. The Square of Opposition.—In dealing with the subject of this chapter it will be convenient to begin with the ancient square of opposition which relates exclusively to the traditional schedule of propositions. It will, however, ultimately be found desirable to give more general accounts of what is to be understood by the terms contradictory, contrary, &c., so that they may be adapted to other schedules of propositions.
Two propositions are technically said to be opposed to each other when they have the same subject and predicate respectively, but differ in quantity or quality or both.[108]
[108] This definition, according to which opposed propositions are not necessarily incompatible with one another, is given by Aldrich (p. 53 in Mansel’s edition). Ueberweg (Logic, § 97) defines opposition in such a way as to include only contradiction and contrariety; and Mansel remarks that “subalterns are improperly classed as opposed propositions” (Aldrich, p. 59). Modern logicians, however, usually adopt Aldrich’s definition, and this seems on the whole the best course. Some term is wanted to signify the above general relation between propositions; and though it might be possible to find a more convenient term, no confusion is likely to result from the use of the term opposition if the student is careful to notice that it is here employed in a technical sense.
Taking the propositions SaP, SiP, SeP, SoP, in pairs, we find that there are four possible kinds of relation between them.
(1) The pair of propositions may be such that they can neither both be true nor both false. This is called contradictory opposition, and subsists between SaP and SoP, and between SeP and SiP. 110
(2) They may be such that whilst both cannot be true, both may be false. This is called contrary opposition. SaP and SeP.
(3) They may be such that they cannot both be false, but may both be true. Subcontrary opposition. SiP and SoP.
(4) From a given universal proposition, the truth of the particular having the same quality follows, but not vice versâ.[109] This is subaltern opposition, the universal being called the subalternant, and the particular the subalternate or subaltern. SaP and SiP. SeP and SoP.
[109] This result and some of our other results may need to be modified when, later on, account is taken of the existential interpretation of propositions. But, as stated in the note at the beginning of the chapter, all complications resulting from considerations of this kind are for the present put on one side.
All the above relations are indicated in the ancient square of opposition.
The doctrine of opposition may be regarded from two different points of view, namely, as a relation between two given propositions; and, secondly, as a process of inference by which one proposition being given either as true or as false, the truth or falsity of certain other propositions may be determined. Taking the second of these points of view, we have the following table:— 111
A being given true, E is false, I true, O false ;
E being given true, A is false, I false, O true ;
I being given true, A is unknown, E false, O unknown;
O being given true, A is false, E unknown, I unknown;
A being given false, E is unknown, I unknown, O true ;
E being given false, A is unknown, I true, O unknown;
I being given false, A is false, E true, O true ;
O being given false, A is true, E false, I true.
80. Contradictory Opposition.—The doctrine of opposition in the preceding section is primarily applicable only to the fourfold schedule of propositions ordinarily recognised. We must, however, look at the question from a wider point of view. It is, in particular, important that we should understand clearly the nature of contradictory opposition whatever may be the schedule of propositions with which we are dealing.
The nature of significant denial will be considered in some detail in the concluding [section] of this chapter. At this point it will suffice to say that to deny the truth of a proposition is equivalent to affirming the truth of its contradictory ; and vice versâ. The criterion of contradictory opposition is that of the two propositions, one must be true and the other must be false ; they cannot be true together, but on the other hand no mean is possible between them. The relation between two contradictories is mutual; it does not matter which is given true or false, we know that the other is false or true accordingly. Every proposition has its contradictory, which may however be more or less complicated in form.
It will be found that attention is almost inevitably called to any ambiguity in a proposition when an attempt is made to determine its contradictory. It has been truly said that we can never fully understand the meaning of a proposition until we know precisely what it denies; and indeed the problem of the import of propositions sometimes resolves itself at least partly into the question how propositions of a given form are to be contradicted.
The nature of contradictory opposition may be illustrated by reference to a discussion entered into by Jevons (Studies in 112 Deductive Logic, p. 116) as to the precise meaning of the assertion that a proposition—say, All grasses are edible—is false. After raising this question, Jevons begins by giving an answer, which may be called the orthodox one, and which, in spite of what he goes on to say, must also be considered the correct one. When I assert that a proposition is false, I mean that its contradictory is true. The given proposition is of the form A, and its contradictory is the corresponding O proposition—Some grasses are not edible. When, therefore, I say that it is false that all grasses are edible, I mean that some grasses are not edible. Jevons, however, continues, “But it does not seem to have occurred to logicians in general to enquire how far similar relations could be detected in the case of disjunctive and other more complicated kinds of propositions. Take, for instance, the assertion that ‘all endogens are all parallel-leaved plants.’ If this be false, what is true? Apparently that one or more endogens are not parallel-leaved plants, or else that one or more parallel-leaved plants are not endogens. But it may also happen that no endogen is a parallel-leaved plant at all. There are three alternatives, and the simple falsity of the original does not shew which of the possible contradictories is true.”
This statement is open to criticism in two respects. In the first place, in saying that one or more endogens are not parallel-leaved plants, we do not mean to exclude the possibility that no endogen is a parallel-leaved plant at all. Symbolically, Some S is not P does not exclude No S is P. The three alternatives are, therefore, at any rate reduced to the two first given. But in the second place, it is incorrect to speak of either of these alternatives as being by itself a contradictory of the original proposition. The true contradictory is the affirmation of the truth of one or other of these alternatives. If the original proposition is false, we certainly know that the new proposition limiting us to such alternatives is true, and vice versâ.
The point at issue may be made clearer by taking the proposition in question in a symbolic form. All S is all P is a condensed expression, resolvable into the form, All S is P and 113 all P is S. It has but one contradictory, namely, Either some S is not P or some P is not S.[110] If either of these alternatives holds good, the original statement must in its entirety be false; and, on the other hand, if the latter is false, one at least of these alternatives must be true. Some S is not P is not by itself a contradictory of All S is all P. These two propositions are indeed inconsistent with one another; but they may both be false.
[110] The contradictory of All S is all P may indeed be expressed in a different form, namely, S and P are not coextensive, but this has precisely the same force as the contradictory given in the text. We go on to shew that two different forms of the contradictory of the same proposition must necessarily be equivalent to one another.
It follows that we must reject Jevons’s further statement that “a proposition of moderate complexity has an almost unlimited number of contradictory propositions, which are more or less in conflict with the original. The truth of any one or more of these contradictories establishes the falsity of the original, but the falsity of the original does not establish the truth of any one or more of its contradictories.”[111] No doubt a proposition which is complicated in form may yield an indefinite number of other non-equivalent propositions the truth of any one of which is inconsistent with its own. It will also be true that its contradictory can be expressed in more than one form. But these forms will necessarily be equivalent to one another, since it is impossible for a proposition to have two or more non-equivalent contradictories. This position may be formally established as follows. Let Q and R be both contradictories of P. They will be equivalent if it can 114 be shewn that if Q then R, and if R then Q. Since P and Q are contradictories, we have If Q then not P, and since P and R are contradictories we have If not P then R. Combining these two propositions we have the conclusion If Q then R. If R then Q follows similarly. Hence we have established the desired result.
[111] It must be admitted that it has not been uncommon for logicians to use the word contradict somewhat loosely. For example, in the Port Royal Logic, we find the following: “Except the wise man (said the Stoics) all men are truly fools. This may be contradicted (1) by maintaining that the wise man of the Stoics was a fool as well as other men; (2) by maintaining that there were others, besides their wise man, who were not fools; (3) by affirming that the wise man of the Stoics was a fool, and that other men were not” (p. 140). The affirmation of any one of these three propositions certainly renders it necessary to deny the truth of the given proposition, but no one of them is by itself the contradictory of the given proposition. The true contradictory is the alternative proposition: Either the wise man of the Stoics is a fool or some other men are not fools.
In connexion with the same point, Jevons raises another question, in regard to which his view is also open to criticism. He says, “But the question arises whether there is not confusion of ideas in the usual treatment of this ancient doctrine of opposition, and whether a contradictory of a proposition is not any proposition which involves the falsity of the original, but is not the sole condition of it. I apprehend that any assertion is false which is made without sufficient grounds. It is false to assert that the hidden side of the moon is covered with mountains, not because we can prove the contradictory, but because we know that the assertor must have made the assertion without evidence. If a person ignorant of mathematics were to assert that ‘all involutes are transcendental curves,’ he would be making a false assertion, because, whether they are so or not, he cannot know it.” We should, however, involve ourselves in hopeless confusion were we to consider the truth or falsity of a proposition to depend upon the knowledge of the person affirming it, so that the same proposition would be now true, now false. It will be observed further that on Jevons’s view both the propositions S is P and S is not P would be false to a person quite ignorant of the nature of S. This would mean that we could not pass from the falsity of a proposition to the truth of its contradictory; and such a result as this would render any progress in thought impossible.
81. Contrary Opposition.—Seeking to generalise the relation between A and E, we might naturally be led to characterize the contrary of a given proposition by saying that it goes beyond mere denial, and sets up a further assertion as far as possible removed from the original assertion; so that, whilst the contradictory of a proposition denies its entire truth, its contrary may be said to assert its entire falsehood. A pair of contraries as thus defined may be regarded as standing at the opposite 115 ends of a scale on which there are a number of intermediate positions.
On this definition, however, the notion of contrariety cannot very satisfactorily be extended much beyond the particular case contemplated in the ordinary square of opposition. For if we have a proposition which cannot itself be regarded as standing at one end of a scale, but only as occupying an intermediate position, such proposition cannot be regarded as forming one of a pair of contraries. Plurative and numerically definite propositions may be taken as illustrations.
Hence if it is desired to define contrariety so that the conception may be generally applicable, the idea of two propositions standing, as it were, furthest apart from each other must be given up, and any two propositions may be described as contraries if they are inconsistent with one another without at the same time exhausting all possibilities. Contraries must on this definition always admit of a mean, but they may not always be what we should speak of as diametrical opposites, and any given proposition is not limited to a single contrary, but may have an indefinite number of non-equivalent contraries. At the same time, it will be observed that this definition still suffices to identify A and E as a pair of contraries, and as the only pair in the traditional scheme of opposition.
82. The Opposition of Singular Propositions.—Taking the proposition Socrates is wise, its contradictory is Socrates is not wise ;[112] and so long as we keep to the same terms, we cannot go beyond this simple denial. The proposition has, therefore, no formal contrary.[113] This opposition of singulars has been called secondary opposition (Mansel’s Aldrich, p. 56).
[112] This must be regarded as the correct contradictory from the point of view reached in the present chapter. The question becomes a little more difficult when the existential interpretation of propositions is taken into account.
[113] We can obtain what may be called a material contrary of the given proposition by making use of the contrary of the predicate instead of its mere contradictory; thus, Socrates has not a grain of sense. This is spoken of as material contrariety because it necessitates the introduction of a fresh term that could not be formally obtained out of the given proposition. It should be added that the distinction between formal and material contrariety might also be applied in the case of general propositions.
116 If, however, there is secondary quantification in a proposition having a singular subject, then we may obtain the ordinary square of opposition. Thus, if our original proposition is Socrates is always (or in all respects) wise, it is contradicted by the statement that Socrates is sometimes (or in some respects) not wise, while it has for its contrary, Socrates is never (or in no respects) wise, and for its subaltern, Socrates is sometimes (or in some respects) wise. It may be said that when we thus regard Socrates as having different characteristics at different times or under different conditions, our subject is not strictly singular, since it is no longer a whole indivisible. This is in a sense true, and we might no doubt replace our proposition by one having for its subject “the judgments or the acts of Socrates.” But it does not appear that this resolution of the proposition is necessary for its logical treatment.
The possibility of implicit secondary quantification, although no such quantification is explicitly indicated, is a not unfruitful source of fallacy in the employment of propositions having singular subjects. If we take such propositions as Browning is obscure, Epimenides is a liar, This flower is blue, and give as their contradictories Browning is not obscure, Epimenides is not a liar, This flower is not blue, shall we say that the original proposition or its contradictory is true in case Browning is sometimes (but not always) obscure, or in case Epimenides sometimes (but not often) speaks the truth, or in case the flower is partly (but not wholly) blue? There is certainly a considerable risk in such instances as these of confusing contradictory and contrary opposition, and this will be avoided if we make the secondary quantification of the propositions explicit at the outset by writing them in the form Browning is always (or sometimes) obscure, &c.[114] The contradictory will then be particular or universal accordingly.
[114] Or we might reduce them to the forms,—All (or some) of the poems of Browning are obscure, All (or some) of the statements of Epimenides are false, All (or some) of the surface of this flower is blue.
83. The Opposition of Modal Propositions.—So far in this chapter our attention has been confined to assertoric propositions. For the present, a very brief reference to the opposition 117 of modals will suffice. The main points involved will come up for further consideration [later on].
We have seen that the unconditionally universal proposition, whether expressed in the ordinary categorical form All S is P, or as a conditional If anything is S it is P, affirms a necessary connexion, by which is meant not merely that all the S’s are as a matter of fact P’s, but that it is inherent in their nature that they should be so. The statement that some S’s are not P’s is inconsistent with this proposition, but is not its contradictory, since both the propositions might be false: the S’s might all happen to be P’s, and yet there might be no law of connexion between S and P. The proposition in question being apodeictic will have for its contradictory a modal of another description, namely, a problematic proposition; and this may be written in the form S need not be P, or If anything is S still it need not be P, according as our original proposition is expressed as a categorical or as a conditional
Similarly, the contradictory of the hypothetical If P is true then Q is true, this proposition being interpreted modally, is If P is true still Q need not be true.
84. Extension of the Doctrine of Opposition.[115]—If we do not confine ourselves to the ordinary square of opposition, but consider any pair of propositions (whatever may be the schedule to which they belong), it becomes necessary to amplify the list of formal relations recognised in the square of opposition, and also to extend the meaning of certain terms. We may give the following classification:
[115] The illustrations given in this section presuppose a knowledge of immediate inferences. The section may accordingly on a first reading be postponed until part of the following chapter has been read.
(1) Two propositions may be equivalent or equipollent, each proposition being formally inferable from the other. Hence if either one of the propositions is true, the other is also true; and if either is false, the other is also false. For example, as will presently be shewn, All S is P and All not-P is not-S stand to each other in this relation.
(2) and (3) One of the two propositions may be formally inferable from the other, but not vice versâ. If we are 118 considering two given propositions Q and R, this yields two cases: for Q may carry with it the truth of R, but not conversely; or R may carry with it the truth of Q, but not conversely. Ordinary subaltern propositions with their subalternants fall into this class; and it will be convenient to extend the meaning of the term subaltern, so as to apply it to any pair of propositions thus related, whether they belong to the ordinary square of opposition or not. It will indeed be found that any pair of simple propositions of the forms A, E, I, O, that are subaltern in the extended sense, are equivalent to some pair that are subaltern in the more limited sense.[116] Thus All S is P and Some P is S, which are subaltern in the extended sense, are equivalent to All S is P and Some S is P. All S is P and Some not-S is not P are another pair of subalterns. Here it is not so immediately obvious in what direction we are to look for a pair of equivalent propositions belonging to the ordinary square of opposition. No not-P is S and Some not-P is not S will, however, be found to satisfy the required conditions.
[116] This will of course not hold good when we apply the term subaltern to compound propositions, e.g., to the pair Some S is not P and some P is not S, Some S is not P or some P is not S.
(4) The propositions may be such that they can both be true together, or both false, or either one true and the other false. For example, All S is P and All P is S. Such propositions may be called independent in their relation to one another.
(5) The propositions may be such that one or other of them must be true while both may be true. A pair of propositions which are thus related—for example, Some S is P and Some not-S is P—may, by an extension of meaning as in the case of the term subaltern, be said to be subcontrary. It can be shewn that any pair of subcontraries of the forms A, E, I, O are equivalent to some pair of subcontraries belonging to the ordinary square of opposition; thus, the above pair are equivalent to Some P is S and Some P is not S.
(6) The two propositions may be contrary to one another, in the sense that they cannot both be true, but can both be false. It can as before be shewn that any pair of contraries of 119 the forms A, E, I, O are equivalent to some pair of contraries in the more ordinary sense. For example, the contraries All S is P and All not-S is P are equivalent to No not-P is S and All not-P is S.
(7) The two propositions may be contradictory to one another according to the definition given in section [80], that is, they can neither both be true nor both false. All S is P and Some not-P is S afford an example outside the ordinary square of opposition. It will be observed that these two propositions are equivalent to the pair All S is P and Some S is not P.
Two propositions, then, may, in respect of inferability, consistency, or inconsistency, be formally (1) equivalent, (2) and (3) subaltern, (4) independent, (5) subcontrary, (6) contrary, (7) contradictory, the terms subaltern, &c., being used in the most extended sense. What pairs of categorical propositions (into which only the same terms or their contradictories enter) actually fall into these categories respectively will be shewn in sections [106] and [107].
These seven possible relations between propositions (taken in pairs) will be found to be precisely analogous to the seven possible relations between classes (taken in pairs) as brought out in a subsequent chapter (section [130]).
85. The Nature of Significant Denial.—It is desirable that, before concluding this chapter, we should briefly discuss a more fundamental question than any that has yet been raised, namely, the meaning and nature of negation and denial.
We observe, in the first place, that negation always finds expression in a judgment, and that it always involves the denial of some other judgment. The question therefore arises whether negation always presupposes an antecedent affirmation. This question must be answered in the negative if it is understood to mean that in order to be able to deny a proposition we must begin by regarding it as true. The proposition which we deny may be asserted or suggested by someone else; or it may occur to us as one of several possible alternatives; or it may be put in the form of a question.
It is, however, to be added that if a denial is to have any value as a statement of matter of fact, the corresponding 120 affirmation must be consistent with the meaning of the terms employed. Thus if A connotes m, n, p, and B connotes not-p, q, r, then the denial that A is B gives no real information respecting A. For the affirmation that A is B cannot be made by anyone who knows what is meant by A and B respectively. The same point may be otherwise expressed by saying that just as the affirmation of a verbal proposition is insignificant regarded as a real affirmation concerning the subject (and not merely as an affirmation concerning the meaning to be attached to the subject-term), so the denial of a contradiction in terms is insignificant from the same point of view. Such a denial yields merely what is tautologous and practically useless.
For example, the denial that the soul is a ship in full sail is insignificant regarded as a statement of matter of fact; for such denial gives no information to anyone who is already acquainted with the meaning of the terms involved.
The nature of logical negation is of so fundamental and ultimate a character that any attempt to explain it is apt to obscure rather than to illumine. It cannot be expressed more simply and clearly than by the laws of contradiction and excluded middle: a judgment and its contradictory cannot both be true; nor can they both be false.
Because every negative judgment involves the denial of some other judgment, it has been argued that a negative judgment such as S is not P is primarily a judgment concerning the positive judgment S is P, not concerning the subject S ; and hence that a negative judgment is not co-ordinate with a positive judgment, but dependent upon it.[117]
[117] Compare Sigwart, Logic, i. pp. 121, 2.
Passing by the point that a positive judgment also involves the denial of some other judgment, we may observe that a distinction must be drawn between “S is P” is not true (which is a judgment about S is P), and S is not P (which is a judgment about S). Denial no doubt presents itself to the mind most simply in the first of these two forms. But in contradicting a given judgment our method usually is to establish another judgment involving the same terms which stands to the given judgment in the relation expressed by the laws of contradiction 121 and excluded middle; and when we oppose the judgment S is not P to the judgment S is P we have reached the less direct mode of denial in which we have again a judgment concerning our original subject.
The example here taken tends perhaps to obscure the point at issue because the distinction between “S is P” is not true and S is not P may appear to be so slight as to be immaterial. That there is a real distinction will, however, appear clear if we take such pairs of propositions as “All S is P” is not true, Some S is not P ; “All S is all P” is not true, Either some S is not P or some P is not S ; “If any P is Q it is R” is not true, P might be Q without being R.
It will be convenient if in general we understand by the contradictory of a proposition P not its simple denial “P is not true,” but the proposition Q involving the same terms, which is formally so related to P, that P and Q cannot both be true or both false.
Sigwart observes that the ground of a denial may be either (a) a deficiency, or (b) an opposition.[118] I may, for example, pronounce that a certain thing does not possess a given attribute either (a) because I fail to discover the presence of the attribute, or (b) because I recognise the presence of some other attribute which I know to be incompatible with the one suggested.
[118] Logic, i. p. 127.
This distinction may be illustrated by one or two further examples. Thus, I may deny that a man travelled by a certain train either (a) because I searched the train through just before it started and found he was not there, or (b) because I know he was elsewhere when the train started,—I may, for instance, have seen him leave the station at the same moment in another train in the opposite direction. Similarly, I may deny a universal proposition either (a) because I have discovered certain instances of its not holding good, or (b) because I accept another universal proposition which is inconsistent with it. Again, I may deny that a given metal, or the metal contained in a certain salt, is copper (a) on the ground of deficiency, namely, that it does not answer to a certain test, or (b) on the ground 122 of opposition, namely, that I recognise it to be another metal, say, zinc.
The ground of denial always involves something positive, for example, the search through the train, or the discovery of individual exceptions. But it is clear that when we establish an opposition we get a result that is itself positive in a way that is not the case when we merely establish a deficiency. This may lead up to a brief examination of a doctrine of the nature of significant denial that is laid down by Mr Bosanquet.
Mr Bosanquet holds that bare denial has in itself no significance, and he apparently denies that the contradictory of a judgment, apart from the grounds on which it is based, conveys any information.[119] For the meaning of significant negation we must, he says, look to the grounds of the negation; or else for contradictory denial we must substitute contrary denial. As a consequence, a judgment can, strictly and properly, “only be denied by another judgment of the same nature; a singular by a singular judgment, a generic by a generic, a hypothetical by a hypothetical”;[120] and, presumably, a particular by a particular, an apodeictic by an apodeictic.
[119] Logic, i. p. 305.
[120] Ibid, p. 383.
It is of course true that every denial must have some kind of positive basis, but it is also necessary that a judgment should be distinguished from the grounds on which it is based. We cannot say that a judgment of given content is different for two people because they accept it on different grounds; and if it is said that this is to beg the question, since a difference in ground constitutes in itself a difference in content, the reply is that such a doctrine must render the content of every judgment so elusive and uncertain as to make it impossible of analysis.
The view that identifies the denial of a judgment with its contrary not only mixes up a judgment with its grounds, but also overlooks one of the two principal grounds of denial. When the ground of negation is an opposition, we may no doubt be said to reach denial through the contrary, though we should still hold that the denial is in itself something less than the contrary; but when the ground of denial is a deficiency, even this cannot be allowed. If, for example, I have arrived 123 at the conclusion that a man did not start by a given train because I searched the train through before its departure and did not find him there; or if I conclude that a given metal is not copper because it does not satisfy a given test; I have obtained no contrary judgment, and yet my denial is justified.
These would be cases of bare denial. I have gained no positive knowledge of the whereabouts of the man in question, nor can I identify the given metal. But surely it cannot be seriously maintained that the denial is meaningless or useless, say, to a detective in the first instance, or to an analytical chemist in the second.
Of course we seldom or never rest content with bare denial. The contrary rather than the contradictory represents our ultimate aim. But it is often the case that, temporarily at any rate, we cannot get beyond bare denial; and we ought not to consider that we have altogether failed to make progress when all that we have achieved is the exclusion of a possible alternative or the overthrow of a false theory. Recent researches, for example, into the origin of cancer have led to no positive results; but it is claimed for them that by destroying preconceived ideas on the subject they have cleared the way for future advance. Will anyone affirm that this was not worth doing or that the time spent on the researches was wasted?
Looking at the question from another point of view, it is surely absurd to say that we cannot deny a universal unless we are able to substitute another universal in its place. Various algebraical formulae have from time to time been suggested as necessarily yielding a prime number. They have all been overthrown, and no valid formula has been established in their place. But knowledge that these formulae are false is not quite appropriately described as ignorance.
Elsewhere Mr Bosanquet says that mere enumerative exceptions are futile and cannot touch the essence of the unconditionally universal judgments they apparently oppose.[121] He appears to have in view cases where nothing more than some modification of the original judgment is shewn to be 124 necessary. But even so the enumerative exceptions have overthrown the original judgment. No doubt a scientific law which has had a great amount of evidence in its favour is likely to contain elements of truth even if it is not altogether true; and the object of a man of science who overthrows a law will be to set up some other law in its place. But, says Mr Bosanquet, even if the first generic judgment were a sheer blunder and confusion, as has been the case from time to time with judgments propounded in science, it is scarcely possible to rectify the confusion except by substituting for it the true positive conceptions that arise out of the cases which overthrew it.” Here it is admitted that the exceptions do overthrow the law, and the rest of the argument is surely an instance of ignoratio elenchi. It is moreover a pure, and in many cases an unjustifiable, assumption that the cases which suffice to overthrow a false law will also suffice as the basis for the establishment of a true law in its place.
[121] Logic, i. p. 313.
EXERCISES.
86. Examine the nature of the opposition between each pair of the following propositions:—None but Liberals voted against the motion; Amongst those who voted against the motion were some Liberals; It is untrue that those who voted against the motion were all Liberals. [K.]
87. If some were used in its ordinary colloquial sense, how would the scheme of opposition between propositions have to be modified? [J.]
88. Explain the technical terms “contradictory” and “contrary” applying them to the following propositions: Few S are P ; He was not the only one who cheated ; Two-thirds of the army are abroad. [V.]
89. Give the contradictory of each of the following propositions:—Some but not all S is P ; All S is P and some P is not R ; Either all S is P or some P is not R ; Wherever the property A is found, either the property B or the property C will be found with it, but not both of them together. [K.]
125 90. Give the contradictory, and also a contrary, of each of the following propositions:
Half the candidates failed;
Wellington was always successful both in beating the enemy and in utilising his victory;
All men are either not knaves or not fools;
All but he had fled;
Few of them are honest;
Sometimes all our efforts fail;
Some of our efforts always fail. [L.]
91. Give the contradictory, and also a contrary, of each of the following propositions:
I am certain you are wrong;
Sometimes when it rains I find myself without an umbrella;
Whatever you say, I shall not believe you. [C.]
92. Define the terms subaltern, subcontrary, contrary, contradictory, in such a way that they may be applicable to pairs of propositions generally, and not merely to those included in the ordinary square of opposition. Do the above exhaust the formal relations (in respect of inferability, consistency, or inconsistency) that are possible between pairs of propositions?
Illustrate your answer by considering the relation (in respect of inferability, consistency, or inconsistency) between each of the following propositions and each of the remainder: S and P are coincident ; Some S is P ; Not all S is P ; Either some S is not P or some P is not S ; Anything that is not P is S. [K.]
93. Given that the propositions X and Z are contradictory, Y and V contradictory, and X and Y contrary, shew (without assuming that X, Y, V, Z belong to the ordinary schedule of propositions) that the relations of V to X, Z to Y, V to Z are thereby deducible. [J.]
94. Prove formally that if two propositions are equivalent, their contradictories will also be equivalent. [K.]
95. Examine the doctrine that a judgment can properly be denied only by another judgment of the same type. Illustrate by reference to (a) universal judgments, (b) particular judgments (c) disjunctive judgments, (d) apodeictic judgments. [K.]