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.]
CHAPTER IV.
IMMEDIATE INFERENCES.122
122 In this chapter we concern ourselves mainly with the traditional scheme of propositions, and except where an explicit statement is made to the contrary we proceed on the assumption that each class represented by a simple term exists in the universe of discourse, while at the same time it does not exhaust that universe. This assumption appears to have been made implicitly in the traditional treatment of logic.
96. The Conversion of Categorical Propositions.—By conversion, in a broad sense, is meant a change in the position of the terms of a proposition.123 Logic, however, is concerned with conversion only in so far as the truth of the new proposition obtained by the process is a legitimate inference from the truth of the original proposition. For example, the change from All S is P to All P is S is not a legitimate logical conversion, since the truth of the latter proposition does not follow from the truth of the former. In other words, logical conversion is a case of immediate inference, which may be defined as the inference of a proposition from a single other proposition.124
123 Ueberweg (Logic, § 84) defines conversion thus. Compare also De Morgan, Formal Logic, p. 58. In geometry, all equiangular triangles are equilateral would be regarded as the converse of all equilateral triangles are equiangular. In this sense of the term conversion, which is its ordinary non-technical sense, we may say—as we frequently do say—“Yes, such and such a proposition is true; but its converse is not true.”
124 In discussing immediate inferences we “pursue the content of an enunciated judgment into its relations to judgments not yet uttered” (Lotze). Instead of “immediate inferences” Professor Bain prefers to speak of “equivalent propositional forms.” It will be found, however, that the new propositions obtained by immediate inference are not always equivalent to the original proposition, e.g., in conversion per accidens. Miss Jones suggests the term eduction as a synonym for immediate inference (General Logic, p. 79); and she then distinguishes between eversions and transversions, an eversion being an eduction from categorical form to categorical, or from hypothetical to hypothetical, &c., and transversion an eduction from categorical form to conditional, or from conditional to categorical, &c. For the present we shall be concerned with eversions only.
127 The simplest form of logical conversion, and that which is understood in logic when we speak of conversion without further qualification, may be defined as a process of immediate inference in which from a given proposition we infer another, having the predicate of the original proposition for subject, and its subject for predicate. Thus, given a proposition having S for its subject and P for its predicate, our object in the process of conversion is to obtain by immediate inference a new proposition having P for its subject and S for its predicate. The original proposition may be called the convertend, and the inferred proposition the converse.
The process will be valid if the two following rules are observed:
(1) The converse must be the same in quality as the convertend
(Rule of Quality);
(2) No term must be distributed in the converse unless it was
distributed in the convertend (Rule of Distribution).
Applying these rules to the four fundamental forms of proposition, we have the following table:—
Convertend. | Converse. |
| All S is P. A. | Some P is S. I. |
| Some S is P. I. | Some P is S. I. |
| No S is P. E. | No P is S. E. |
| Some S is not P. O. | (None) |
It is desirable at this stage briefly to call attention to a point which will receive fuller consideration later on in connexion with the reading of propositions in extension and intension, namely, that, generally speaking, in any judgment we have naturally before the mind the objects denoted by the 128 subject, but the qualities connoted by the predicate. In the process of converting a proposition, however, the extensive force of the predicate is made prominent, and an import is given to the predicate similar to that of the subject. At the same time the distribution of the predicate has to be made explicit in thought. It is in passing from the predicative to the class reading (e.g. from all men are mortal to all men are mortals), that the difficulty sometimes found in correctly converting propositions probably consists. We shall at any rate do well to recognise that conversion and other immediate inferences usually involve a distinct mental act of the above nature.
It follows from what has been said above that some propositions lend themselves to the process of conversion much more readily than others. When the predicate of a proposition is a substantive little or no effort is required in order to convert the proposition; more effort is necessary when the predicate is an adjective; and still more when in the original proposition the logical predicate is not expressed separately at all, as in propositions secundi adjacentis. Compare for purposes of conversion the propositions, Whales are mammals, Lions are carnivorous, A stitch in time saves nine. In some cases, in consequence of the awkwardness of changing adjectives and verbal predicates into substantives, the conversion of a proposition appears to be a very artificial production.125
125 Compare Sigwart, Logic, i. p. 340.
97. Simple Conversion and Conversion per accidens.—It will be observed that in the case of I and E, the converse is of the same form as the original proposition; moreover we do not lose any part of the information given us by the convertend, and we can pass back to it by re-conversion of the converse. The convertend and its converse are accordingly equivalent propositions. The conversion under these conditions is said to be simple.
In the case of A, it is different; we cannot pass by immediate inference from All S is P to All P is S, inasmuch as P is distributed in the latter of these propositions but undistributed in the former. Hence, although we start with a universal proposition, we obtain by conversion a particular 129 proposition only,126 and by no means of operating upon the converse can we regain the original proposition. The convertend and its converse are accordingly non-equivalent propositions. The conversion in this case is called conversion per accidens,127 or conversion by limitation.128
126 The failure to recognise or to remember that universal affirmative propositions are not simply convertible is a fertile source of fallacy.
127 The conversion of A is said by Mansel to be called conversion per accidens ‘because it is not a conversion of the universal per se, but by reason of its containing the particular. For the proposition ‘Some B is A’ is primarily the converse of ‘Some A is B,’ secondarily of ‘All A is B’” (Mansel’s Aldrich, p. 61). Professor Baynes seems to deny that this is the correct explanation of the use of the term (New Analytic of Logical Forms, p. 29); but however this may be, we certainly need not regard the converse of A as necessarily obtained through its subaltern. It is possible to proceed directly from All A is B to Some B is A without the intervention of Some A is B.
128 Simple conversion and conversion per accidens are also called respectively conversio pura and conversio impura. Compare Lotze, Logic, § 79.
For concrete illustrations of the process of conversion we may take the propositions,—A stitch in time saves nine; None but the brave deserve the fair. The first of these may be written in the form,—All stitches in time are things that save nine stitches. This, being an A proposition, is only convertible per accidens, and we have for our converse,—Some things that save nine stitches are stitches in time. The second of the given propositions may be written,—No one who is not brave is deserving of the fair. This, being an E proposition, may be converted simply, giving, No one deserving of the fair is not brave. Our results may be expressed in a more natural form as follows: One way of saving nine stitches is by a stitch in time; No one deserving of the fair can fail to be brave.
No difficulty ought ever to be found in converting or performing other immediate inferences upon any given proposition when once it has been brought into the traditional logical form, its quantity and quality being determined, its subject, copula, and predicate being definitely distinguished from one another, and its predicate as well as its subject being read in extension. If, however, this rule is neglected, mistakes are pretty sure to follow.
130 98. Inconvertibility of Particular Negative Propositions.—It follows immediately from the rules of conversion given in section 96 that Some S is not P does not admit of ordinary conversion; for S which is undistributed in the convertend would become the predicate of a negative proposition in the converse, and would therefore be distributed.129 It will be shewn presently, however, that although we are unable to infer anything about P in this case, we are able to draw an inference concerning not-P.
Jevons considers that the fact that the particular negative proposition is incapable of ordinary conversion “constitutes a blot in the ancient logic” (Studies in Deductive Logic, p. 37). There is, however, no sufficient justification for this criticism. We shall find subsequently that just as much can be inferred from the particular negative as from the particular affirmative (since the latter unlike the former does not admit of contraposition). No logic, symbolic or other, can actually obtain more from the given information than the ancient logic does. It has been suggested that what Jevons means is that the inconvertibility of O results in a want of symmetry and that logicians ought specially to aim at symmetry. With this last contention we may heartily agree. The want of symmetry, however, in the case before us is apparent only and results from taking an incomplete view. It will be found that symmetry reappears later on.130
99. Legitimacy of Conversion.—Aristotle proves the conversion of E indirectly, as follows;131 No S is P, therefore, No P is S ; for if not, Some individual P, say Q, is S ; and hence Q is both S and P ; but this is inconsistent with the original proposition.
131 “By the method called ἔκθεσις, i.e., by the exhibition of an individual instance.” See Mansel’s Aldrich, pp. 61, 2.
Having shewn that the simple conversion of E is legitimate, we can prove that the conversion per accidens of A is also legitimate. All S is P, therefore, Some P is S ; for, if not, No P is S, and therefore (by conversion) No S is P ; but this 131 is inconsistent with the original supposition. The legitimacy of the simple conversion of I follows similarly.
The above proof appears to involve nothing beyond the principles of contradiction and excluded middle. The proof itself, however, is not satisfactory; for it practically assumes the validity of the very process that it seeks to justify, that is to say, it assumes the equivalence of the propositions S is Q and Q is S.
A better justification of the process of conversion may be obtained by considering the class relations involved in the propositions concerned. Thus, taking an E proposition, it is self-evident that if one class is entirely excluded from another class, this second class is entirely excluded from the first.132 In the case of an A proposition it is clear on reflection that the statement All S is P is consistent with either of two relations of the classes S and P, namely, S and P coincident, or P containing S and more besides, and further that these are the only two possible relations with which it is consistent. It is self-evident that in each of these cases Some P is S ; and hence the inference by conversion from an A proposition is shewn to be justified.133 In the case of an O proposition, if we consider all the relationships of classes in which it holds good, we find that nothing is true of P in terms of S in all of them. Hence O is inconvertible.134 The inconvertibility of O can also be established 132 by shewing that Some S is not P is compatible with every one of the following propositions—All P is S, Some P is S, No P is S, Some P is not S.
132 It is impossible to agree with Professor Bain, who would establish the rules of conversion by a kind of inductive proof. He writes as follows:—“When we examine carefully the various processes in Logic, we find them to be material to the very core. Take Conversion. How do we know that, if No X is Y, No Y is X? By examining cases in detail, and finding the equivalence to be true. Obvious as the inference seems on the mere formal ground, we do not content ourselves with the formal aspect. If we did, we should be as likely to say, All X is Y gives All Y is X ; we are prevented from this leap merely by the examination of cases” (Logic, Deduction, p. 251). But no one would on reflection maintain it to be self-evident that the simple conversion of A is legitimate; for when the case is put to us we recognise immediately that the contradictory of All P is S is compatible with All S is P. On the other hand, no one can deny that in the case of E the legitimacy of the process of conversion is self-evident.
133 Compare section 126, where this and other similar inferences are illustrated by the aid of the Eulerian diagrams.
100. Table of Propositions connecting any two terms.—There are—connecting any two terms S and P—eight propositions of the forms A, E, I, O, namely, four with S as subject, and four with P as subject. The results at which we have arrived concerning the conversion of propositions shew that of these eight, the two E propositions are equivalent to one another, and that the same is true of the two I propositions, E and I being simply convertible; also that these are the only equivalences obtainable. We have, therefore, the following table of propositions connecting any two terms S and P:—
| SaP, |
| PaS, |
| SeP = PeS, |
| SiP = PiS, |
| SoP, |
| PoS. |
The pair of propositions SaP and PaS are independent (see section 84); and the same is true of the pairs SoP and PoS, SaP and PoS, PaS and SoP. The first pair taken together indicate that the classes S and P are coextensive, and they may be called complementary propositions. The second pair taken together indicate that the classes S and P are neither coextensive nor either included within the other; they may be called sub-complementary propositions. The third pair taken together indicate that the class S is included within the class P but that it does not exhaust that class; they may be called contra-complementary propositions. The fourth pair taken together indicate that the class P is included within the class S but that it does not exhaust that class; they are, therefore, also contra-complementary.135
135 The new technical terms here introduced have been suggested by Mr Johnson.
The above table will be supplemented in section 106 by a table of propositions connecting any two terms and their 133 contradictories, S, P, not-S, not-P. It will then be found that we have a symmetry that is at present wanting.
101. The Obversion of Categorical Propositions.136—Obversion is a process of immediate inference in which the inferred proposition (or obverse), whilst retaining the original subject, has for its predicate the contradictory of the predicate of the original proposition (or obvertend). This process is legitimate for a proposition of any form if at the same time the quality of the proposition is changed. The inferred proposition is, moreover, in all cases equivalent to the original proposition, so that we can always pass back from the obverse to the obvertend.
136 The process of immediate inference discussed in this section has been called by a good many different names. The term obversion, which is used by Professor Bain, is the most convenient. Other names which have been used are permutation (Fowler), aequipollence (Ueberweg), infinitation (Bowen), immediate inference by private conception (Jevons), contraversion (De Morgan), contraposition (Spalding). Professor Bain distinguishes between formal obversion and material obversion. By formal obversion is meant the kind of obversion discussed in the above section, and this is the only kind of obversion that can properly be recognised by the formal logician. Material obversion is described as the process of making “obverse inferences which are justified only on an examination of the matter of the proposition” (Logic, vol. i., p. 111); and the following are given as examples—“Warmth is agreeable; therefore, cold is disagreeable. War is productive of evil; therefore, peace is productive of good. Knowledge is good; therefore, ignorance is bad.” It is very doubtful if these are legitimate inferences, formal or otherwise. The conclusions appear to require quite independent investigations to establish them. Apart from this, however, it is a mistake to regard the process as analogous to formal obversion. In the latter, the inferred proposition has the same subject as the original proposition, whilst its quality is different; but neither of these conditions is fulfilled in the above examples. The process is really more akin to the immediate inference presently to be discussed under the name of inversion.
We have the following table:—
Obvertend. | Obverse. |
| All S is P. A. | No S is not-P. E |
| Some S is P. I. | Some S is not not-P. O. |
| No S is P. E. | All S is not-P. A. |
| Some S is not P. O. | Some S is not-P. I. |
134 It will be observed that the obversion of All S is P depends upon the principle of contradiction, which tells us that if anything is P then it is not not-P; but that we pass back from No S is not-P to All S is P by the principle of excluded middle, which tells us that if anything is not not-P then it is P. The remaining inferences by obversion also depend upon one or other of these two principles.
102. The Contraposition of Categorical Propositions.137—Contraposition may be defined as a process of immediate inference in which from a given proposition another proposition is inferred having for its subject the contradictory of the original predicate. Thus, given a proposition having S for its subject and P for its predicate, we seek to obtain by immediate inference a new proposition having not-P for its subject.
137 This form of immediate inference is called by some logicians conversion by negation ; Miss Jones suggests the name contraversion. More strictly we might speak of conversion by contraposition. The word contrapositive was used by Boethius for the opposite of a term (e.g., not-A), the word contradictory being confined to propositional forms; and the passage from All S is P to All not-P is not-S was called Conversio per contrapositionem terminorum. In this usage Boethius was followed by the medieval logicians. Compare Minto, Logic, pp. 151, 153.
It will be observed that in the above definition it is left an open question whether the contrapositive of a proposition has the original subject or the contradictory of the original subject for its predicate; and every proposition which admits of contraposition will accordingly have two contrapositives, each of which is the obverse of the other. For example, in the case of All S is P there are the two forms No not-P is S and All not-P is not-S. For many purposes the distinction may be practically neglected without risk of confusion. It will be observed, however, that when not-S is taken as the predicate of the contrapositive, the quality of the original proposition is preserved and there is greater symmetry.138 On the other hand, 135 if we regard contraposition as compounded out of obversion and conversion in the manner indicated in the following paragraph, the form with S as predicate is the more readily obtained. Perhaps the best solution (in cases in which it is necessary to mark the distinction) is to speak of the form with not-S as predicate as the full contrapositive, and the form with S as predicate as the partial contrapositive.139
138 The following is from Mansel’s Aldrich, p. 61,—“Conversion by contraposition, which is not employed by Aristotle, is given by Boethius in his first book, De Syllogismo Categorico. He is followed by Petrus Hispanus. It should be observed, that the old logicians, following Boethius, maintain that in conversion by contraposition, as well as in the others, the quality should remain unchanged. Consequently the converse of ‘All A is B’ is ‘All not-B is not-A,’ and of ‘Some A is not B,’ ‘Some not-B is not not-A.’ It is simpler, however, to convert A into E, and O into I, (‘No not-B is A,’ ‘Some not-B is A’), as is done by Wallis and Archbishop Whately; and before Boethius by Apuleius and Capella, who notice the conversion, but do not give it a name. The principle of this conversion may be found in Aristotle, Top. II. 8. 1, though he does not employ it for logical purposes.”
139 In previous editions the form with S as predicate was called the contrapositive, and the form with not-S as predicate was called the obverted contrapositive.
The following rule may be adopted for obtaining the full contrapositive of a given proposition:—Obvert the original proposition, then convert the proposition thus obtained, and then once more obvert. For given a proposition with S as subject and P as predicate, obversion will yield an equivalent proposition with S as subject and not-P as predicate; the conversion of this will make not-P the subject and S the predicate; and a repetition of the process of obversion will yield a proposition with not-P as subject and not-S as predicate.
Applying this rule, we have the following table:—
| Original Proposition | Obverse | Partial Contrapositive | Full Contrapositive |
| All S is P. A. | No S is not-P. E. | No not-P is S. E. | All not-P is not-S A. |
| Some S is P. I. | Some S is not not-P. O. | (None.) | (None.) |
| No S is P. E. | All S is not-P. A. | Some not-P is S. I. | Some not-P is not not-S. O. |
| Some S is not P. O. | Some S is not-P. I. | Some not-P is S. I. | Some not-P is not not-S. O. |
It will be observed that in the case of A and O, the contrapositive is equivalent to the original proposition, the quantity 136 being unchanged, whereas in the case of E we pass from a universal to a particular.140 In order to emphasize this difference, and following the analogy of ordinary conversion, the contraposition of A and O has been called simple contraposition, and that of E contraposition per accidens.141