246 



THE POPULAR EDUCATOR. 



language is metaphorical, there will bo ambiguity ; if too con- 

 cise, obscurity ; and if too prolix, confusion. 



Two Propositions are said to be opposed to one another when, 

 having the same Subject and the same Predicate, they differ in 

 quantity or quality, or both. Hence there must be four 

 different kinds of Opposition. If we take the same subject 

 and the same Predicate, we can obviously make out of them 

 four different Propositions, which are represented by the four 

 symbols A, E, I, and O. Thus, let X represent the Subject, 

 and Y the Predicate, we shall have the several propositions, 

 "All X is Y," "No X is Y," "Some X is Y," and "Some X 

 is not Y," any two of which are said to be mutually opposed. 

 Hence there result the following four kinds of Opposition : 



(1) Contradictomj . Where the two propositions differ both 

 in quantity and quality, they are called Contradictories. These 

 will, of course, be A and O, or E and I. 



(2) Contrary. This takes place between propositions which 

 differ in quality only, and which are both universal, i.e., be- 

 tween A and E. 



(3) Sub-contrary. Where propositions differ in quality only, 

 but are both particular, they are called Sub-contraries. Thia 

 kind of Opposition, therefore, exists only between I and O. 



(4) Sub-alternate. This kind of Opposition is between those 

 propositions which differ in quantity only. It may, conse- 

 quently, be either between A and I, or between E and O. 



Certain Rules or Canons of Opposition have been laid down 

 by logicians, in reference to what may be inferred from the 

 truth or falsehood of one of two opposed propositions as to the 

 truth or falsehood of the other. These are most conveniently 

 enumerated as four, one in reference to each species of oppo- 

 sition. 



(1) Contradictories cannot be both true or both false at the 

 same time ; one of them must be true, and the other false. If 

 the negative be true, the affirmative must be false ; and if 

 "the negative be false, the affirmative must be true ; and vice 

 versa. This will appear manifest, if we recollect that every- 

 thing (whether individual or species), without exception, must 

 either belong to any given class or not, must possess a given 

 attribute or be destitute of it. Every A, as this is sometimes 

 expressed, must either be B or not B. 



(2) Of Contraries, both at the same time may be false, but 

 cannot be true. It is not necessary that either all or none 

 of the members of a species must possess a certain attribute ; 

 for example, the two propositions, " All men have the right to 

 freedom" and "No men have the right to freedom," are loth 

 false. Contraries cannot, however, at the same time both be 

 true. If it be true that a given predicate may be asserted of 

 the whole of a class, the same predicate cannot with truth be 

 denied of the whole. If, for instance, we lay down as true 

 that "all men have a right to freedom," we cannot with con- 

 sistency maintain also that "no men have such a right." 

 Hence, although if we are told that one of two Contraries is 

 false, this does not enable us to determine whether the other 

 is false or not, yet, if we know that one is true, we are certain 

 that the other must be false. 



(3) Sub-contraries may be at the same time both true, or one 

 of them false and the other true, but not both false. Where an 

 attribute belongs to part of a class, and does not belong to the 

 rest, e.g., "some men are wise," "some men are not wise," 

 there the Sub-contraries are both true. If, on the other hand, 

 such a proposition as " some men are stones " is false, it cannot 

 be so unless " some men are not stones " is true. If, therefore, 

 we are given the falsehood of one Sub-contrary, we may infer 

 the truth of the other ; but by our being given the truth of one 

 ye are not given anything as to the truth or falsehood of the 

 other, as they may, as we have seen, be both true. 



(4) Lastly, in Sub-alternation the two propositions may be at 

 the same time one true and one false, or both false, or both 

 true. There are thus four cases which may arise, and in two 

 of these we have grounds for inference, while in the remaining 

 two we are without them. If the Universal (generally called 

 the Sub-alternans) is true, the Particular (generally called the 

 Bub-alterna) is true also. If "all men are mortal" and "no 

 jnen are stones " are true, so also must be the corresponding 

 Particulars, "some men are mortals" and "some men are 

 not stones." We cannot, however, rever.ie the process, and 

 infer the truth of the Universal from our knowledge of the truth 

 of the Particular. It doea not follow from "some men are 



mad," for example, that " all men are mad." Secondly, where 

 we have ascertained that the Particular is false, we know that 

 the Universal also is false. That " some men are stones " could 

 not be false, unless it was also false that " all men are stones." 

 Thirdly, if, however, what we are given is the falsehood of tho 

 Universal, we cannot, merely from knowing this much, say 

 whether the Particular is true or not. To learn that we cannot 

 truly say that oil the individuals of a class do or do not, as the 

 case may be, possess a certain attribute, is not to learn that we 

 can truly say that some of them do or do not possess it. In 

 certain cases, but not in all, the Particular is true, even when 

 the Universal is false. Nor, lastly, are we warranted in assert- 

 ing the truth of the Universal because we may be certain of the 

 truth of the Particular. If the Subject and Predicate are " man " 

 and " mortal," both the Sub-alterna and Sub-alternans will be 

 true ; but the former may be true : for example, in the propo- 

 sition " some women are foolish," where the latter is evidently 

 false. 



It should be remarked, before passing from this branch of 

 the subject, that some logicians have refused to regard Sub- 

 contrariety as a species of opposition at all. And, speaking 

 strictly, it would seem as if they were right, as according to 

 the definition of Opposition above given, the subject in the 

 two Sub-contraries is not always exactly the same. In the 

 propositions " some men are wise " and " some men are not 

 wise," it is not really the same individual men which we are 

 speaking of in each. We mean in the one " some men," and 

 in the other "some other men" different from those spoken 

 of in the former proposition. No confusion, however, need 

 arise from following the ordinary classification, if this obser- 

 vation is kept in mind. 



We must next consider Conversion. This, unlike Oppo- 

 sition, which is a mere species of relation borne to one another 

 by propositions of a certain kind, is a process actually per- 

 formed, by which one proposition is changed into another, 

 which then bears a certain relation to the former. This will 

 naturally, being a process of inference, lead us on to the 

 theory and use of the Syllogism ; indeed, some writers have 

 considered Conversion as at bottom a process of reasoning, 

 capable of being reduced to a syllogistic form. 



A proposition, then, is said to be converted when its extremes 

 (or terms) are transposed, i.e., when the subject is put into the 

 place of the predicate, and the predicate into the place of the 

 subject, so as to form a new proposition. The name of Con- 

 vertend is given to the proposition to be converted, and that 

 of Converse to the new one which results from the transposition. 

 Logicians differ widely as to whether the judgment expressed by 

 the converse is a new judgment, or merely the old one expressed 

 in another form ; but this would be a question of too great 

 detail and difficulty to consider for the purpose we have in view. 



Conversion may be effected in various ways, but those princi- 

 pally employed in Logical treatises are two Simple and Per 

 Accidens. 



Simple conversion is that in which both the quantity and the 

 quality of the converse are the same as those of the convertend, 

 in which case, of course, the operation does not change the 

 symbol by which the proposition is to be designated. It will be 

 found that the only propositions which can be thus dealt with 

 are E and I. " No virtuous man is a rebel " may be converted 

 into " no rebel is a virtuous man," and " some boasters are 

 cowards" into " some cowards are boasters ;" and in each of 

 these cases the conversion is said to be illative, i.e., the truth of 

 the converse follows from, may bo inferred from, the truth of 

 the convertend. The one cannot be true unless the other is. 



We cannot, however, deal with A in the same manner. In it, 

 as we have already seen, the predicate is undistributed. Conse- 

 quently, if we simply transposed the terms, and let the quantity 

 of the proposition still remain universal, we should have the 

 term, which as predicate of the convertend was undistributed, 

 distributed when used as subject of the converse. Of course 

 this is an operation which may actually be performed ; but the 

 process will not be illative. We are not able to infer the truth 

 of the new proposition from the truth of the old ; and this 

 plainly, because the fact that a part only of all the individuals 

 or objects signified by the term used as predicate in the latter 

 proposition were spoken about, cannot warrant us in making an 

 assertion in the former about tho vjhole of those individuals or 

 objects. It may, indeed, happen accidentally that the new pro- 



