4 I is represented by the 1st, 2nd, 3rd, and 5th diagrams. From the 3rd and also from the 5th follows by contraposition I, Some not-P is S. But from the 1st and 2nd, I does not follow. Hence from all the forms of I, that is, from I in every case, I (Some not-P is S) cannot be inferred by contraposition. Again, from the 1st and 2nd follows O (Some not-P is not S); but it does not follow from the other two diagrams1, and therefore O (Some not-P is not S) cannot be inferred from all the forms of I. Two diagrams (3rd and 5th) allow I, and two others (1st and 2nd) allow O; but from each of them neither I nor O can be inferred. Hence I cannot be contraposed. Recapitulation. The contrapositive of A is E, of E I, and of O I, while I cannot be contraposed. The student should care 1 In the 3rd diagram, a part of P coincides with a part of S, and some not-P, which lies outside P and consequently outside the coinciding part of P, lies outside the coinciding part of S and not outside the whole of S,—that is, all that is known certainly is that some not-P is excluded from a part, and not from the whole, of S; or, in other words, the proposition "Some not-P is not S" is not true. In the 5th diagram, P coincides with a part of S, and therefore some not-P, which lies outside P, lies outside the coinciding part of S; but whether some not-P lies outside the remaining part of S is not known, —that is, it is not known if some not-P is excluded from the whole of S. We know only that it is excluded from a part. Hence the proposition "Some not-P is not S" is not true. This proposition means that at least one not-P is excluded from the whole of S; but this cannot be inferred, as we have seen, from these two diagrams. fully note that I cannot be contraposed, and that O cannot be converted. An hypothetical proposition may be contraposed by taking the antecedent and the contradictory of the consequent in the proposition as the consequent and the antecedent respectively in the inference, and then changing the quality in the case of A and O, and also the quantity in the case of E. (1) If A is, B is: its contrapositive is 'If B is not, A never is,' 'Wherever B is not, A never is.' (2) If A is, B is not: its contrapositive is 'In some cases if B is not, A is.' (3) In some cases if A is, B is not: its contrapositive is 'In some cases if B is not, A is.' NOTE.-Contraposition is also called Conversion by Negation. The older logicians converted O by this process. We have seen that the process is applicable also to A and E, and inapplicable to I only. The contrapositive of a given proposition may be regarded as the converted obverse of it; and contraposition as consisting in obversion and in conversion of the obverse. Some logicians have indeed regarded the inference as double and the process as two-fold, including obversion and conversion, and have accordingly excluded contraposition from Immediate inference. But we have seen that, with the aid of the diagrams, the contrapositive of a proposition can be inferred as immediately as its obverse or its converse. In contraposing a proposition according to the older method, first obvert it, and then take the converse of the obverse. (1) All S is P. Examples. Its obverse is 'No S is not-P'; the converse of this obverse is 'No not-P is S,' and this last is the contrapositive of the given proposition (All S is P). (2) No S is P. Its obverse is 'All S is not P'; the converse of this obverse is 'Some not-P is S,' which is the contrapositive of the given proposition (No S is P). (3) Some S is not P. Its obverse is 'Some S is not-P'; the converse of this obverse is 'Some not-P is S,' and this last is the contrapositive of the given proposition (Some S is not P). (4) Some S is P. Its obverse is 'Some S is not not-P,' which is O, and O cannot be converted as we have seen before (vide pp. 127-8). Exercise. Contrapose the following propositions: 1. All animals are mortal. 2. No created being is perfect. 3. All gases can be liquefied. 4. Some plants are not devoid of the power of locomotion. 5. Some animals are insentient. 6. Some substances have no cause, 7. All bodies that have inertia have weight. 8. If mercury is heated, it expands. 9. In some cases if a body is heated, its temperature does not rise. 10. In some cases a sensation is followed by a perception. 11. If A is B, C is D. 12. If A is B, C is not D. 13. In some cases if A is B, C is not D. 14. In some cases if A is B, C is D. 15. In all cases if A is not B, C is D. § 5. IV. Of Subalternation. This process of immediate inference consists in passing from the universal to the particular, and from the particular to the universal, with the same subject and predicate, and of the same quality. By subalternation follows: (1) From the truth of A, the truth of I, and from the truth of E, the truth of 0; but not conversely from the latter the former. Thus, if 'All S is P' be true, 'Some S is P' will also be true; but if the latter be true, the former will not necessarily be true. (2) From the falsity of I, the falsity of A, and from the falsity of O, the falsity of E; but not conversely the former from the latter. If 'Some S is P' be false, then 'All S is P' must also be false; if 'Some S is not P' be false, then 'No S is P' must be false; but not conversely, that is, the falsity of the particular does not follow from the falsity of the corresponding universal. 'All S is P' may be false, and still 'Some S is P' may be true. Similarly, E may be false, and the corresponding O true. The proof lies in the fact (1) that I or O simply repeats what is already recognized as true by A or E, and (2) that what fails even in one case can not be universally true, or what holds good even in one case can not be universally denied. The proof of the converse lies in the fact (1) that something may be true or false in some cases, in at least one case, though not universally, and (2) that what is not true or false in all cases, may yet be true or false in some cases, in at least one case. The rules of inference given above may be easily proved also from the diagrams. § 6. V.-Of Opposition. In a previous chapter (vide p. 78) we have seen that A and O, and E and I, are called, in relation to each other, Contradictory Opposites, that A and E are called, in relation to each other, Contrary Opposites, and that I and O are called Subcontrary Opposites. In consequence of the opposition which exists among A, E, I, and O, having the same subject and predicate, but differing in quality, or in both quality and quantity, when any one is given as true or false, the others are necessarily either true, false, or unknown. We shall now inquire into these necessary connections among them, and lay down certain general rules of immediate inference by opposition :— (1) Given the truth of A (All S is P). From the truth of A, as illustrated by the 1st and 2nd diagrams, it follows that E is false and also that O (Some S is not P) is false. (2) Given the falsity of A (All S is P). From the falsity of A, as represented by one or other of the 3rd, 4th, and 5th diagrams1, follow the truth of O (Some S is not P); and also the truth in one case (4th), and the falsity in the other cases (3rd and 5th) of E, or, in other words, the doubtfulness or uncertainty of E (No S is P). (3) Given the truth of E (No S is P). From the 4th diagram representing E, follows at once the falsity of A, and also the falsity of I (Some S is P). (4) Given the falsity of E (No S is P). The falsity of E is represented by one or other of the 1st, 2nd, 3rd, and 5th diagrams, from which follow the truth of I, and also the truth of A in two cases (1st and 2nd), and the falsity of A in two others (3rd and 5th), or, in other words, the doubtfulness of A. (5) Given the truth of I (Some S is P). From the 1st, 2nd, 3rd, and 5th diagrams representing I, follow at once the falsity of E and also the truth of O (Some S is not P) in two cases (3rd and 5th), and the falsity of O in the other two (1st and 2nd), or, in other words, the doubtfulness of O. (6) Given the falsity of I (Some S is P). This is represented by the 4th diagram, from which follows at once the truth of E (No S is P), and also the truth of O. (7) Given the truth of O (Some S is not P). This is represented by the 3rd, 4th, and 5th diagrams, from which follows at once the falsity of A, and also the doubtfulness of I. (8) Given the falsity of O (some S is not P). This is represented by one or other of the 1st and 2nd diagrams, from which follows at once the truth of A (All S is P), and also the truth of L 1 The falsity of A means that the relation between the subject and the predicate can not be represented by the 1st and 2nd diagrams, and that it must be represented by one or other of the remaining three diagrams. The falsity of E, I, or O may similarly be represented by diagrams. |