if this form of inference were called Subalternation or Subordination. Nor, strictly speaking, can the relation between I and O be called one of opposition, for they may both be true together. Accordingly, Aristotle says that in reality (kar' aλýbear) there are three forms of opposition (those between A and E, A and O, E and I), though in language (Karà Thy λég) there are four (adding that between I and O). What is called a Subaltern Opposition he does not recognise. $3. On Conversions. A proposition is said to be converted when its terms are transposed, so that the subject becomes the predicate, and the predicate the subject. A Conversion may be defined as an immediate inference in which from one proposition we infer another having the same terms as the original proposition, but their order reversed. This inference in some cases necessitates a change of quantity in passing from one proposition to the other, and then it is called a Conversion per accidens; when it necessitates no such change, it is called a Simple Conversion2. I and E may both be converted simply. Thus, from 'Some X is Y,' or 'Some poets are philosophers,' I may infer Some Y is X,' or 'Some philosophers are poets.' From 'No X is Y,' or 'No savages are trustworthy,' I may infer 'No Y is X,' or 'No trustworthy persons are savages.' 2 It is proposed by Sir W. Hamilton to call the original proposition the "Convertend," the inferred proposition the "Converse." A can only be converted per accidens. For though it may sometimes happen that the subject and predicate of an A proposition are co-extensive, and therefore convertible, this is not implied in the form of the proposition, and it is with what is implied in the form of the proposition that we are alone concerned. Thus, if I assert the proposition 'All triangles are three-sided rectilineal figures,' it happens in this particular case that I am justified, without any change of quantity, in stating the converse, 'All three-sided rectilineal figures are triangles.' But if I state that All triangles are rectilineal figures,' I am only justified in inferring that 'Some rectilineal figures are triangles.' As, therefore, the general form of an A proposition does not imply the simple convertibility of the subject and predicate, I am only justified in inferring from All X is Y,' that 'Some Y is X.' In those cases, however, in which the form of the proposition implies that the subject and predicate are coextensive, the proposition, though an A proposition, may be converted simply. Thus, from the propositions 'The second legion is the only legion quartered in Britain,' 'Virtue is the condition of Happiness,' 'All triangles may be defined as three-sided rectilineal figures,' it may be inferred by simple conversion that 'The only legion quartered in Britain is the second legion,' 'The condition of Happiness is Virtue,' 'All three-sided rectilineal figures may be called triangles.' An O proposition cannot be converted at all. From 'Some X is not Y,' it does not follow that 'Some Y is to a genus. not X,' for Y may stand to X in the relation of a species Thus from the proposition 'Some Europeans are not Frenchmen,' I cannot infer that 'Some Frenchmen are not Europeans.' A Permutation may be defined as an immediate Inference in which from one proposition we infer another differing in quality, and having, therefore, instead of the original predicate its contradictory. Thus :— From All X is Y, we may infer that No X is not-Y. The legitimacy of these inferences is apparent from the fact that contradictory terms (A and not-A) admit of no medium, so that, if I predicate the one affirmatively, I may always predicate the other negatively, and vice versâ. The O proposition, when permuted from Some X is not Y,' into 'Some X is not-Y,' may of course be converted into Some not-Y is X.' This combination of permutation and conversion is improperly described by Whately and many previous logicians as a single inference, and styled "Conversion by Contra-Position or Negation." 3 The term Permutation is borrowed from Mr. Karslake's Aids to Logic. The same inference is sometimes called Infinitation, from the Nomen Infinitum, or, more properly, Nomen Indefinitum (not -Y, as the contradictory of Y), which is employed as the predicate. It may assist the student if we add some further instances of permutations :— All men are fallible, ... No men are infallible. No men are infallible, ... All men are fallible. Some poets are reflective, ... Some poets are not unreflective. Some poets are not unreflective, ... Some poets are reflective. All poets are men of genius,.. (by permutation) No poets are not-men-of-genius; ... (by conversion) No not-men-of-genius ( None but men of genius) are poets. = Note. We have here employed an expression Contradictory Terms,' which in most works on Logic is explained in the first part, as included under the doctrine of Opposition of Terms. It seemed, however, desirable to introduce only those distinctions of terms which were likely to be frequently required in the sequel of the work. We may here state that 'Contradictory Terms,' such as white and not-white, lawful and un-lawful, are terms which admit of no medium, i. e. terms which are not both predicable of the same thing, and one or other of which must be predicable. Contrary Terms,' like good and bad, black and white, are terms which are most opposed under the same genus; they are not both predicable of the same thing, but it is not necessary that one or other of them should be predicable. CHAPTER III. On Mediate Inference or Syllogism. § 1. The Structure of the Syllogism. A SYLLOGISM may be defined as a combination of two propositions, necessitating a third in virtue of their mutual connection; or as an inference in which one proposition is inferred from two others conjointly, the inferred proposition being virtually contained in the propositions from which it is inferred. This is obviously a definition of a legitimate syllogism. There may (as will appear below) be apparent syllogisms, which do not fulfil the conditions of this definition. We may give as instances of syllogisms: (I) (2) All B is A, All C is B; ... All C is A. All sovereign powers are invested with su preme authority over their subjects, All republics are sovereign powers; ... All republics are invested with supreme autho rity over their subjects. |