Deductive Logic Part 9

-- 247. Very often the matter of an indefinite proposition is such as clearly to indicate to us its quant.i.ty. When, for instance, we say 'Metals are elements,' we are understood to be referring to all metals; and the same thing holds true of scientific statements in general. Formal logic, however, cannot take account of the matter of propositions; and is therefore obliged to set down all indefinite propositions as particular, since it is not evident from the form that they are universal.

-- 248. Particular propositions, therefore, are sub-divided into such as are Indefinite and such as are Particular, in the strict sense of the term.

-- 249. We must now examine the sub-division of universal propositions into Singular and General.

-- 250. A Singular proposition is one which has a singular term for its subject, e.g. 'Virtue is beautiful.'

-- 251. A General proposition is one which has for its subject a common term taken in its whole extent.

-- 252. Now when we say 'John is a man' or 'This table is oblong,' the proposition is quite as universal, in the sense of the predicate applying to the whole of the subject, as when we say 'All men are mortal.' For since a singular term applies only to one thing, we cannot avoid using it in its whole extent, if we use it at all.

-- 253. The most usual signs of generality in a proposition are the words 'all,' 'every,' 'each,' in affirmative, and the words 'no,'

'none,' 'not one,' &c. in negative propositions.

-- 254. The terminology of the division of propositions according to quant.i.ty is unsatisfactory. Not only has the indefinite proposition to be set down as particular, even when the sense manifestly declares it to be universal; but the proposition which is expressed in a particular form has also to be construed as indefinite, _so_ that an unnatural meaning is imparted to the word 'some,' as used in logic. If in common conversation we were to say 'Some cows chew the cud,' the person whom we were addressing would doubtless imagine us to suppose that there were some cows which did not possess this attribute. But in logic the word 'some' is not held to express more than 'some at least, if not all.' Hence we find not only that an indefinite proposition may, as a matter of fact, be strictly particular, but that a proposition which appears to be strictly particular may be indefinite. So a proposition expressed in precisely the same form 'Some A is B' may be either strictly particular, if some be taken to exclude all, or indefinite, if the word 'some' does not exclude the possibility of the statement being true of all. It is evident that the term 'particular' has become distorted from its original meaning. It would naturally lead us to infer that a statement is limited to part of the subject, whereas, by its being opposed to universal, in the sense in which that term has been defined, it can only mean that we have nothing to show us whether part or the whole is spoken of.

-- 255. This awkwardness of expression is due to the indefinite proposition having been displaced from its proper position. Formerly propositions were divided under three heads--

(1) Universal,

(2) Particular,

(3) Indefinite.

But logicians anxious for simplification asked, whether a predicate in any given case must not either apply to the whole of the subject or not? And whether, therefore, the third head of indefinite propositions were not as superfluous as the so-called 'common gender' of nouns in grammar?

-- 256. It is quite true that, as a matter of fact, any given predicate must either apply to the whole of the subject or not, so that in the nature of things there is no middle course between universal and particular. But the important point is that we may not know whether the predicate applies to the whole of the subject or not. The primary division then should be into propositions whose quant.i.ty is known and propositions whose quant.i.ty is unknown. Those propositions whose quant.i.ty is known may be sub-divided into 'definitely universal' and 'definitely particular,' while all those whose quant.i.ty is unknown are cla.s.sed together under the term 'indefinite.' Hence the proper division is as follows--

Proposition __________|____________ | | Definite Indefinite _____|_______ | | Universal Particular.

-- 257. Another very obvious defeat of terminology is that the word 'universal' is naturally opposed to 'singular,' whereas it is here so used as to include it; while, on the other hand, there is no obvious difference between universal and general, though in the division the latter is distinguished from the former as species from genus.

_Affirmative and Negative Propositions._

-- 258. This division rests upon the Quality of propositions.

-- 259. It is the quality of the form to be affirmative or negative: the quality of the matter, as we saw before (-- 204), is to be true or false. But since formal logic takes no account of the matter of thought, when we speak of 'quality' we are understood to mean the quality of the form.

-- 260. By combining the division of propositions according to quant.i.ty with the division according to quality, we obtain four kinds of proposition, namely--

(1) Universal Affirmative (A).

(2) Universal Negative (E).

(3) Particular Affirmative (I).

(4) Particular Negative (O).

-- 261. This is an exhaustive cla.s.sification of propositions, and any proposition, no matter what its form may be, must fall under one or other of these four heads. For every proposition must be either universal or particular, in the sense that the subject must either be known to be used in its whole extent or not; and any proposition, whether universal or particular, must be either affirmative or negative, for by denying modality to the copula we have excluded everything intermediate between downright a.s.sertion and denial. This cla.s.sification therefore may be regarded as a Procrustes' bed, into which every proposition is bound to fit at its proper peril.

-- 262. These four kinds of propositions are represented respectively by the symbols A, E, I, O.

-- 263. The vowels A and I, which denote the two affirmatives, occur in the Latin words 'affirmo' and 'aio;' E and O, which denote the two negatives, occur in the Latin word 'nego.'

_Extensive and Intensive Propositions._

-- 264. It is important to notice the difference between Extensive and Intensive propositions; but this is not a division of propositions, but a distinction as to our way of regarding them. Propositions may be read either in extension or intension. Thus when we say 'All cows are ruminants,' we may mean that the cla.s.s, cow, is contained in the larger cla.s.s, ruminant. This is reading the proposition in extension. Or we may mean that the attribute of chewing the cud is contained in, or accompanies, the attributes which make up our idea of 'cow.' This is reading the proposition in intension. What, as a matter of fact, we do mean, is a mixture of the two, namely, that the cla.s.s, cow, has the attribute of chewing the cud. For in the ordinary and natural form of proposition the subject is used in extension, and the predicate in intension, that is to say, when we use a subject, we are thinking of certain objects, whereas when we use a predicate, we indicate the possession of certain attributes. The predicate, however, need not always be used in intension, e.g. in the proposition 'His name is John' the predicate is not intended to convey the idea of any attributes at all. What is meant to be a.s.serted is that the name of the person in question is that particular name, John, and not Zacharias or Abinadab or any other name that might be given him.

-- 265. Let it be noticed that when a proposition is read in extension, the predicate contains the subject, whereas, when it is read in intension, the subject contains the predicate.

_Exclusive Propositions._

-- 266. An Exclusive Proposition is so called because in it all but a given subject is excluded from partic.i.p.ation in a given predicate, e.g. 'The good alone are happy,' 'None but the brave deserve the fair,' 'No one except yourself would have done this.'

-- 267. By the above forms of expression the predicate is declared to apply to a given subject and to that subject only. Hence an exclusive proposition is really equivalent to two propositions, one affirmative and one negative. The first of the above propositions, for instance, means that some of the good are happy, and that no one else is so. It does not necessarily mean that all the good are happy, but a.s.serts that among the good will be found all the happy. It is therefore equivalent to saying that all the happy are good, only that it puts prominently forward in addition what is otherwise a latent consequence of that a.s.sertion, namely, that some at least of the good are happy.

-- 268. Logically expressed the exclusive proposition when universal a.s.sumes the form of an E proposition, with a negative term for its subject

No not-A is B.

-- 269. Under the head of exclusive comes the strictly particular proposition, 'Some A is B,' which implies at the same time that 'Some A is not B.' Here 'some' is understood to mean 'some only,' which is the meaning that it usually bears in common language. When, for instance, we say 'Some of the gates into the park are closed at nightfall,' we are understood to mean 'Some are left open.'

_Exceptive Propositions._

-- 270. An Exceptive Proposition is so called as affirming the predicate of the whole of the subject, with the exception of a certain part, e.g. 'All the jury, except two, condemned the prisoner.'

-- 271. This form of proposition again involves two distinct statements, one negative and one affirmative, being equivalent to 'Two of the jury did not condemn the prisoner; and all the rest did.'

-- 272. The exceptive proposition is merely an affirmative way of stating the exclusive--

No not-A is B = All not-A is not-B.

No one but the sage is sane = All except the sage are mad.