Deductive Logic Part 10

_Tautologous or Identical Propositions_

-- 273. A Tautologous or Identical proposition affirms the subject of itself, e.g. 'A man's a man,' 'What I have written, I have written,'

'Whatever is, is.' The second of these instances amounts formally to saying 'The thing that I have written is the thing that I have written,' though of course the implication is that the writing will not be altered.

CHAPTER IV.

_Of the Distribution of Terms._

-- 274. The treatment of this subject falls under the second part of logic, since distribution is not an attribute of terms in themselves, but one which they acquire in predication.

-- 275. A term is said to be distributed when it is known to be used in its whole extent, that is, with reference to all the things of which it is a name. When it is not so used, or is not known to be so used, it is called undistributed.

-- 276. When we say 'All men are mortal,' the subject is distributed, since it is apparent from the form of the expression that it is used in its whole extent. But when we say 'Men are miserable' or 'Some men are black,' the subject is undistributed.

-- 277. There is the same ambiguity attaching to the term 'undistributed' which we found to underlie the use of the term 'particular.' 'Undistributed' is applied both to a term whose quant.i.ty is undefined, and to one whose quant.i.ty is definitely limited to a part of its possible extent.

-- 278. This awkwardness arises from not inquiring first whether the quant.i.ty of a term is determined or undetermined, and afterwards proceeding to inquire, whether it is determined as a whole or part of its possible extent. As it is, to say that a term is distributed, involves two distinct statements--

(1) That its quant.i.ty is known;

(2) That its quant.i.ty is the greatest possible.

The term 'undistributed' serves sometimes to contradict one of these statements and sometimes to contradict the other.

-- 279. With regard to the quant.i.ty of the subject of a proposition no difficulty can arise. The use of the words 'all' or 'some,' or of a variety of equivalent expressions, mark the subject as being distributed or undistributed respectively, while, if there be nothing to mark the quant.i.ty, the subject is for that reason reckoned undistributed.

-- 280. With regard to the predicate more difficulty may arise.

-- 281. It has been laid down already that, in the ordinary form of proposition, the subject is used in extension and the predicate in intension. Let us ill.u.s.trate the meaning of this by an example. If someone were to say 'Cows are ruminants,' you would have a right to ask him whether he meant 'all cows' or only 'some.' You would not by so doing be asking for fresh information, but merely for a more distinct explanation of the statement already made. The subject being used in extension naturally a.s.sumes the form of the whole or part of a cla.s.s. But, if you were to ask the same person 'Do you mean that cows are all the ruminants that there are, or only some of them?' he would have a right to complain of the question, and might fairly reply, 'I did not mean either one or the other; I was not thinking of ruminants as a cla.s.s. I wished merely to a.s.sert an attribute of cows; in fact, I meant no more than that cows chew the cud.'

-- 282. Since therefore a predicate is not used in extension at all, it cannot possibly be known whether it is used in its whole extent or not.

-- 283. It would appear then that every predicate is necessarily undistributed; and this consequence does follow in the case of affirmative propositions.

-- 284. In a negative proposition, however, the predicate, though still used in intension, must be regarded as distributed. This arises from the nature of a negative proposition. For we must remember that in any proposition, although the predicate be not meant in extension, it always admits of being so read. Now we cannot exclude one cla.s.s from another without at the same time wholly excluding that other from the former. To take an example, when we say 'No horses are ruminants,' the meaning we really wish to convey is that no member of the cla.s.s, horse, has a particular attribute, namely, that of chewing the cud. But the proposition admits of being read in another form, namely, 'That no member of the cla.s.s, horse, is a member of the cla.s.s, ruminant.' For by excluding a cla.s.s from the possession of a given attribute, we inevitably exclude at the same time any cla.s.s of things which possess that attribute from the former cla.s.s.

-- 285. The difference between the use of a predicate in an affirmative and in a negative proposition may be ill.u.s.trated to the eye as follows. To say 'All A is B' may mean either that A is included in B or that A and B are exactly co-extensive.

[Ill.u.s.tration]

-- 286. As we cannot be sure which of these two relations of A to B is meant, the predicate B has to be reckoned undistributed, since a term is held to be distributed only when we know that it is used in its whole extent.

-- 287. To say 'No A is B,' however, is to say that A falls wholly outside of B, which involves the consequence that B falls wholly outside of A.

[Ill.u.s.tration]

-- 288. Let us now apply the same mode of ill.u.s.tration to the particular forms of proposition.

-- 289. If I be taken in the strictly particular sense, there are, from the point of view of extension, two things which may be meant when we say 'Some A is B'--

(1) That A and B are two cla.s.ses which overlap one another, that is to say, have some members in common, e.g. 'Some cats are black.'

[Ill.u.s.tration]

(2) That B is wholly contained in A, which is an inverted way of saying that all B is A, e.g. 'Some animals are men.'

[Ill.u.s.tration]

-- 290. Since we cannot be sure which of these two is meant, the predicate is again reckoned undistributed.

-- 291. If on the other hand 1 be taken in an indefinite sense, so as to admit the possibility of the universal being true, then the two diagrams which have already been used for A must be extended to 1, in addition to its own, together with the remarks which we made in connection with them (---- 285-6).

-- 292. Again, when we say 'Some A is not B,' we mean that some, if not the whole of A, is excluded from the possession of the attribute B. In either case the things which possess the attribute B are wholly excluded either from a particular part or from the whole of A. The predicate therefore is distributed.

[Ill.u.s.tration]

From the above considerations we elicit the following--

-- 293. Four Rules for the Distribution of Terms.

(1) All universal propositions distribute their subject.

(2) No particular propositions distribute their subject,

(3) All negative propositions distribute their predicate.

(4) No affirmative propositions distribute their predicate.

-- 294. The question of the distribution or non-distribution of the subject turns upon the quant.i.ty of the proposition, whether universal or particular; the question of the distribution or non-distribution of the predicate turns upon the quality of the proposition, whether affirmative or negative.

CHAPTER V.

_Of the Quantification of the Predicate._

-- 295. The rules that have been given for the distribution of terms, together with the fourfold division of propositions into A, E, 1, 0, are based on the a.s.sumption that it is the distribution or non-distribution of the subject only that needs to be taken into account in estimating the quant.i.ty of a proposition.

-- 296. But some logicians have maintained that the predicate, though seldom quantified in expression, must always be quantified in thought--in other words, that when we say, for instance, 'All A is B,'

we must mean either that 'All A is all B' or only that 'All A is some B.'