Imágenes de páginas
PDF
EPUB

If one were interested in describing John the facts would naturally be stated in one way; if he were interested in describing the horse they would be stated in the other way. But the mode of statement was settled before the logical process began. It was with the description and classification of things, not with their outer relations, that logic was concerned. It therefore never occurred to the early logicians that it was a part of their business to show how the state of relations of one thing at any particular time involved corresponding conditions in something else. Except as they contained data for classification the mere spatial and temporal and causal relations of things had no logical import and were not worth analyzing. It was with descriptions of things, not with events, that the logical process began.

Let us assume for a time, with the old logicians, that our only logical interest lies in the description and classification of things, and that every proposition with which logic deals must contain a subject, a predicate, and a copula whose sole function is to affirm or deny an equality or identity between the subject and predicate. If a proposition has not such a form it must be given one before it is dealt with logically, so that instead of saying “ John runs’, and ‘ Ducks like water ’, we must say “John is running ’, or more properly * John is a creature who is running', and 'Ducks are creatures who like water '.

The first thing to be noticed from this standpoint is that description merges insensibly into classification. When we say that ducks like water we undoubtedly describe one of their characteristics; but when we say that they are creatures who like water' we may be regarded as classing them with other creatures who like it (if such exist) as distinguished from those who do not. Thus the many relations really expressed in propositions are reduced for logical purposes to one; when the traditional logician says that any object or class of objects, S, is P, all he means is that the object or class S is contained in the class P.

What does this statement about S enable us to say about the class P?_This is the question of conversion. “A proposition is said to be converted when its terms are transposed, so that the subject becomes the predicate and the predicate the subject ” (Fowler, p. 80). Converting the proposition 'S is P’in this way we get ‘P is S'. But how many of the P's are S ? From the fact that all ducks like · water it does not neces

cessariiy follow that every creature that likes water is a duck.

Sir, I admit your general rule,
That every poet is a fool ;
But you yourself will serve to show it

That every fool is not a poet. The mediæval logician sought mechanical rules for manipulating words, and so he asked ‘Is there any rule by which we can tell the quantity and quality of propositions that have been converted ? ', and he found two which could always be followed with safety:

1. The quality of the proposition (affirmative or negative) must be preserved, and

“2. No term must be distributed in the Converse, unless it was distributed in the Convertend” (Jevons, p. 82).

The convertend means, of course, the proposition that is to be converted; and the converse that obtained by converting it.' A term is said to be distributed when used in such a way as to necessarily include all and not merely some of the members of the class it denotes. The subjects of the propositions A and E are thus said to be distributed; the subjects of I and 0 to be undistributed. But how about the predicates ?

When anybody who has not studied logic says that ducks like water, he uses the term 'ducks' demonstratively, and he can tell fairly well whether he means to speak of all, or of only some or most, of the creatures that the name denotes. The other two words in the sentence—like water '—he uses descriptively to tell something about the ducks. If he is

.

now told that he was really talking not only about ducks but also about a class of 'creatures that like water', and is asked whether he refers to all or only some of this class of creatures, he cannot help being puzzled, for the thought of such a class probably never entered his mind, That is the objection to what Sir William Hamilton and others have called the Quantification of the Predicate. But if he is compelled to answer, the only safe thing to say is that he means that ducks are at least some of the creatures that like water. Then his statement about the ducks will be true whether other things, such as gulls and frogs and fishes, happen to like water or not. To put the matter generally: Affirmative propositions, whether universal or particular, do not distribute their predicates.

To convert an affirmative proposition we must therefore reverse the subject and predicate, retain the affirmative copula (rule 1), and see that the subject is undistributed, i.e., that the proposition is particular (rule 2). In other words, the converse of A or I is always I.

When a universal proposition is converted into a universal or a particular into a particular it is said to be converted simply; but when a universal is converted into a particular it is said to be converted per accidens or by limitation. It will be noticed that when A is converted into I it cannot be converted back again into A, but only into I. From the fact that “all Rhode Islanders are Americans', it follows that ' Some Americans are Rhode Islanders’; but all that can be inferred from the latter proposition is that 'Some Rhode Islanders are Americans'.

In contrast with affirmatives, negative propositions always distribute their predicates. The statement that no cats like water means that cats and creatures that like water form two wholly distinct classes, and that no individual belongs to them both. We can therefore be as sure. that no single creature that likes water is a cat as that no single cat is a creature that likes water. Thus the converse of E (e.g., No

cats like water) is always E (No creatures that like water are cats).

The other negative proposition, O, is harder to deal with.

In attempting to convert the proposition O we encounter a peculiar difficulty, because its subject is undistributed; and yet the subject should become by conversion the predicate of a negative proposition, which distributes its predicate” (Jevons, p. 83). If certain boys, A, B, and C, do not like water we can be quite sure that no creature that likes water is one of these boys A, B, and C.

Here it is easy enough to convert; but the statement that A, B, and C do not like water is equivalent to three singular propositions, not to a particular. If it had been forgotten who these boys were, one might still be sure that some boys do not like water (proposition O); but all that could be said on the strength of this about creatures that like water is that none of them are some boys or other. We could not say that none of them are boys. We might say, to be sure, that some of them are not boys, but this would be on the strength of what we know about cats and monkeys, not on the strength of the statement that some boys do not like water. Since the statement that no P is some S or other conveys practically no information whatever about P, we must conclude that the proposition cannot be converted: nothing definite can be said about either some or all of the objects denotable * by its predicate.

Many logicians say in substance that though O cannot be converted in the usual way we can “apply a new process, which may be called conversion by negation, and which consists in first changing the convertend into an affirmative proposition, and then converting it simply” (Jevons, p. 83). In this way

Some boys do not like water' becomes Some boys dislike water', and converting this we get

* I say denotable rather than denoted, because the predicate of a proposition is usually used to describe the things pointed out by the subject, not to point out new ones.

[ocr errors]
[ocr errors]
[ocr errors]

'Some creatures that dislike water are boys'.

(Some S isn't P = Some S is non-P Some non-P is S.)

This so-called conversion by negation consists simply in converting the obverse; and it is a process which can be applied just as well to A and E (though not to I, whose obverse is O and inconvertible) as to O. The process

of alternate obversion and conversion can be carried through various stages as follows; but it is valid only if the existence of all the objects named is presupposed :

PROPOSITION A.
Beginning with Obversion. Beginning with Conversion.
A. All S is P =

A. All S is P
E. No S is non-P

1. Some P is S
E. No non-P is S

O. Some P isn't non-S.
A. All non-P is non-S =
1. Some non-S is non-P =

s Some non-S isn't non-non-P, or
0.
Some non-S isn't P.

PROPOSITION E. Beginning with Obversion. Beginning with Conversion. E. No S is P.

E. No S is P A. All S is non-P.

E. No P is S=
I. Some non-P is S.

A. All P is non-S =
O. Some non-P isn't non-S. I. Some non-S is P =

O. Some non-S isn't non-P.

PROPOSITION I.
Beginning with Obversion.

Beginning with Conversion. 1. Some S is P.

1. Some S is P. O. Some S isn't non-P. I. Some P is S.

O. Some P isn't non-S.

PROPOSITION O.
Beginning with Obversion.

Cannot be Converted.
O. Some S isn't P =
1. Some S is non-P =
1. Some non-P is S =
O. Some non-P isn't non-S.

« AnteriorContinuar »