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Experience teaches that every real thing, indeed every object of which it is possible to think at all, has qualities and relations of some kind. If it is not large it is small, if it is not here it is elsewhere; or else it is a spirit or some other kind of immaterial object existing without space-relations, but not without the moral or other relations which that kind of immaterial objects possess. In general, if any object, S, has not a certain quality or relation, P, it must have some other quality or relation incompatible with P, and which we may therefore call not-P or non-P. To put it more briefly : If S isn't P it is non-P; every S must be either P or non-P; and to say that S isn't P is equivalent to saying that it is non-P.
This is the secondary sense of the law; and it does not depend, like the primary, wholly upon the mere nature of judgment, but partly also upon the fact that we have never found and cannot think of an object that does not possess some definite relation or other.
But there is often still a third implication which the law seems to cover.
From earliest childhood we find those about us dividing objects into various kinds, and in the course of experience we learn to take it for granted that however much they may differ from each other all objects of the same kind are determined in the same respects ; have the same kind of qualities and relations. Every man has some moral standing. Hence if he is not good he must be bad. He has some color. Hence if he is not white he must be black, brown, yellow, or red. And so of size, weight, and all the other gener:: qualities which every man possesses.
Now, as it happens, it is just as easy to find out that some object, say the soul or the square root of 33, has no color or weight or shape at all as to find out that it is not red or heavy or round, and in ninety-nine cases out of a hundred it is a great deal more serviceable to say so. Consequently when any one says that some object, X, is not red his hearer takes it for granted that he is talking about an object that has some color or other, since otherwise he would have said it was colorless. * And so of every other quality and relation : we interpret the statement 'S isn't P' to mean that S has some quality P' or P'', which, though different from P, belongs to the same general class.
The third matter which the law of excluded middle seems to cover is thus a rule for the interpretation of language. But such a rule is by no means infallible. It is quite possible to say that a triangle is not virtuous without the slightest intention of implying that it is vicious.
Assuming the truth of the law in the secondary sense, and understanding that the phrase non-P is used to indicate some
quality or relation incompatible with P, we can Obversion.
turn every affirmative proposition into an equivalent negative, and vice versa : • All S is P'(A) into “No S is non-P' (E); “Some S isn't P' (O) into “Some S is non-P' (I); · A'll men are mortal' into • No men are iinmortal'; and vice versa.
The manipulation of propositions in this way is called Obversion or Immediate Inference by Privitive Conception. It is less commonly known as Permutation or Infinitation. If we wish a formal definition we can say :
To OBVERT a proposition is to deny or affirm the absence or presence of a relation whose presence or absence the original proposition affirmed or denied.
* "No one would be so foolish as to deny what no one could have the slightest temptation to affirm. If I say, then, that is not Y, I imply that there are certain elements in X, by which, if they were taken alone, it might be confounded with Y. Of course the elements of resemblance may be comparatively few, but something in this case must have occurred to bring it into prominence.” C. C. Everett, “ Fichte's Science of Knowledge”, p. 103.
IMMEDIATE INFERENCE, OR INFERENCE FROM A SINGLE
WHEN an inference is drawn from a single premise it is called immediate, when from several premises taken together, mediate. The terms immediate and mediate as thus used have only a secondary reference to time. Their main object is to indicate the absence or presence of some intermediate process.
Where an inference rests upon several premises and cannot be drawn from one of them alone, the intermediate process consists in constructing a notion of a total state of affairs according to specifications part of which are laid down by one premise and part by another. It is this total state of affairs in which alone both or all the premises can be realized that implicates the conclusion. A state of affairs in which none or only a part of the premises were realized might implicate it, but we know that this other must.
We are at present concerned with immediate inference, or the cases in which the state of affairs described by a single premise necessarily implicates that described in the conclusion.
The most interesting and important kind of immediate inference is called · Conversion'. In discussing obversion we saw how it was possible to pass from one statement to another about the same object ; in conversion we feel warranted in passing from a statement
If he were
about one object or kind of object to a statement about another object or kind of object about which something has been implied, though not directly said in the statement about the first object. If, for example, we should happen to know that some white things are square we should be able to infer that some square things are white. To be white and to be square are two very different matters; but yet the world and our minds are so constituted that a statement about the one class of objects may serve as the basis for a statement about the other.* In like manner, if a person interested in the city of Cleveland should be told that it is 183 miles west of Buffalo and he should afterwards have occasion to tell all that he knew about Buffalo, he would be able to say that it is 183 miles east of Cleveland. interested in places 183 miles west of Buffalo he would be able to say that at least one of them was Cleveland. And if he were interested in the distance of 183 miles he could say that it is as far as from Buffalo to Cleveland. - The first statement was about Cleveland ; but the fact asserted was of such a nature that it could not be true unless the other statements, not about Cleveland but about Buffalo and places 183 miles west of Buffalo and the distance of 183 miles, were also true.
The doctrine of conversion found in most text-books on logic provides for only one of the three inferences which is
here drawn, namely: a place 183 miles west of The traditional Buffalo is Cleveland. The reason for this lies in
the traditional way of dividing every proposition into subject, predicate, and copula, and of regarding the copula as a perfectly colorless sign of affirmative or negative predication. Every proposition was regarded as a kind of
* The constitution of the world involved in this particular case is the fact that several attributes (e.g., whiteness and squareness) may be possessed with equal intimacy by the same object; in the case about to be mentioned it is the fact that relations are reciprocal--facts of so familiar
kind that we forget the debt our logic owes them.
equation of which the copula supplied only the idea of equality or non-equality. Our original proposition, divided in this way, would read
Subject. Copula. Predicate. Cleveland is (a place) 183 miles west of Buffalo, and it would be regarded as meaning nothing more than that Cleveland was identical with a-place-183-miles-west-of-Buffalo. Regarded in this way the proposition says nothing whatever about Buffalo or about 183 miles, but only about Cleveland and a place 183 miles west of Buffalo. These are
the terms, and they cannot be broken up, consequently the * only inference to be drawn, if inference it is, except by obversion or by way of opposition is found by reversing the equation, and saying a place 183 miles west of Buffalo is Cleveland.
According to the same way of regarding things the proposition · John is riding a horse' can be converted into 'a person riding a horse is John', but the traditional rules of logic make no provision for any inference about the horse. *
* This limited view of the meaning of propositions can be easily explained. In an age like that of Plato and Aristotle when scientific knowledge was thought to consist merely in description and classification, it was natural enough to overlook all the relations asserted in propositions except those of substance and attribute and individual identity, for the causal and other outer relations between things had no scientific significance except as indicating qualities of the things, by which they could be identified and classified. From this point of view the statement that John is rid. ing a horse is of value only as it tells one of John's accidents or occasional states by which he might perhaps be identified or distinguished from other people who never ride or at least did not ride at the time referred to. From the same point of view the converse statement that a horse is being ridden by John has scientific significance only as indicating that horses, or at least this particular horse, can be described as capable of being ridden or as having been ridden. This standpoint lent itself only too easily to the purely mechanical and verbal treatment of propositions which is still common. Certainly with modern writers this purely verbal treatment of logical processes is only a pedagogic device.
Yet it seems to me that it limits the usefulness of logic and that the subject is capable of being treated more directly.