Imágenes de páginas
PDF
EPUB

It is plain that any positive term and its corresponding negative divide between them the whole universe of thought: whatever does not fall into one must fall into the other, by the third fundamental Law of Thought, the Law of Duality. It follows at once that there are two modes of representing a difference. Supposing that the things represented by A and B are found to differ, we may indicate (see p. 17) the result of the judgment by the notation.

[blocks in formation]

We may now represent the same judgment by the assertion that A agrees with those things which differ from B, or that A agrees with the not-B's. Using our notation for negative terms (see p. 14), we obtain

A = Ab

as the expression of the ordinary negative proposition. Thus if we take A to mean quicksilver, and B solid, then we have the following proposition:

Quicksilver Quicksilver not-solid.

=

There may also be several other classes of negative propositions, of which no notice was taken in the old logic. We may have cases where all A's are not-B's, and at the same time all not-B's are A's; there may, in short, be a simple identity between A and not-B, which may be expressed in the form

[blocks in formation]

We shall also frequently have to deal as results of deduction, with simple, partial, or limited identities between negative terms, as in the forms

[blocks in formation]

It would be possible to represent affirmative propositions in the negative form. Thus "Iron is solid," might be expressed as "Iron is not not-solid," or " Iron is not fluid;" or, taking A and b for the terms " iron," and "not-solid," the form would be A

~

b.

But there are very strong reasons why we should employ all propositions in their affirmative form. All inference proceeds by the substitution of equivalents, and a proposition expressed in the form of an identity is ready to yield all its consequences in the most direct manner. As will be more fully shown, we can infer in a negative proposition,

but not by it. Difference is incapable of becoming the ground of inference; it is only the implied agreement with other differing objects which admits of deductive reasoning; and it will always be found advantageous to employ propositions in the form which exhibits clearly the implied agreements.

Conversion of Propositions.

The old books of logic contain many rules concerning the conversion of propositions, that is, the transposition of the subject and predicate in such a way as to obtain a new proposition which will be true when the original proposition is true. The reduction of every proposition to the form of an identity renders all such rules and processes needless. Identity is essentially reciprocal. If the colour of the Atlantic Ocean is the same as that of the Pacific Ocean, that of the Pacific must be the same as that of the Atlantic. Sodium chloride being identical with common salt, common salt must be identical with sodium chloride. If the number of windows in Salisbury Cathedral equals the number of days in the year, the number of days in the year must equal the number of the windows. Lord Chesterfield was not wrong when he said, "I will give anybody their choice of these two truths, which amount to the same thing; He who loves himself best is the honestest man; or, The honestest man loves himself best." Scotus Erigena exactly expresses this reciprocal character of identity in saying, "There are not two studies, one of philosophy and the other of religion; true philosophy is true religion, and true religion is true philosophy."

=

A mathematician would not think it worth while to mention that if x then also y = X. y He would not consider these to be two equations at all, but one equation accidentally written in two different manners. In written symbols one of two names must come first, and the other second, and a like succession must perhaps be observed in our thoughts: but in the relation of identity there is no need for succession in order (see p. 33), each is simultaneously equal and identical to the other. These remarks will hold true both of logical and mathematical identity; so that I shall consider the two forms

[blocks in formation]

to express exactly the same identity differently written. All need for rules of conversion disappears, and there will be no single proposition in the system which may not be written with either end foremost. Thus A: AB is the same as AB = A, aC = bC is the same as bC

forth.

~

=

aC, and so

The same remarks are partially true of differences and inequalities, which are also reciprocal to the extent that one thing cannot differ from a second without the second differing from the first. Mars differs in colour from Venus, and Venus must differ from Mars. The Earth differs from Jupiter in density; therefore Jupiter must differ from the Earth. Speaking generally, if AB we shall also have B A, and these two forms may be considered expressions of the same difference. But the relation of differing things is not wholly reciprocal. The density of Jupiter does not differ from that of the Earth in the same way that that of the Earth differs from that of Jupiter. The change of sensation which we experience in passing from Venus to Mars is not the same as what we experience in passing back to Venus, but just the opposite in nature. The colour of the sky is lighter than that of the ocean; therefore that of the ocean cannot be lighter than that of the sky, but darker. In these and all similar cases we gain a notion of direction or character of change, and results of immense importance may be shown to rest on this notion. For the present we shall be concerned with the mere fact of identity existing or not existing.

Twofold Interpretation of Propositions.

Terms, as we have seen (p. 25), may have a meaning either in extension or intension; and according as one or the other meaning is attributed to the terms of a proposition, so may a different interpretation be assigned to the proposition itself. When the terms are abstract we must read them in intension, and a proposition connecting such terms must denote the identity or non-identity of the qualities respectively denoted by the terms. Thus if we

say

Equality Identity of magnitude,

=

the assertion means that the circumstance of being equal exactly corresponds with the circumstance of being identical in magnitude. Similarly in

Opacity= Incapability of transmitting light,

the quality of being incapable of transmitting light is declared to be the same as the intended meaning of the word opacity.

When general names form the terms of a proposition we may apply a double interpretation. Thus

Exogens Dicotyledons

=

means either that the qualities which belong to all exogens are the same as those which belong to all dicotyledons, or else that every individual falling under one name falls equally under the other. Hence it may be said that there are two distinct fields of logical thought. We may argue either by the qualitative meaning of names or by the quantitative, that is, the extensive meaning. Every argument involving concrete plural terms might be converted into one involving only abstract singular terms, and vice versa. But there are reasons for believing that the intensive or qualitative form of reasoning is the primary and fundamental one. It is sufficient to point out that the extensive meaning of a name is a changeable and fleeting thing, while the intensive meaning may nevertheless remain fixed. Very numerous additions have been lately made to the extensive meanings both of planet and element. Every iron steam-ship which is made or destroyed adds to or subtracts from the extensive meaning of the name steam-ship, without necessarily affecting the intensive meaning. Stage coach means as much as ever in one way, but in extension the class is nearly extinct. Chinese railway, on the other hand, is a term represented only by a single instance; in twenty years it may be the name of a large class.

Tires for only semen, alternative and camp meg. proposters have hown whresenta, hence we

вели

to have inly live argumen?

[ocr errors]

CHAPTER IV.

DEDUCTIVE REASONING.

THE general principle of inference having been explained in the previous chapters, and a suitable system of symbols provided, we have now before us the comparatively easy task of tracing out the most common and important forms of deductive reasoning. The general problem of deduction is as follows:-From one or more propositions called premises to draw such other propositions as will necessarily be true when the premises are true. By deduction we investigate and unfold the information contained in the premises; and this we can do by one single rule-For any term occurring in any proposition substitute the term which is asserted in any premise to be identical with it. To obtain certain deductions, especially those involving negative conclusions, we shall require to bring into use the second and third Laws of Thought, and the process of reasoning will then be called Indirect Deduction. In the present chapter, however, I shall confine my attention to those results which can be obtained by the process of Direct Deduction, that is, by applying to the premises themselves the rule of substitution. It will be found that we can combine into one harmonious system, not only the various moods of the ancient syllogism, but a great number of equally important forms of reasoning, which had no recognised place in the old logic. We can at the same time dispense entirely with the elaborate apparatus of logical rules and mnemonic lines, which were requisite so long as the vital principle of reasoning was not clearly expressed.

E

« AnteriorContinuar »