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and if that is all we know, we must remain in doubt about the truth or falsity of E. So likewise with I; so long as we do not know whether those unmarked circles should really be marked with a plain stroke or with a crossed stroke we cannot say whether it is true or false that some S's are not P. We can tell, however, about the truth or falsity of A; for if A were true and all S’s were P, all the circles would have to be marked with a plain stroke, and that is not possible so long as at least some of them are marked with a crossed stroke. Hence we can see from the symbols that represent the truth of O that A must be false. And so of the rest,

So far no particular S has been definitely and individually designated. To indicate some particular individual or subgroup of individuals use a small black dot or blacken the circle. All the remaining categorical propositions can then be symbolized.

Singular A: Socrates is P
Singular E: Plato is not P
Exceptive A: All the S’s but B are P ΦΦ or 0 0

The first of these figures indicates · All the S’s but B are
P (and B is not)', the second indicates · All the S’s but B
are P (and it is not said whether B is P or not)'.
Exceptive E: No S but B is P ΦΦ

or Exclusive A: B is the only S which is P Exclusive E: B is the only S which is not P . φφ

It is to be noticed that a person interpreting these diagrams could not distinguish between a proposition and its 'obverse’; for example, between the affirmative proposition All Sis P and the negative No S is non-P, or between the negative No S is P and the affirmative All S is non-P. This is an advantage rather than a defect; indeed the whole value of the symbols rests upon such facts, for the difference between a proposition and its obverse expresses a difference of shading or accent in the thought, but not a difference in the objects thought about. The relations of the objects remain the same whether they are told about in one way or in another, and the diagrams symbolize these relations as they are supposed to exist in the objects. They point to the reality with which thought is concerned and to which it must always conform whatever its shading, rather than to the particular shading which the thought may happen to take or the words in which it happens to be expressed, and they can be used to test the thought no matter what its shading or form of expression.

The fact that these diagrams express no difference between a proposition and its obverse suggests the question that is sometimes discussed whether proposition A is not after all negative rather than affirmative. When we say that every nation prefers its own interests to the good of humanity (All S is P), do we have in mind all the nations that do this, or the fact that none can be found which does not ? Certainly we cannot be sure that the statement is true until we find that there is no nation which does not (No S is non-P). Perhaps we can say that when proposition A expresses a hasty and unverified generalization it is affirmative, when it is derived deductively from general considerations it may also be affirmative, but when it is reached cautiously in the absence of general considerations it is usually negative. When we seek to verify a general statement, we do not count the cases in which it holds, but we look for exceptions.*

* This system of diagrams seems to me to indicate the opposition of propositions better than Euler's (explained elsewhere), partly because it provides a diagram for every proposition, while his only provides for the first four, partly because the same diagram represents a proposition and its obverse, partly because the diagrams for all the propositions that express different facts are distinctly different, but mainly because it preserves the distinction between things and attributes, and represents the presence or absence of the latter in the former rather than the partial or complete inclusion or exclusion of one class by another.

CHAPTER X.

INFERENCE AND THE SO-CALLED LAWS OF THOUGHT.

In previous chapters we have given examples of good and bad inference; we have said that all inference involves judgments about real or supposed objects of thought different from the judgments themselves; we have said that these judgments can be expressed in propositions, and in the chapter on the Opposition of Propositions we have had practical examples of the relation between the facts and the propositions about them. We must now inquire more fully what inference really is. In doing so we turn, though never altogether, from the question of words and their meanings, and fix our attention more fully upon things and their relations.

We infer when we suppose that because one state of affairs exists another exists also. The real or supposed facts that we reason from are called premises; those that

Inferencewe reason to, conclusions ; and we may say that what. the conclusion of any argument is true because the premises are true, or that the premises are true and therefore the conclusion is true.

Clear as this matter seems it is not fully understood until we distinguish the relation of premise and conclusion from two other relations each of which may likewise be indicated by the words ' because' and 'therefore ', namely, the relation of cause and effect and that of motive and act. A man may say, for example, that he believes in Christianity because he was born and bred in a Christian community, or

some

cause.

because he wants to go to heaven, or because the four gospels and the sacraments of the Church must have had

The first because' indicates a cause, the second a motive, the third a premise. The knowledge that such and such causes or motives exist may enable us to infer the existence of the corresponding effects or acts. If it is raining we know that people will put up their umbrellas. Similarly the knowledge that the effects or acts exist may enable us to infer the existence of the causes or motives. If people have up their umbrellas we know that it is raining. But with causes and motives as such, inference has no more to do than with any other relations.

Inferences are usually divided into two classes : Deductive, or those in which conclusions follow so necessarily from

their premises that their truth is as certain as that Deduction.

of the premises themselves; and Inductive, or those in which the conclusion follows from the premises with more or less probability, but by no means so inevitably—so that the premises might sometimes be true and yet the conclusion be false. To illustrate the latter first : from the presence of dark clouds and a moist atmosphere we can infer that it will rain, but we cannot be certain of it. On the other hand, if we know that on cloudy days it always rains and that to-day it is cloudy, we can be quite certain that there has been or will be rain to-day.

Deductive inference is the only kind that logicians discussed for two thousand years.

All that we shall have to say in the next seven chapters has direct reference to it. Induction will be discussed afterwards; something also will be said about the relation between the two kinds of infer

In the meantime we need only say that there is not nearly so much difference between them as has often been supposed

Deduction, or the absolutely indisputable kind of inference, does not depend, as most logicians have assumed, upon any special relation between our thoughts, but-like

ence.

the other kind—it depends upon the nature and inner relations of the objects thought about. It can be drawn only when the state of affairs asserted by the premise or premises could not possibly exist without the state of affairs asserted by the conclusion, or in other words, only when what is asserted by the premise or premises and what is asserted by the conclusion are different aspects of some wider system in which the former could not exist without the latter. If the line A cuts the line B we can infer with absolute certainty that the line B cuts the line A, because one cannot cut the other without making some such figure as this t, in which the other also cuts the one.

To take another example: If any one is told that three athletic teams, A, B, and C, each played three games with each of the others, that there were no drawn games, and that A won twice and twice only from B and twice and twice only from C, while B won only once from C, he has been told enough to enable him to construct a general scheme of things that includes also the number of games won by C and the relative standing of the teams: facts about which he was told nothing, and which even now some reader may not take the trouble to work out. That it can be worked out is not due primarily to any relation between the ideas merely as ideas of the person working it out; but it is due to the fact which he knows, and which would exist whether he knew it or not, that when a contestant plays a game which is not drawn he must either win it or lose it, and if he does not win it his opponent does, and to the known relations of number. If the rules of the game made it possible for both sides to win or for both to lose, or if two from three left two instead of one, the reasoning to be correct would have to take these facts into account and the conclusion would be different.

A curved surface cannot be concave on one side without being convex on the other; and in general the nature of things is such that every variation in one aspect of a complex

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