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are not more moral than Pagans,” but it would be absurd to suppose that it would be necessary to go to the contrary extreme, and shew that “No Christians are more moral than Pagans.". In short A is sufficiently and best

l disproved by 0, and E by I. It will be easily apparent that, vice versa, o is disproved by A, and I by E; nor is there, indeed, any other mode at all of disproving these particular propositions.

When we compare together the propositions I and o we find that they are in a certain sense contrary in nature, one being affirmative and the other negative, but that they are still consistent with each other. It is true both that “Some metals are brittle,” for instance Antimony, Bismuth and Arsenic ; but it is also true that “ Some metals are not brittle.” And the reader will observe that when I affirm “Some metals are elements," there is nothing in this to prevent the truth of “Some metals are not elements," although on other grounds we know that this is not true. The propositions I and 0 are called subcontraries each of the other, the name connoting a less degree of contrariety than exists between A and E.

As regards the relation of A to I and E to 0, it is plain that the truth of the universal includes and necessitates the truth of the particular. What may be affirmed or denied of all parts of a class may certainly be affirmed or denied similarly of some part of the class. From the truth of the particular we have no right to infer either the truth or falsity of the universal of the same quality. These pairs of propositions are called subalterns, i. e. one under the other (Latin sub under, and alter the other of two), or we may say more exactly that I and o are respectively the subalternates of A and E, each of which is a subalternans,

The relations of the propositions just described are all clearly shown in the following scheme :

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2.

It is so highly important to apprehend completely and readily the consistency or opposition of propositions, that I will put the matter in another form. Taking any two propositions having the same subject and predicate, they must come under one of the following statements :

1. Of contradictory propositions, one must be true and one false.

Of contrary propositions, both cannot be true, and both may be false.

3. Of subcontrary propositions, one only can be false, and both may be true.

4. Of subalterns, the particular is true if the universal be true; but the universal may or may not be true when the particular is true.

I put the same matter in yet another form in the following table, which shows how the truth of each of A, E I, and 0, affects the truth of each of the others.

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It will be evident that from the affirmation of universals more information is derived than from the affirmation of particulars. It follows that more information can be derived from the denial of particulars than from the denial of universals, that is to say, there are less cases left doubtful, as in the above table.

The reader may well be cautioned, however, against an ambiguity which has misled some even of the most eminent logicians. In particular propositions the adjective some is to be carefully interpreted as some, and there may or may not be more or all. Were we to interpret it as some, not more nor all, then it would really give to the proposition the force of I and o combined. If I say “some men are sincere," I must not be taken as implying that

some men are not sincere ;" I must be understood to predicate sincerity of some men, leaving the character of the remainder wholly unaffected. It follows from this that, when I deny the truth of a particular, I must not be understood as implying the truth of the universal of the same quality. To deny the truth of “ some men are mortal” might seemn very natural, on the ground that not some but all men are mortal; but then the proposition denied would really be some men are not mortal, i. e. O not I. Hence when I deny that “ some men are immortal” I mean that “no men are immortal ;" and when I deny that

some men are not mortal," I mean that “all men are mortal.”.

It has long been usual to compare propositions as

regards the quality of the subject matter to which they refer, and what is technically called the matter was distinguished into three kinds, necessary, contingent, and impossible. Necessary matter consists of any subject in which the proposition A may be affirmed; impossible in which I may be affirmed. Any subject or branch of knowledge in which universal statements cannot usually be made is called contingent matter, and it implies the truth of I and 0. Thus “comets are subject to gravitation," though an indefinite or indesignate proposition (p. 65), may be interpreted as A, because it refers to a part of natural science where such general laws obtain. But “ men are sincere” would be properly interpreted as particular or I, because the matter is clearly contingent. The truth of the following statements is evident.

In necessary matter A and I are true; E and o false.
In contingent matter I and 0 are true; A and E false,
Inimpossible matter E and 0 are true ; A and I false.

In reality, however, this part of logical doctrine is thoroughly illogical, because in treating a proposition we have no right, as already explained (p. 70), to assume ourselves acquainted with the science to which it refers, Our duty is to elicit the exact consequences of any statements given to us. We must learn in logic to transform information in every possible way, but not to add extraneous facis,

LESSON X.

CONVERSION OF PROPOSITIONS, AND

IMMEDIATE INFERENCE.

WE are said to infer whenever we draw one truth from another truth, or pass from one proposition to another. As Sir W. Hamilton says, Inference is “the carrying out into the last proposition what was virtually contained in the antecedent judgments.” The true sphere of the science of logic indeed is to teach the principles on which this act of inference must be performed, and all the previous consideration of terms and propositions is only useful or pertinent so far as it assists us to understand the processes of inference. We have to consider in succession all the modes in which the same information may be moulded into different forms of expression often implying results of an apparently different character, Logicians are not agreed exactly as to what we may include under the name Inference, and what we should not. All would allow that there is an act of inference when we see drops of water on the ground and believe that it has rained. This is a somewhat complicated act of inference, which we shall consider in later lessons under the subject of Induction. Few or none would say that there is an act of inference in passing from "The Duke of Cambridge is the Commander-in-chief,” to “The Commander-inchief is the Duke of Cambridge.” But without paying much regard to the name of the process I shall in this

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