First Notions of Logic (preparatory to the Study of Geometry)

Part 1

FIRST NOTIONS OF LOGIC (PREPARATORY TO THE STUDY OF GEOMETRY)

AUGUSTUS DE MORGAN,

OF TRINITY COLLEGE, CAMBRIDGE, PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.

The root of all the mischief in the sciences, is this; that falsely magnifying and admiring the powers of the mind, we seek not its real helps.—BACON.

LONDON:

PRINTED FOR TAYLOR AND WALTON,

BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE.

28 UPPER GOWER STREET.

M.DCCC.XXXIX.

⁂ This Tract contains no more than the author has found, from experience, to be much wanted by students who are commencing with Euclid. It will ultimately form an Appendix to his Treatise on Arithmetic.

The author would not, by any means, in presenting the minimum necessary for a particular purpose, be held to imply that he has given enough of the subject for all the ends of education. He has long regretted the neglect of logic; a science, the study of which would shew many of its opponents that the light esteem in which they hold it arises from those habits of inference which thrive best in its absence. He strongly recommends any student to whom this tract may be the first introduction of the subject, to pursue it to a much greater extent.

University College, Jan, 8, 1839.

LONDON:—PRINTED BY JAMES MOYES, Castle Street, Leicester Square.

FIRST NOTIONS OF LOGIC.

What we here mean by Logic is the examination of that part of reasoning which depends upon the manner in which inferences are formed, and the investigation of general maxims and rules for constructing arguments, so that the conclusion may contain no inaccuracy which was not previously asserted in the premises. It has nothing to do with the truth of the facts, opinions, or presumptions, from which an inference is derived; but simply takes care that the inference shall certainly be true, if the premises be true. Thus, when we say that all men will die, and that all men are rational beings, and thence infer that some rational beings will die, the logical truth of this sentence is the same whether it be true or false that men are mortal and rational. This logical truth depends upon the structure of the sentence, and not on the particular matters spoken of. Thus,

Instead of, Write, All men will die. Every A is B. All men are rational beings. Every A is C. Therefore some rational beings will die. Therefore some Cs are Bs.

The second of these is the same proposition, logically considered, as the first; the consequence in both is virtually contained in, and rightly inferred from, the premises. Whether the premises be true or false, is not a question of logic, but of morals, philosophy, history, or any other knowledge to which their subject-matter belongs: the question of logic is, does the conclusion certainly follow if the premises be true?

Every act of reasoning must mainly consist in comparing together different things, and either finding out, or recalling from previous knowledge, the points in which they resemble or differ from each other. That particular part of reasoning which is called inference, consists in the comparison of several and different things with one and the same other thing; and ascertaining the resemblances, or differences, of the several things, by means of the points in which they resemble, or differ from, the thing with which all are compared.

There must then be some propositions already obtained before any inference can be drawn. All propositions are either assertions or denials, and are thus divided into affirmative and negative. Thus, A is B, and A is not B, are the two forms to which all propositions may be reduced. These are, for our present purpose, the most simple forms; though it will frequently happen that much circumlocution is needed to reduce propositions to them. Thus, suppose the following assertion, ‘If he should come to-morrow, he will probably stay till Monday’; how is this to be reduced to the form A is B? There is evidently something spoken of, something said of it, and an affirmative connexion between them. Something, if it happen, that is, the happening of something, makes the happening of another something probable; or is one of the things which render the happening of the second thing probable.

A │is│ B

The happening of his arrival │ │an event from which it may be to-morrow │is│ inferred as probable that he │ │ will stay till Monday.

The forms of language will allow the manner of asserting to be varied in a great number of ways; but the reduction to the preceding form is always possible. Thus, ‘so he said’ is an affirmation, reducible as follows:

What you have just said (or │is│the thing which he said. whatever else ‘so’ refers to) │ │

By changing ‘is’ into ‘is not,’ we make a negative proposition; but care must always be taken to ascertain whether a proposition which appears negative is really so. The principal danger is that of confounding a proposition which is negative with another which is affirmative of something requiring a negative to describe it. Thus ‘he resembles the man who was not in the room,’ is affirmative, and must not be confounded with ‘he does not resemble the man who was in the room.’ Again, ‘if he should come to-morrow, it is probable he will not stay till Monday,’ does not mean the simple denial of the preceding proposition, but the affirmation of the directly opposite proposition. It is,

A │is│ B

The happening of his arrival │ │an event from which it may be to-morrow, │is│ inferred to be improbable │ │ that he will stay till Monday,

whereas the following,

The happening of his arrival │ │an event from which it may be to-morrow, │is not│ inferred as probable that he │ │ will stay till Monday,

Part 2

would be expressed thus: ‘If he should come to-morrow, that is no reason why he should stay till Monday.’

Moreover, the negative words not, no, &c., have two kinds of meaning which must be carefully distinguished. Sometimes they deny, and nothing more: sometimes they are used to affirm the direct contrary. In cases which offer but two alternatives, one of which is necessary, these amount to the same thing, since the denial of one, and the affirmation of the other, are obviously equivalent propositions. In many idioms of conversation, the negative implies affirmation of the contrary in cases which offer not only alternatives, but degrees of alternatives. Thus, to the question, ‘Is he tall?’ the simple answer, ‘No,’ most frequently means that he is the contrary of tall, or considerably under the average. But it must be remembered, that, in all logical reasoning, the negation is simply negation, and nothing more, never implying affirmation of the contrary.

The common proposition that two negatives make an affirmative, is true only upon the supposition that there are but two possible things, one of which is denied. Grant that a man must be either able or unable to do a particular thing, and then not unable and able are the same things. But if we suppose various degrees of performance, and therefore degrees of ability, it is false, in the common sense of the words, that two negatives make an affirmative. Thus, it would be erroneous to say, ‘John is able to translate Virgil, and Thomas is not unable; therefore, what John can do Thomas can do,’ for it is evident that the premises mean that John is so near to the best sort of translation that an affirmation of his ability may be made, while Thomas is considerably lower than John, but not so near to absolute deficiency that his ability may be altogether denied. It will generally be found that two negatives imply an affirmative of a weaker degree than the positive affirmation.

Each of the propositions, ‘A is B,’ and ‘A is not B,’ may be subdivided into two species: the universal, in which every possible case is included; and the particular, in which it is not meant to be asserted that the affirmation or negation is universal. The four species of propositions are then as follows, each being marked with the letter by which writers on logic have always distinguished it.

A Universal Affirmative Every A is B E Universal Negative No A is B I Particular Affirmative Some A is B O Particular Negative Some A is not B

In common conversation the affirmation of a part is meant to imply the denial of the remainder. Thus, by ‘some of the apples are ripe,’ it is always intended to signify that some are not ripe. This is not the case in logical language, but every proposition is intended to make its amount of affirmation or denial, and no more. When we say, ‘Some A is B,’ or, more grammatically, ‘Some As are Bs,’ we do not mean to imply that some are not: this may or may not be. Again, the word some means, ‘one or more, possibly all.’ The following table will shew the bearing of each proposition on the rest.

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