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word has already been identified in his mind with some one single definite object, the mention of the word will suffice to recall that object, and with certainty inform him of what it is that we speak.

In precise language, therefore, to avoid ambiguity, only one term should be used to express the same idea, and only one idea should be comprehended under the same term; and further, the precise idea to which the term was attached should be distinctly and positively defined. Now, the definition of a term is the explanation of the meaning of that term, expressed by other terms which are already understood, and not synonymous with the term to be defined.

There are two kinds of ideas, namely, simple and complex; the first are those which exist in the mind single and independent of every other idea, and can only be expressed by naming the term employed to denote them; such are some of the terms employed in Geometry, as point, space, &c.: the second are such as result from the combination or comparison of two or more simple ideas, and can be expressed in either of two ways, namely, firstly, by naming the term expressing them, or, secondly, by expressing, in the proper terms, the manner in which the complex idea is formed from other simple ideas; that is, in other words, by defining the complex idea.

The simplest form in which a definition can be expressed is, a statement of the class to which the term to be defined belongs, and of that property which distinguishes it from every other of that class; the first is called the genus, and the second the differentia of the definition. For example, the definition of a parallelogram, given at page 5, is as follows:-" A PARALLELOGRAM is a quadrilateral figure, whose opposite sides are parallel; here the genus or class to which a parallelogram belongs is that of quadrilateral figures, and the differentia or peculiar property which distinguishes a parallelogram from every other quadrilateral figure is, that its opposite sides are parallel. In the definitions throughout this work, the thing to be defined, the genus, and the differentia, are all distinguished by a difference of type; thus the thing to be defined is printed in small capitals, the genus in Roman, and the differentia in Italics.

Now, the genus and differentia must each involve a distinct idea, and since every simple idea is entirely independent of every other idea, it is evidently impossible to define it logically; hence (as is stated at page 1) it is that no logical definition can be given of such things as a point, line, or surface.

The thing to be defined can only belong to one class, but it may be distinguished from every other in that class by more than one property peculiar to itself. Hence, in the definition of a term, there can be but one genus, but there may be several differentiæ. It would not, however, be necessary to state more than one differentia in order to give a correct and sufficient logical defini

tion. And in the mathematics, especially, only one differentia should be stated in the definition, and that one should be the easiest to be expressed and understood; its other differentiæ (or peculiar properties) should be afterwards shown, by logical deductions founded on the previous definition and some other admitted truths, to belong to the thing defined.

When, in the definition of a term, more than one peculiar property or differentia is stated, it ceases then to be a logical definition, and becomes a description. The object of a description is to convey to the mind a complete notion of everything included in the complex idea described, so that the mind may perceive or take in, at one view, the whole complex idea in its full extent and generality; that is, may perceive, at one time, every other idea which it is meant to include. While, on the other hand, the object of a definition is rather to limit the mental perception, and fix it upon some one peculiar property or differentia, and to none other; to enable the mind with certainty to separate that idea from every other, and view it distinctly with reference to any one (but only one) of its distinguishing features.

By thus attending to the precise meaning of the terms employed in the subject and predicate of our propositions, we are enabled to fulfil the first requirement which we laid down as necessary to the attainment of a true conclusion, namely, "that the propositions employed as premises are not ambiguous, are correctly understood, and are true."

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We have said that every assertion involves a proposition, but it is seldom that the propositions so involved are explicitly stated in such a form as to enable us to distinguish at once the subject, predicate, and copula. Whatever the form, however, of the original proposition, it may always be so expressed that the subject and predicate shall be separated by the copula, which latter may always be reduced to the present tense of the substantive-verb To BE, namely, either "is" or "is not," "are or "are not." Thus if the original proposition were, "I have learned geometry," it might be reduced to the form "Geometry Is a science which I have learned;" or if it were "Proficiency in mathematics can only be attained by long study," it might be reduced to "Proficiency in mathematics Is only to be attained by long study." In these examples the subject is distinguished by being printed in Roman, the copula by being in small capitals, and the predicate by being in Italics, and this mode of distinguishing the several parts of every proposition will be used in the following pages.

As regards the truth or falsity of a proposition, logic has no more to do than to see that the terms employed in its subject and predicate are distinctly defined in meaning, so that the assertion made may be clearly and correctly understood; whether the assertion so made be true or false, it is not the province of logic to inquire, but of that particular branch of science or knowledge to which the subject of the proposition relates or belongs. In the propositions

which we shall employ in the following pages, to explain their use in logical reasoning, we shall therefore substitute letters as symbols, to represent the subject and predicate. Thus, if we put the letter W to represent the science of geometry and distinguish all the sciences which I have learned by the letter Y, the above proposition, "Geometry is a science which I have learned," may be expressed by "W Is Y;" and if X be made to stand for " Proficiency in mathematics," and Z for "anything which is only to be attained by long study," the second proposition may be more briefly expressed by "X IS Z."

Propositions regarded as sentences may be divided according to their grammatical structure into categorical and hypothetical; a categorical proposition makes a simple assertion, as "Every square Is a parallelogram ;" an hypothetical proposition may be either conditional, that is, when the assertion is made under a condition, as, "If a triangle is equilateral, it is equiangular," or disjunctive; that is, when the assertion involves an alternative, as, A rightangled triangle must be either isosceles or scalene." It should be observed, that most of the theorems in Euclid are in the form of conditional hypothetical propositions.

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Again, as it is the essential quality of an assertion to affirm or deny, propositions are divided according to their quality into affirmative and negative: thus, "The opposite sides of a square ARE equal to each other," is an affirmative proposition, and One side of a triangle is NOT equal to the sum of the other two," is a negative proposition.

The third division of propositions is according to their quantity into universal and particular. A proposition is said to be universal when the predicate refers to the whole of the subject, as, "Every equiangular triangle is equilateral;" here the quality of being equilateral is predicated of the whole of the class of equiangular triangles; if, however, the predicate refers only to a portion of the subject, then the proposition is said to be particular, as, "Some triangles ARE isosceles;" here the property of being isosceles is only predicated of certain particular triangles included in the term some, and there may be other triangles of which that property could not be predicated.

The following table exhibits at one view the threefold division explained above, and affords an example of each kind of proposition :

PROPOSITIONS,

Considered as sentences*, may be divided into :

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*This is said to be a division of propositions according to their substance.

According to quality, may be divided into :—

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And according to quantity, may be divided into :—

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As every proposition must be either affirmative or negative, and also either universal or particular, with the same subject and predicate four different propositions may be framed, which, for convenience, are distinguished by the four vowels, A, E, Í, and O the following table enumerates them, and gives an example of each :

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When the terms of a proposition, that is, the subject or predicate, include or relate to everything which can be referred to by such terms, they are said to be distributed, and if the contrary, non-distributed; thus in the proposition, "Every man is mortal," the subject, 66 every man," is said to be distributed, because the quality of mortality is asserted to belong to everything to which the name of man can be applied; but the predicate, "mortal," is not distributed, because the proposition does not make any assertion with regard to everything that can be called mortal. The distribution of the subject depends upon the quantity of the proposition; in a universal proposition the subject is always distributed, but never in a particular one; thus in the universal affirmative proposition, "Every X IS Y," or the universal negative, "No X IS I," the subject is distributed, because the assertion is in each case made of everything to which the symbol X can be applied; whereas in the particular affirmative proposition, "Some XS ARE YS," or the particular negative, "Some XS ARE NOT Ys," the subject is not distributed, because the assertions are only made of a portion,-some, of the things to which the symbol X refers. The distribution of the predicate depends upon the quality of the proposition, the predicate being always distributed in a negative proposition, and never in an affirmative one; thus in the universal negative proposition, "No X Is Y," or the particular negative, "Some XS ARE NOT Ys," the predicate is distributed, because the assertion made is of everything which can be represented by the symbol Y; and in the universal affirmative proposition, "Every X Is Y," or the particular affirmative, "Some XS ARE YS," the predicate is not distributed, because no assertion is made of the whole of the class which the symbol Y represents.

When two propositions which have the same subject and predicate differ in quality or quantity, or both, they are said to be

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If two universal propositions differ in quality they are called Contraries, as

A Every X 18 Y.

E No X is Y.

If two particular propositions differ in quality they are called Subcontraries, as

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Some XS ARE YS.
10 Some XS ARE NOT Y8.

If two proportions agree in quality but differ in quantity, they are called subalterns, but they are not actually opposed to each other, as

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10 Some XS ARE NOT Ys.

As regards opposed propositions, it may be observed that
In Contradictories, one must be true and the other false.
In Contraries, both may be false, but they cannot both be true.
And in Subcontraries, both may be true, but they cannot
both be false.

The following scheme will exhibit at one view the various kinds of opposition alluded to above :

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