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In speaking of division by dichotomy we have already introduced the notion of sub-division. For scientific accuracy, and often even for practical purposes, it may be necessary to sub-divide the dividing members, again to sub-divide the results of this sub-division, and SO on. The relation of these various divisions and sub-divisions, one to another, has given rise to several logical terms, and is of sufficient importance in itself to require a brief treatment. It is best to commence with an example, and we select a mathematical one, as being of the simplest kind :

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In a series of divisions and sub-divisions like the foregoing, the term at the head of the series (in this case Figure) is called the Summum Genus. The terms at the bottom of the series (equilateral triangles, circles, &c.) are called Infimæ Species. The intermediate terms are called Subaltern Genera, or Subaltern Species, viz. subaltern genera with reference to the terms immediately below them, and subaltern species with reference to the terms immediately above them. Thus triangle would be a subaltern genus with reference to equilateral triangle, a subaltern species with reference to rectilinear figure. Species which fall immediately under the same genus, as e.g. triangle, quadrilateral, and polygon, are called Cognate

Species. A Cognate Genus is any one of the ascending genera under which the species falls. Thus triangle, rectilinear figure, figure, are all genera cognate to equilateral triangle. A differentia which constitutes, as is said, an infima species is called a Specific Difference, one which constitutes a subaltern species a Generic Difference. Thus 'equilateral' is a specific difference, 'three-sided' a generic difference of an equilateral triangle. Or three-sided' would be regarded as a specific difference, and 'rectilinear' a generic difference of a triangle. Lastly, a property which is derived from an attribute or attributes connoted by a summum or subaltern genus is called a Generic Property; a property derived from an attribute or attributes connoted by an infima species is called a Specific Property. It is, for instance, a property of all rectilinear figures that the sum of their angles is equal to twice as many right angles as the figure has sides, minus four right angles. Thus the angles of a triangle are together equal to two right angles, those of a quadrilateral to four, those of a pentagon to six, and so on. It is also a property of a triangle that it may be generated by the section of a cone, but this is not a property common to other rectilinear figures. Hence the latter would be called a specific, the former a generic property.

The instance we have given is one of the simplest that could be selected. If we had taken instead of it, say, the division of animals into vertebrate and invertebrate, of vertebrate animals into birds, reptiles, fishes, amphibia, and mammals, of mammals into the various species of men,

horses, oxen, &c., it would have required a long scientific discussion to distinguish the various species and genera, to state the specific and generic differences, and to give instances of specific and generic properties. And yet it is exactly in such a case as this that divisions and sub-divisions (or, in one word, Classifications) are most important. In fact, the sciences of Botany and Zoology (in the vulgar acceptation of these words) consist entirely of classifications. To give rules for so important and complicated a process as scientific classification, or even to attempt any precise definition of the word, would be to go beyond the scope of an elementary work like this. It may be sufficient to suggest that where, as in the case of plants and animals, species are separated from one another by an indefinite number of attributes, and may be separated by many attributes of which we are yet ignorant, our classifications, like our definitions, should always be regarded as provisional. To this we may add two plain rules, which meet with universal acceptance: first, that our classifications should proceed as gradually as possible; and second, that we should select as principles of division attributes the most fruitful in their consequences, i. e. attributes from which the largest number of important properties can be derived1. Thus the natural system of Botany, founded, in its main division, on differences in the seed-vessels of plants, is far more

1 Such attributes are called by the French physiologists caractères dominateurs. See Milne Edwards, Cours Elementaire d'Histoire Naturelle, edition septième, § 367.

instructive than the Linnæan system, founded on differences in the numbers of the pistils and stamens.

Note. We have employed the expressions summum genus' and 'infima species,' as if they were entirely relative to any particular classification. But in the Isagoge of Porphyry, and by the Scholastic logicians who, for the most part, adopted his account, both summa genera and infimæ species, as well as all the subaltern genera and species, were regarded as unalterably fixed by nature. Thus the ten Categories of Aristotle (Substance, Quantity, Quality, Relation, &c.) were regarded as the summa genera, and terms like man, horse, &c. were regarded as expressing infimæ species. Classes like 'black men,' 'Arabian horses,' &c. would not have been admitted to be species at all. We, on the contrary, conceive that there is no limit to our power of making classes; however specialized a group may be, we can almost always think of some attribute, the addition of which will make it more special still.

PART III.-OF INFERENCES.

CHAPTER I.

On the various kinds of Inferences.

THE third and most important part of Logic treats of Inferences1. Wherever we assert a proposition in consequence of one or more other propositions, or, in other words, wherever we regard one or more propositions as justifying us in asserting a proposition distinct from any that has preceded, the combination of propositions may be regarded as an inference. Thus defined, inferences may be divided into inductive and deductive, and

1 The word 'inference' is employed in no less than three different senses. It is sometimes used to express the conclusion in conjunction with the premiss or premisses from which it is derived, as when we speak of a syllogism or an induction as an inference, sometimes it is used to express the conclusion alone, sometimes the process by which the conclusion is derived from the premisses, as when we speak of Induction or Deduction as inferences or inferential processes. Except where the meaning is obvious from the context, we shall endeavour to confine the word to the first-named signification. The terms Induction and Deduction will be appropriated to express processes which result, the former in inductions or inductive inferences, the latter in deductions or deductive inferences, these being subdivided into syllogism and immediate inferences.

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