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Demonstration deals with necessary truths. The definition of ‘distributively true,’ ‘essential,’ ‘universal.’

Now since the object of absolute knowledge can never undergo change, the objects of demonstrative knowledge must be necessary. Knowledge becomes demonstrative when we possess a demonstration of it, and hence demonstration is a conclusion drawn from necessary premises. We must now then state from what premises and conclusions demonstrations may be drawn; and first let us define what we mean by ‘Distributively true,’ ‘Essential’ and ‘Universal.’

By ‘Distributively true,’ I mean a quality which is not merely present in some instances and absent in others, or present at some times and absent at others; e.g. if the quality ‘Animal’ be distributively predicable of man, if it be true to say ‘this is a man,’ it must also be true to say ‘this is an animal’; and if he be the one now, then he must be the other now; so too if ‘Point’ be true of every line. An empirical proof of this is the fact that when the question is raised whether one thing is true of another distributively, our objections take the form of asserting that it is not true of some particular instance or at some particular time.

I. ‘Essential’ qualities are all those which enter into the essence of a thing, (as ‘line’ does into that of ‘triangle,’ and ‘point’ into that of ‘line’; for ‘line’ and ‘point’ belong to the essence of ‘triangle’ and line respectively), and are mentioned in their definition.

II. Essential qualities are, further, attributes of subjects in the definition of which the subject is mentioned, thus ‘Straight’ or ‘Curved’ are essential attributes of ‘Line’; ‘Odd’ or ‘Even’ of ‘Number’; as also ‘Prime’ and ‘Compound,’ ‘Equilateral’ (as 3) and ‘Scalene’ (as 6); in all these cases the things form part of the definition of the real nature of the attributes mentioned, these things being in the first instance ‘Line,’ in the second ‘Number.’ So too in other instances I call attributes which inhere in either of these ways ‘essential,’ while attributes which do not belong to the subject in either of these ways I call ‘accidental’; e.g. ‘Musical’ or ‘White’ as applied to ‘Animal.’

III. Thirdly, essential is that which is not predicated of anything other than itself as attribute of subject; thus if I say, ‘the walking thing,’ some other independent thing is ‘walking’ or is ‘white.’ On the other hand substances and everything which denotes a particular object are not what they are in virtue of being anything else but what they are. Things then which are not predicable of any subject I call ‘essential,’ those which are so predicable ‘accidental’ [in the sense of dependent].

IV. In a fourth sense the attribute which exists in a subject as a result of itself is essential, while that which is not self-caused is accidental. E.g. Suppose lightning to appear while a person is walking. This is accidental, for the lightning is not caused by his walking, but, as we say, ‘it was a coincidence.’ If, however, the attribute be self-caused it is essential: e.g. if someone is wounded and dies, his death is an essential consequence of the wound, since it has been caused by it:—the wound and death are not an accidental coincidence. In the case then of the objects of absolute knowledge, that which is called ‘essential’ in the sense of inhering in the attributes or of having the latter inhering in it is self-caused and necessary; for it must inhere either unconditionally or as one of a pair of contraries, e.g. as either straight or curved inhere in line, odd or even in number. Contrariety consists in either the privation or the contradiction of a quality in the case of homogeneous subjects: e.g. in the case of numbers ‘even’ is that which is not ‘odd,’ in so far as one of these qualities is necessarily present in a subject. Hence, if one of these qualities must be either affirmed or denied, essential attributes are necessary. This then may suffice for the definition of Distributive and Essential.

By ‘Universal’ I mean that which is true of every case of the subject and of the subject essentially and as such. It is clear then that all universal attributes inhere in things necessarily. Now ‘essentially’ and ‘as such’ are identical expressions: e.g. Point and Straight are essential attributes of line, in that they are attributes of it as such. Or again the possession of two right angles is an attribute of triangle as such, for the angles of a triangle are essentially equal to two right angles. The condition of universality is satisfied only when it is proved to be predicable of any member that may be taken at random of the class in question, but of no higher class; e.g. the possession of two right angles is not a universal attribute of figure, for though one may demonstrate of a particular figure that it has two right angles, it cannot be done of any and every figure, nor does the demonstrator make use of any and every figure, for a square is a figure, but its angles are not equal to two right angles. Any and every isosceles triangle has its angles equal to two right angles, but it is not a primary, ‘triangle’ standing yet higher. Thus any primary taken at random which is shewn to have its angles equal to two right angles, or to possess any other quality, is the primary subject of the universal predicate, and it is to that demonstration primarily and essentially applies; to everything else it applies only in a sense. Nor is this quality of having its angles equal to two right angles a universal attribute of isosceles triangle, but is of a wider application.