12. A class is defined by stating enough properties to decide whether a thing belongs to it or not. Thus, “rational animal" was given as a definition of "man." 13. If we denote by the class possessing any given property, all things not possessing this property form another class, which is called the contradictory of the first, and is denoted by non-r, meaning "not x;" e.g., the contradictory of animate is inanimate. 14. Any one thing belongs either to the class r or to the class non-r, but no thing belongs to both. It follows that r is just as much the contradictory of non-r as non-x is of x. So any class y and the class non-y are mutually the contradictories of each other, and both together include all things in the universe; e.g., unconscious and conscious. III. The Universe of Discourse. 15. In most investigations, we are not really considering all things in the world, but only the collection of all objects which are contemplated as objects about which assertion or denial may take place in the particular discourse. This collection we call our universe of discourse, leaving out of consideration, for the time, every thing not belonging to it. Thus, in talking of geometry, our terms have no reference. to perfumes. 16. Within the universe of discourse, whether large or small, the classes r and non-r are still mutually contradictory, and every thing is in one or the other; e.g., within the universe mammals, every thing is man or brute. 17. The exhaustive division into x and non-r is applicable to any universe, and so is of particular importance in logic. But a special universe of discourse may be capable of some entirely different division into contradictories, equally exhaustive. Thus, with reference to any particular magnitude, all magnitudes of that kind may be exhaustively divided into the contradictories, Greater than, equal to, less than. IV. Contranominal, Converse, Inverse, Obverse. 18. If x and y are classes, our typical statement x is y means, if a thing belongs to class x, then it also belongs to class y; e.g., Man is mortal, means, to be in the class men, is to be in the class mortals. If the typical statement is true, then every individual belongs to the class y: hence no x belongs to the class non-y, or no thing not y is a thing x; that is, every non-y is non-x: e.g., the immortals are not-human. The statements x is y, and non-y is non-x, are called each the contranominal form of the other. Though both forms express the same fact, it is, nevertheless, often of importance to consider both. One form may more naturally connect the fact with others already in our mind, and so show us an unexpected depth and importance of meaning. 19. Since x is y means all the x's are y's, the class y thus contains all the individuals of the class r, and may contain. others, besides. Some of the y's, then, must be x's. Thus, from "a crystal is solid" we infer "some solids are crystals." This guarded statement, some y is x, is called the logical converse of x is y. It is of no importance in geometry. 20. If, in the true statement is y, we simply interchange the subject and predicate, without any restriction, we get the inverse statement y is x, which may be false. In geometry it often happens that inverses are true and important. When the inverse is not true, this arises from the circumstance that the subject of the direct statement has been more closely limited than was requisite for the truth of the statement. 21. The contranominal of the inverse, namely, non-x is non-y, is called the obverse of the original proposition. Of course, if the inverse is true, the obverse is true, and vice versa. To prove the obverse, amounts to the same thing as proving the inverse. They are the same statement, but may put the meaning expressed, in a different light to our minds. 22. If the original statement is x is y, its contranominal is non-y is non-x, its inverse is y is x, its obverse is non-x is non-y. The first two are equivalent, and the last two are equivalent. Thus, of four such associated theorems it will never be necessary to demonstrate more than two, care being taken that the two selected are not contranominal. 23. From the truth of either of two inverses, that of the other cannot be inferred. If, however, we can prove them both, then the classes x and y are identical. A perfect definition is always invertable. V. On Theorems. 24. A theorem is a statement usually capable of being inferred from other statements previously accepted as true. 25. The process by which we show that it may be so inferred is called the proof or the demonstration of the theorem. 26. A corollary to a theorem is a statement whose truth follows at once from that of the theorem, or from what has been given in the demonstration of the theorem. 27. A theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. 28. A geometric theorem usually relates to some figure, and says that a figure which has a certain property has of necessity also another property; or, stating it in our typical form, x is y, "a figure which has a certain property" is "a figure having another specified property." The first part names or defines the figure to which the theorem relates; the last part contains an additional property. The theorem is first stated in general terms, but in the proof we usually help the mind by a particular figure actually drawn on the page; so that, before beginning the demonstration, the theorem is restated with special reference to the figure to be used. 29. Type. - Beginners in geometry sometimes find it difficult to distinguish clearly between what is assumed and what has to be proved in a theorem. It has been found to help them here, if the special enunciation of what is given is printed in one kind of type; the special statement of what is required, in another sort of type; and the demonstration, in still another. In the course of the proof, the reason for any step may be indicated in smaller type between that step and the next. 30. When the hypothesis of a theorem is composite, that is, consists of several distinct hypotheses, every theorem formed by interchanging the conclusion and one of the hypotheses is an inverse of the original theorem. VI. On Proving Inverses. 31. Often in geometry when the inverse, or its equivalent, the obverse, of a theorem, is true, it has to be proved geometrically quite apart from the original theorem. But if we have proved that every r is y, and also that there is but one individual in the class y, then we infer that y is x. The extra-logical proof required to establish an inverse is here contained in the proof that there is but one y. RULE OF IDENTITY. 32. If it has been proved that x is y, that no two r's are the same, and that there are as many individuals in class x as in class y, then we infer y is x. RULE OF INVERSION. 33. If the hypotheses of a group of demonstrated theorems. exhaustively divide the universe of discourse into contradictories, so that one must be true, though we do not know which, and the conclusions are also contradictories, then the inverse of every theorem of the group will necessarily be true. Examples occur in geometry of the following type: — If a is equal to b, then c is equal to d. If a is less than b, then c is less than d. Three such theorems having been demonstrated geometrically, the inverse of each is always and necessarily true. Take, for instance, the inverse of the first; namely, when c is greater than d, then a is greater than b. This must be true; for the second and third theorems imply that if a is not greater than b, then c is not greater than d. |