No C is B All A is B therefore No A is C, may be reduced as follows. The proposition, No C is B, being a universal negative, admits of simple conversion, and may be changed into No B is C, which, as we showed, is the very same assertion in other words—the same fact differently expressed. This transformation having been effected, the argument assumes the following form: No B is C All A is B therefore No A is C, which is a good syllogism in the second mood of the first figure. Again, an argument in the first mood of the third figure must resemble the following: All B is C All B is A therefore Some A is C, where the minor premise, All B is A, conformably to what was laid down in the last chapter respecting universal affirmatives, does not admit of simple conversion, but may be converted _per accidens_, thus, Some A is B; which, though it does not express the whole of what is asserted in the proposition All B is A, expresses, as was formerly shown, part of it, and must therefore be true if the whole is true. We have, then, as the result of the reduction, the following syllogism in the third mood of the first figure: All B is C Some A is B, from which it obviously follows, that Some A is C. In the same manner, or in a manner on which after these examples it is not necessary to enlarge, every mood of the second, third, and fourth figures may be reduced to some one of the four moods of the first. In other words, every conclusion which can be proved in any of the last three figures, may be proved in the first figure from the same premises, with a slight alteration in the mere manner of expressing them. Every valid ratiocination, therefore, may be stated in the first figure, that is, in one of the following forms: Every B is C No B is C All A is B, All A is B, Some A is B, Some A is B, therefore therefore All A is C. No A is C. Some A is C. Some A is not C. Or, if more significant symbols are preferred: To prove an affirmative, the argument must admit of being stated in this form: All animals are mortal; All men/Some men/Socrates are animals; therefore All men/Some men/Socrates are mortal. To prove a negative, the argument must be capable of being expressed in this form: No one who is capable of self-control is necessarily vicious; No one who is capable of self-control is necessarily vicious; All negroes/Some negroes/Mr. A’s negro are capable of self-control; therefore No negroes are/Some negroes are not/Mr. A’s negro is not necessarily vicious.