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; 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.
Although all ratiocination admits of being thrown into one or the other of these forms, and sometimes gains considerably by the transformation, both in clearness and in the obviousness of its consequence; there are, no doubt, cases in which the argument falls more naturally into one of the other three figures, and in which its conclusiveness is more apparent at the first glance in those figures, than when reduced to the first. Thus, if the proposition were that pagans may be virtuous, and the evidence to prove it were the example of Aristides; a syllogism in the third figure,
Aristides was virtuous, Aristides was a pagan, therefore Some pagan was virtuous,
would be a more natural mode of stating the argument, and would carry conviction more instantly home, than the same ratiocination strained into the first figure, thus--
Aristides was virtuous, Some pagan was Aristides, therefore Some pagan was virtuous.
A German philosopher, Lambert, whose _Neues Organon_ (published in the year 1764) contains among other things one of the most elaborate and complete expositions ever yet made of the syllogistic doctrine, has expressly examined what sorts of arguments fall most naturally and suitably into each of the four figures; and his solution is characterized by great ingenuity and clearness of thought.(29) The argument, however, is one and the same, in whichever figure it is expressed; since, as we have already seen, the premisses of a syllogism in the second, third, or fourth figure, and those of the syllogism in the first figure to which it may be reduced, are the same premisses in everything except language, or, at least, as much of them as contributes to the proof of the conclusion is the same. We are therefore at liberty, in conformity with the general opinion of logicians, to consider the two elementary forms of the first figure as the universal types of all correct ratiocination; the one, when the conclusion to be proved is affirmative, the other, when it is negative; even though certain arguments may have a tendency to clothe themselves in the forms of the second, third, and fourth figures; which, however, cannot possibly happen with the only class of arguments which are of first-rate scientific importance, those in which the conclusion is an universal affirmative, such conclusions being susceptible of proof in the first figure alone.