Sec. 10.  Is Reduction necessary, however; or have not all the Figures equal and independent validity?  In one sense not only every Figure but each Mood has independent validity:  for any one capable of abstract thinking sees its validity by direct inspection; and this is true not only of the abstract Moods, but very frequently of particular concrete arguments.  But science aims at unifying knowledge; and after reducing all possible arguments that form categorical syllogisms to the nineteen Moods, it is another step in the same direction to reduce these Moods to one form.  This is the very nature of science:  and, accordingly, the efforts of some Logicians to expound separate principles of each Figure seem to be supererogatory.  Grant that they succeed; and what can the next step be, but either to reduce these principles to the Dictum, or the Dictum and the rest to one of these principles?  Unless this can be done there is no science of Formal Logic.  If it is done, what is gained by reducing the principles of the other Figures to the Dictum, instead of the Moods of the other Figures to those of the first Figure?  It may, perhaps, be said that to show (1) that the Moods of the second, third, and fourth Figures flow from their own principles (though, in fact, these principles are laboriously adapted to the Moods); and (2) that these principles may be derived from the Dictum, is the more uncompromisingly gradual and regular method:  but is not Formal Logic already sufficiently encumbered with formalities?

Sec. 11.  Euler’s diagrams are used to illustrate the syllogism, though not very satisfactorily, thus: 

Barbara—­

[Illustration:  FIG. 5.]

Celarent—­

[Illustration:  FIG. 6.]

Darii—­

[Illustration:  FIG. 7.]

Remembering that ‘Some’ means ‘It may be all,’ it is plain that any one of these diagrams in Fig. 7, or the one given above for Barbara, may represent the denotative relations of P, M and S in Darii; though no doubt the diagram we generally think of as representing Darii is No. 1 in Fig. 7.

Remembering that A may be U, and that, therefore, wherever A occurs there may be only one circle for S and P, these syllogisms may be represented by only two circles, and Barbara by only one.

Ferio—­

[Illustration:  FIG. 8.]

Here, again, probably, we generally think of No. 1 as the diagram representing Ferio; but 2, or 3, or that given above for Celarent, is compatible with the premises.

If instead of dealing with M, P, and S, a concrete example be taken of Darii or Ferio, a knowledge of the facts of the case will show what diagram is suitable to it.  But, then, surely it must be possible to do without the diagram.  These diagrams, of course, can be used to illustrate Moods of the other Figures.

CHAPTER XI

ABBREVIATED AND COMPOUND ARGUMENTS