In his excellent Logic: The Right Use of Reason in the Inquiry After Truth, Isaac Watts identifies 14 forms of syllogism that he considers valid and useful:

Figure 1:  AAA, EAE, AII, EIO
Figure 2: EAE, AEE, EIO, AOO
Figure 3: AAI, EAO, IAI, AII, OAO, EIO

From the list that I consider valid, he omits these ten as either invalid or “useless”
(note that the moods of figure 4 given in italics were accepted by most Medieval logicians):

Figure 1: AAI, EAO
Figure 2: AEO, EAO
Figure 4: AAI, EAO, AEE, IAI, EIO, AEO

Watts’ argument for rejecting figure 4 is that “this is a very indirect manner of concluding, and is never used in the sciences, nor in human life, and therefore I call it useless.”

I find it intriguing, however, that his own argument can be readily understood as an AEE-4 syllogism, one of the forms he rejects (his assumed premise is given in parentheses).

(All useful syllogisms are forms used in the sciences or in human life)
No forms used in the sciences or human life are figure 4 syllogisms.
Therefore, no figure 4 syllogisms are useful syllogisms.

I can appreciate the Medieval scholars’ rejection of AAI-1, EAO-1, AEO-2, EAO-2, and AEO-4, given that the conclusions are “weaker” (subimplication) versions of their corresponding valid forms, even if they can be shown to be equivalent to other forms they do not reject. But I cannot understand Watts’ reasoning here.

Oh well.