“The syllogism in the first figure alone is logically valid. In the second, there is an undistributed middle term; in the third, an illicit process of the minor.”[400]

[400] On this subject the student may be referred to the remainder of the note from which the above extract is taken, and to Hamilton, Discussions, pp. 152 to 156. Compare also Karslake, Aids to the Study of Logic, Book II.

An enthymeme is now usually defined as a syllogism incompletely stated, one of the premisses or the conclusion being understood but not expressed.[401] The arguments of everyday life are to a large extent enthymematic in this sense; and the same may be said of fallacious arguments, which are seldom completely stated, or their want of cogency would be more quickly recognised.

[401] This account of the enthymeme appears to have been originally based on the erroneous idea that the name signified the retention of one premiss in the mind, ἐν θυμῷ. Thus, in the Port Royal Logic, an enthymeme is described as “a syllogism perfect in the mind, but imperfect in the expression, since some one of the propositions is suppressed as too clear and too well known, and as being easily supplied by the mind of those to whom we speak” (p. 229). As regards the true origin of the name enthymeme, see Mansel’s Aldrich, p. 218.

An enthymeme is said to be of the first order when the major premiss is suppressed; of the second order when the minor premiss is suppressed; and of the third order when the conclusion is suppressed.

Thus, “Balbus is avaricious, and therefore, he is unhappy,” is an enthymeme of the first order; “All avaricious persons are unhappy, and therefore, Balbus is unhappy,” is an enthymeme of the second order; “All avaricious persons are unhappy, and Balbus is avaricious,” is an enthymeme of the third order.

323. The Polysyllogism and the Epicheirema.—A chain of syllogisms, that is, a series of syllogisms so linked together that the conclusion of one becomes a premiss of another, is called a polysyllogism. In a polysyllogism, any individual syllogism 369 the conclusion of which becomes the premiss of a succeeding one is called a prosyllogism, any individual syllogism one of the premisses of which is the conclusion of a preceding syllogism is called an episyllogism. Thus,—

All C is B,⎫
All B is C,⎬ prosyllogism,
therefore, All B is D,⎭⎫
but All A is B,⎬ episyllogism.
therefore, All A is D,⎭

The same syllogism may of course be both an episyllogism and a prosyllogism, as would be the case with the above episyllogism if the chain were continued further.

A chain of reasoning[402] is said to be progressive (or synthetic or episyllogistic) when the progress is from prosyllogism to episyllogism. Here the premisses are first given, and we pass on by successive steps of inference to the ultimate conclusion which they yield. A chain of reasoning is, on the other hand, said to be regressive (or analytic or prosyllogistic) when the progress is from episyllogism to prosyllogism. Here the ultimate conclusion is first given and we pass back by successive steps of proof to the premisses on which it may be based.[403]