CHAPTER VI.

OF TRAINS OF SYLLOGISTIC REASONING.

§ 1. A Train of Syllogistic Reasoning is a combination of two or more syllogisms so connected with one another as to establish a single conclusion. When each of the component syllogisms is fully expressed, it has either of these two typical forms:

(1) That in which the single conclusion is stated last, and the conclusion in one syllogism forms a premiss in the next.

(2) That in which the single conclusion is stated first, and a premiss in one syllogism forms the conclusion in the next, or both premisses form conclusions in two distinct syllogisms.

In the example of the first form the single conclusion is "All A is E" stated last, and the conclusion of the first syllogism is a premiss in the second, and the conclusion of the second a premiss in the third.

In the example of the second form, the single conclusion is the same (All A is E), but it is stated first, and the two premisses of the 1st syllogism form the conclusions in the 2nd and 3rd, i.e., are proved by them.

The first syllogism in the first form is called a Prosyllogism in relation to the 2nd, and the 2nd in relation to the 1st is called an Episyllogism; that is, a Prosyllogism is a syllogism in a train of reasoning, whose conclusion forms a premiss in another, and an Episyllogism is a syllogism which has for one of its premisses the conclusion of another. These two terms are relative, and the same syllogism may be a prosyllogism in relation to one, and an episyllogism in relation to another. For example, the 2nd syllogism stands in the twofold relation to the 3rd and the 1st respectively.

In the example of the second form, the 1st syllogism is an episyllogism in relation to the 2nd and 3rd, and both these are prosyllogisms in relation to the 1st.

The first form is called Synthetic, Progressive, or Episyllogistic, because the advance in the reasoning is from a prosyllogism to an episyllogism, from certain premisses to the conclusion which follows from them. The second form is called Analytic, Regressive, or Prosyllogistic, because the advance in the reasoning is from an episyllogism to a prosyllogism, from a conclusion to the premisses which prove it.

§ 2. The synthetical train of syllogistic reasoning gives rise to the Synthetical Method, and the analytical train of syllogistic reasoning to the Analytical Method in Deductive Logic.

In the Synthetical Method we start with certain principles as premisses; and by comparing and combining them in various ways, we deduce the conclusions which follow necessarily from them. In the Analytical Method, on the contrary, we start with the conclusions, and proceed regressively to the principles from which they follow deductively. It is by the former method that Euclid proves his propositions; he starts with the axioms, postulates, and definitions as premisses, and proves progressively the propositions which follow from them.

§ 3. An episyllogistic or synthetic train of reasoning in which all the conclusions, except the last, are suppressed, is called a Sorites. Thus, omitting the conclusions of the 1st two syllogisms, and consequently also the minor premisses of the 2nd. and 3rd in the example given above, we get a Sorites of the following form:

All A is B,

All B is C,

All C is D,

All D is E,

.. All A is E,

in which the conclusion of the prosyllogism forms the minor premiss in the next episyllogism. This is called the Aristotelian Sorites. When the conclusion of the Prosyllogism forms, on the other hand, the major premiss in the next Episyllogism, we have a sorites of a different form, called, after its discoverer, the Goclenian Sorites. In the fully expressed form the corresponding train of syllogistic reasoning is as follows:

Suppressing all the conclusions except the last, and consequently also all the major premisses except the first, we have the following Goclenian Sorites:—

All B is C,

All A is B,

All D is A,

All E is D,

All E is C.

and suppressing all the conclusions except the last, and therefore also all the major premisses except the first, we have the following example of the Goclenian Sorites:

All D is E,

All C is D,

All B is C,

All A is B,

.. All A is E.

Both the Goclenian and the Aristotelian Sorites are abridged trains of syllogistic reasoning, and both are synthetic, progressive, or episyllogistic, the advance in the reasoning being from a prosyllogism to an episyllogism.

An Epicheirema is a prosyllogistic, analytical, or regressive train of reasoning with some of its premisses suppressed. It consists of a syllogism with a reason or reasons for one or both of its premisses being given. For example, the train of reasoning "All A is B; and all C is A, because all C is D: therefore all C is B" is an Epicheirema, in which a reason is given for one premiss, and which may be thus fully expressed :—

For the minor premiss the reason given is that ‘All C is D.'

This with that premiss evidently constitutes an enthymeme, whose major premiss is suppressed, thus :—

In the following example reasons are given for both the premisses: "All A is B, because all A is G; all C is A, because all F is A; therefore all C is B." When fully expressed it consists of the following three syllogisms :

The major premiss is proved by an enthymeme, whose major premiss is suppressed :—

The minor premiss is also proved by an enthymeme, whose minor premiss is suppressed :—

The Epicheirema is thus an abridged train of syllogistic reasoning, in which the argument proceeds analytically, from an episyllogism to a prosyllogism.

The analytic train of syllogistic reasoning which we have given at the beginning of this chapter may give rise to any of the following Epicheiremas by suppressing different premisses :