Deductive Logic Part 45

-- 799. Again the complex destructive may be read thus--

If A is B, C is D; and if E is F, G is H, .'. It not being true that C is D and G is H, it is not true that A is B and E is F,

which may be resolved into two steps of immediate inference, namely, conversion by contraposition followed by subalternation--

All cases of A being B and E being F are cases of C being D and G being H.

.'. Whatever is not a case of C being D and G being H is not a case of A being B and E being F.

.'. Some case which is not one of C being D and G being H is not a case of A being B and E being F.

CHAPTER XXIX.

_Of Trains of Reasoning._

-- 800. The formal logician is only concerned to examine whether the conclusion duly follows from the premisses: he need not concern himself with the truth or falsity of his data. But the premisses of one syllogism may themselves be conclusions deduced from other syllogisms, the premisses of which may in their turn have been established by yet earlier syllogisms. When syllogisms are thus linked together we have what is called a Train of Reasoning.

-- 801. It is plain that all truths cannot be established by reasoning. For the attempt to do so would involve us in an infinite regress, wherein the number of syllogisms required would increase at each step in a geometrical ratio. To establish the premisses of a given syllogism we should require two preceding syllogisms; to establish their premisses, four; at the next step backwards, eight; at the next, sixteen; and so on ad infinitum. Thus the very possibility of reasoning implies truths that are known to us prior to all reasoning; and, however long a train of reasoning may be, we must ultimately come to truths which are either self-evident or are taken for granted.

-- 802. Any syllogism which establishes one of the premisses of another is called in reference to that other a Pro-syllogism, while a syllogism which has for one of its premisses the conclusion of another syllogism is called in reference to that other an Epi-syllogism.

_The Epicheirema_.

-- 803. The name Epicheirema is given to a syllogism with one or both of its premisses supported by a reason. Thus the following is a double epicheirema--

All B is A, for it is E.

All C is B, for it is F.

.'. All C is A.

All virtue is praiseworthy, for it promotes the general welfare.

Generosity is a virtue, for it prompts men to postpone self to others.

.'. Generosity is praiseworthy.

-- 804. An epicheirema is said to be of the first or second order according as the major or minor premiss is thus supported. The double epicheirema is a combination of the two orders.

-- 805. An epicheirema, it will be seen, consists of one syllogism fully expressed together with one, or, it may be, two enthymemes (-- 557). In the above instance, if the reasoning which supports the premisses were set forth at full length, we should have, in place of the enthymemes, the two following pro-syllogisms--

(i) All E is A.

All B is E.

.'. All B is A.

Whatever promotes the general welfare is praiseworthy.

Every virtue promotes the general welfare.

.'. Every virtue is praiseworthy.

(2) All F is B.

All C is F.

.'. All C is B.

Whatever prompts men to postpone self to others is a virtue.

Generosity prompts men to postpone self to others.

.'. Generosity is a virtue.

-- 806. The enthymemes in the instance above given are both of the first order, having the major premiss suppressed. But there is nothing to prevent one or both of them from being of the second order--

All B is A, because all F is.

All C is B, because all F is.

.'. All C is A.

All Mahometans are fanatics, because all Monotheists are.

These men are Mahometans, because all Persians are.

.'. These men are fanatics.

Here it is the minor premiss in each syllogism that is suppressed, namely,

(1) All Mahometans are Monotheists.

(2) These men are Persians.

_The Sorites_.

-- 807. The Sorites is the neatest and most compendious form that can be a.s.sumed by a train of reasoning.

-- 808. It is sometimes more appropriately called the chain-argument, and map be defined as--

A train of reasoning, in which one premiss of each epi-syllogism is supported by a pro-syllogism, the other being taken for granted.

This is its inner essence.

-- 809. In its outward form it may be described as--A series of propositions, each of which has one term in common with that which preceded it, while in the conclusion one of the terms in the last proposition becomes either subject or predicate to one of the terms in the first.

-- 810. A sorites may be either--

(1) Progressive,

or (2) Regressive.

_Progressive Sorites_.

All A is B.

All B is C.