_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 assumed 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.
All C is D.
All D is E.
.'. All A is E.
_Regressive Sorites_.
All D is E.
All C is D.
All B is C.
All A is B.
.'. All A is E.
§ 811. The usual form is the progressive; so that the sorites is
commonly described as a series of propositions in which the predicate
of each becomes the subject of the next, while in the conclusion the
last predicate is affirmed or denied of the first subject. The
regressive form, however, exactly reverses these attributes; and would
require to be described as a series of propositions, in which the
subject of each becomes the predicate of the next, while in the
conclusion the first predicate is affirmed or denied of the last
subject.
§ 812. The regressive sorites, it will be observed, consists of the
same propositions as the progressive one, only written in reverse
order. Why then, it may be asked, do we give a special name to it,
though we do not consider a syllogism different, if the minor premiss
happens to precede the major? It is because the sorites is not a mere
series of propositions, but a compressed train of reasoning; and the
two trains of reasoning may be resolved into their component
syllogisms in such a manner as to exhibit a real difference between
them.
§ 813. The Progressive Sorites is a train of reasoning in which the
minor premiss of each epi-syllogism is supported by a pro-syllogism,
while the major is taken for granted.
§ 814. The Regressive Sorites is a train of reasoning in which the
major premiss of each epi-syllogism is supported by a pro-syllogism,
while the minor is taken for granted.
_Progressive Sorites_.
(i) All B is C.
All A is B.
.'. All A is C.
(2) All C is D.
All A is C.
.'. All A is D.
(3) All D is E.
All A is D.
.'. All A is E.
_Regressive Sorites_.
(1) All D is E.
All C is D.
.'. All C is E.
(2) All C is E.
All B is C.
.'. All B is E.
(3) All B is E.
All A is B.
.'. All A is E.
§ 815. Here is a concrete example of the two kinds of sorites,
resolved each into its component syllogisms--
_Progressive Sorites_.
All Bideford men are Devonshire men.
All Devonshire men are Englishmen.
All Englishmen are Teutons.
All Teutons are Aryans.
.'. All Bideford men are Aryans.
(1) All Devonshire men are Englishmen.
All Bideford men are Devonshire men.
.'. All Bideford men are Englishmen.
(2) All Englishmen are Teutons.
All Bideford men are Englishmen.
.'. All Bideford men are Teutons.
(3) All Teutons are Aryans.
All Bideford men are Teutons.
.'. All Bideford men are Aryans.
_Regressive Sorites._
All Teutons are Aryans.
All Englishmen are Teutons.
All Devonshiremen are Englishmen.
All Bideford men are Devonshiremen.
.'. All Bideford men are Aryans.
(1) All Teutons are Aryans.
All Englishmen are Teutons.
.'. All Englishmen are Aryans.
(2) All Englishmen are Aryans.
All Devonshiremen are Englishmen.
.'. All Devonshiremen are Aryans.
(3) All Devonshiremen are Aryans.
All Bideford men are Devonshiremen.
.'. All Bideford men are Aryans.
§ 816. When expanded, the sorites is found to contain as many
syllogisms as there are propositions intermediate between the first
and the last. This is evident also on inspection by counting the
number of middle terms.
§ 817. In expanding the progressive form we have to commence with the
second proposition of the sorites as the major premiss of the first
syllogism. In the progressive form the subject of the conclusion is
the same in all the syllogisms; in the regressive form the predicate
is the same. In both the same series of means, or middle terms, is
employed, the difference lying in the extremes that are compared with
one another through them.
[Illustration]
§ 818. It is apparent from the figure that in the progressive form we
work from within outwards, in the regressive form from without
inwards. In the former we first employ the term 'Devonshiremen' as a
mean to connect 'Bideford men' with 'Englishmen'; next we employ
'Englishmen' as a mean to connect the same subject 'Bideford men' with
the wider term 'Teutons'; and, lastly, we employ 'Teutons' as a mean
to connect the original subject 'Bideford men' with the ultimate
predicate 'Ayrans.'
§ 819. Reversely, in the regressive form we first use 'Teutons' as a
mean whereby to bring 'Englishmen' under 'Aryans'; next we use
'Englishmen' as a mean whereby to bring 'Devonshiremen' under the dame
predicate 'Aryans'; and, lastly, we use 'Devonshiremen' as a mean
whereby to bring the ultimate subject 'Bideford men' under the
original predicate 'Aryans.'
§ 820. A sorites may be either Regular or Irregular.
§ 821. In the regular form the terms which connect each proposition in
the series with its predecessor, that is to say, the middle terms,
maintain a fixed relative position; so that, if the middle term be
subject in one, it will always be predicate in the other, and vice
versâ. In the irregular form this symmetrical arrangement is violated.
§ 822. The syllogisms which compose a regular sorites, whether
progressive or regressive, will always be in the first figure.
In the irregular sorites the syllogisms may fall into different
figures.
§ 823. For the regular sorites the following rules may
be laid down.
(1) Only one premiss can be particular, namely, the first, if the
sorites be progressive, the last, if it be regressive.
(2) Only one premiss can be negative, namely, the last, if the
sorites be progressive, the first, if it be regressive.
§ 824. _Proof of the Rules for the Regular Sorites_.
(1) In the progressive sorites the proposition which stands first is
the only one which appears as a minor premiss in the expanded
form. Each of the others is used in its turn as a major. If any
proposition, therefore, but the first were particular, there would
be a particular major, which involves undistributed middle, if the
minor be affirmative, as it must be in the first figure.
In the regressive sorites, if any proposition except the last were
particular, we should have a particular conclusion in the syllogism
in which it occurred as a premiss, and so a particular major in the
next syllogism, which again is inadmissible, as involving
undistributed middle.
(2) In the progressive sorites, if any premiss before the last were
negative, we should have a negative conclusion in the syllogism in
which it occurs. This would necessitate a negative minor in the next
syllogism, which is inadmissible in the first figure, as involving
illicit process of the major.
In the regressive sorites the proposition which stands first is the
only one which appears as a major premiss in the expanded form.
Each of the others is used in its turn as a minor. If any premiss,
therefore, but the first were negative, we should have a negative
minor in the first figure, which involves illicit process of the
major.
§ 825. The rules above given do not apply to the irregular sorites,
except so far as that only one premiss can be particular and only one
negative, which follows from the general rules of syllogism. But there
is nothing to prevent any one premiss from being particular or any one
premiss from being negative, as the subjoined examples will show. Both
the instances chosen belong to the progressive order of sorites.
(1) Barbara.
All B is A.
All C is B.
All C is A.
All B is A.
All C is B.
Some C is D.
All D is E
.'. Some A is E
[Illustration]
(2) Disamis.
Some C is D.
All C is A.
Some A is D.
(3) Darii.
All D is E
Some A is D.
Some A is E.
(1) Barbara.
All B is C.
All A is B.
All A is C.
All A is B.
All B is C.
No D is C.
All E is D.
.'. No A is E.
[Illustration]
(2) Cesare.
No D is C.
All A is C.
.'. No A is D.
(3) Camestres.
All E is D.
No A is D.
.'. No A is E.
§ 826. A chain argument may be composed consisting
of conjunctive instead of simple propositions. This is
subject to the same laws as the simple sorites, to which
it is immediately reducible.
_Progressive._ _Regressive._
If A is B, C is D. If E is F, G is H.
If C is D, E is F. If C is D, E is F.
If E is F, G is H. If A is B, C is D.
.'. If A is B, G is H. .'. If A is B, G is H.