Deductive Logic Part 46

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.

[Ill.u.s.tration]

-- 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.