All A is B A biped is an animal, All C is A A man is a biped, All D is C A teacher is a man, All E is D Thomas Arnold was a teacher, Hence all E is B Hence Thomas Arnold was an animal. When regarded from the viewpoint of extension, the progressive sorites proceeds from the smaller to the larger while the regressive is the converse of this. The point may be illustrated by circles: Illustration: FIG. 15. Circle 1 stands for Thomas Arnold. Circle 2 stands for teacher. Circle 3 stands for man. Circle 4 stands for biped. Circle 5 stands for animal. The progressive sorites proceeds from the smaller circle to the larger, thus: All of circle 1 belongs to 2 All of circle 2 belongs to 3 All of circle 3 belongs to 4 All of circle 4 belongs to 5 Hence, All of circle 1 belongs to 5 The regressive sorites proceeds from the larger to the smaller; i. e.: All of circle 4 belongs to 5 All of circle 3 belongs to 4 All of circle 2 belongs to 3 All of circle 1 belongs to 2 Hence, All of circle 1 belongs to 5 Other differences become apparent when the omitted conclusions are expressed. _Progressive_ _Symbolized_ _Word Form_ All A is B T. Arnold was a teacher, (_A_) All B is C A teacher is a man, (_A_) ∴ All A is C ∴ T. Arnold was a man. (_A_) All C is D A man is a biped, (_A_) ∴ All A is D ∴ T. Arnold was a biped. (_A_) All D is E A biped is an animal, (_A_) ∴ All A is E ∴ T. Arnold was an animal. (_A_) In the three completed syllogisms it becomes evident that the progressive sorites uses the minor as its first premise and in consequence takes the form of the fourth figure, though the reasoning is according to the first figure. The progressive sorites must conform to the following rules: (1) The first premise may be universal or particular, all the others _must_ be universal. (2) The last premise may be affirmative or negative; all the others _must_ be affirmative. A violation of the first rule would result in undistributed middle; whereas a violation of the second rule would give illicit major. These rules may be illustrated by giving attention to the symbols of the foregoing completed syllogisms. The first completed syllogism of the sorites is: All A is B All B is C ∴ All A is C Securing a logical arrangement by interchanging the major and minor premises gives: (M) (G) (A) All B is C (First premise universal) ― (S) (M) (A) All A is B (S) (G) (A) ∴ All A is C ― Applying the rules we find this syllogism valid, or we may recall that A A is valid in the first figure. A Let us now make the first premise of the sorites _particular_ and test. Some A is B All B is C ∴ Some A is C _Arranged logically_: