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.