(S) (G) (E) ∴ No A is E ― ― _Proof_: “G” is distributed in the conclusion but not in the major premise. Fallacy of illicit major. Hence no other premise can be negative. We may now consider the completed syllogisms of the _regressive_ sorites. All A is B All C is A ∴ All C is B All D is C ∴ All D is B All E is D ∴ All E is B By examining the foregoing it becomes apparent that the regressive sorites, both in form and in the reasoning, adapts itself to the first figure. The rules of the regressive sorites are just the reverse of the progressive. These are: (1) The first premise may be negative; all the others must be affirmative. (2) The last premise may be particular; all the others must be universal. It would be a valuable exercise for the student to test these rules according to the plan pursued in treating the progressive sorites. =5. IRREGULAR ARGUMENTS.= It has been intimated that a syllogistic argument, in order to be logical, should be made to conform to the _rules of the syllogism_. It must not be inferred from this, however, that all deductive reasoning is included by the logical forms here treated. There seem to be arguments which yield valid conclusions, and yet which are not logical in the strict sense of the word. The following illustrate some of these forms: (1) _ Quantitative Arguments._ John is taller than James, Albert is taller than John, ∴ Albert is taller than James. Here, apparently, is a fallacy of four terms: these four terms are (1) John, (2) taller than James, (3) Albert, (4) taller than John. Yet we know that the argument is valid. There is not a particle of doubt in the mind relative to the truth of the conclusion that “Albert is taller than James.” We are consequently forced to the inference that such quantitative arguments lie outside the field of syllogistic reasoning. The argument involves this new principle, “Whatever is greater than a _second thing_ which is greater than a _third thing_ is itself greater than a third thing.” There are many other arguments similar to this which are not syllogistic in nature. To wit: A equals B, B equals C, C equals D; A equals D. A is a brother of B, B is a brother of C, C is a brother of D; A is a brother of D. A is west of B, B is west of C, C is west of D; A is west of D. (2) _Plurative Arguments._ These are arguments in which the propositions are introduced by _more_ or _most_; e. g.: Most (more than half) of the team are seniors, Most (at least half) of the team are under twenty, ∴ Some students under twenty are seniors. I Here we have an I which is evidently valid. No term distributed and yet I the conclusion is unquestionably true. This is due to the fact that the propositions are so worded as to force an overlapping of the major and minor terms. The student may illustrate this relation by circles. =6. OUTLINE.=