(M) (G) (A) All B is C ― (S) (M) (I) Some A is B (S) (G) (I) ∴ Some A is C _Proof_: Since one premise is particular the conclusion must be particular. (Rule 7) As there are no negatives in the argument, only one conclusion is possible; namely, a particular affirmative (I). Thus, instead of the conclusion, “All A is C,” which is an (A), it must be, “Some A is C,” or an (I). Underscoring the distributed term, it is seen that the middle term is distributed in the major premise and that no term is distributed in the conclusion. Thus the mood is valid. This is “checked” when we recall that AII is always valid in the first figure. We have now shown that the first premise of a progressive sorites may be _universal_ or _particular_. Let us further proceed to prove that all the other premises must be universal. _Data_: Given the first completed syllogism of the sorites: All A is B All B is C ∴ All A is C _Proof_: Let any other premise, such as the second, be particular; this gives the following: All A is B Some B is C ∴ Some A is C _Arranged logically_: Mood, figure, and distribution indicated. (M) (G) (I) Some B is C (S) (M) (A) All A is B ― (S) (G) (I) ∴ Some A is C We note at once that the middle term is undistributed, hence the I mood A is invalid in the first figure; reference to the valid moods in I figure _one_ “checks” this conclusion. Since no premise, other than the first, can be particular, then all save the first must be universal. The truth of the first rule has been demonstrated, and now we may follow a similar plan to prove the truth of the second rule. _Problem_: To prove that the last premise may be negative.[11] _Data_: Given the _last_ completed syllogism: { All A is D { All D is E { ∴ All A is E Let us make the last premise negative (E) and test the result. (As all but the first must be universal we cannot use an O.) All A is D No D is E ∴ No A is E _Arranged logically and symbolized_: (M) (G) (E) No D is E ― ― (S) (M) (A) All A is D ― ― (S) (G) (E) ∴ No A is E ― ― _Proof_: Negative premise; negative conclusion. No particulars. Middle term distributed in major premise. No term distributed in conclusion which is not distributed in premise where it occurs. Syllogism valid. We must now prove that all the other premises must be affirmative. _Problem_: To prove that no other premise can be negative, or that all others must be affirmative. _Data_: Given last syllogism of sorites with the first premise negative. (Any other may be taken.) No A is D All D is E ∴ No A is E _Arranged logically and symbolized_: (M) (G) (A) All D is E ― (S) (M) (E) No A is D