4. SORITES.
A sorites is a series of syllogisms in which all of the conclusions are omitted except the last one.
Just as the epicheirema is a combination of enthymemes of the first and second orders, so the sorites is a combination of enthymemes of the third order. If each conclusion were written, the sorites would take the form of prosyllogisms and episyllogisms. Two forms of the sorites are recognized by logicians. These are the progressive or Aristotelian, and the regressive or Goclenian.
Illustrations.
Progressive
Symbolized. Put in Word Form.
All A is B Thomas Arnold was a teacher,
All B is C A teacher is a man,
All C is D A man is a biped,
All D is E A biped is an animal,
Hence all A is E Hence Thomas Arnold was an animal.
Regressive
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:
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:
(A) All (M)
B is (G)
C (First premise universal)
(A) All (S)
A is (M)
B
(A) ∴ All (S)
A is (G)
C
Applying the rules we find this syllogism valid, or we may recall that A
A
A is valid in the first figure.
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:
(A) All (M)
B is (G)
C
(I) Some (S)
A is (M)
B
(I) ∴ Some (S)
A is (G)
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 furtherproceed 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.
(I) Some (M)
B is (G)
C
(A) All (S)
A is (M)
B
(I) ∴ Some (S)
A is (G)
C
We note at once that the middle term is undistributed, hence the mood I
A
I is invalid in the first figure; reference to the valid moods in 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:
(E) No (M)
D is (G)
E
(A) All (S)
A is (M)
D
(E) ∴ No (S)
A is (G)
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:
(A) ∴ All (M)
D is (G)
E
(E) No (S)
A is (M)
D
(E) ∴ No (S)
A is (G)
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.