The Second Figure: Conditions of Validity and Four Productive Moods

1. Structure of the second figure

In the first figure the middle term was predicate of the minor premise and subject of the major. In the second figure, the middle is the predicate in both premises. Both premises therefore relate their subjects to the same middle term.

Minor premise: minor term — middle term
Major premise: major term — middle term
Conclusion: minor term — major term

Example:
All humans are living beings.
No stone is a living being.
Therefore no human is a stone.

“Living being” is the middle in both predicates, “human” is the minor, and “stone” is the major.

2. A shared predicate is not enough

Consider this invalid inference:
All cats are animals.
All dogs are animals.
Therefore all cats are dogs.

Both premises may be true while the conclusion is false. Sharing a broad predicate does not make the subjects identical: two different kinds can both fall within “animal.” The argument has the second-figure form but is not a valid syllogism.

Identifying a figure concerns where the terms occur. Establishing validity requires further conditions governing how the premises relate.

3. The condition concerning quality

ويشترط في إنتاج الشكل الثاني بحسب الكيف أي: الإيجاب والسلب اختلاف المقدمتين
For the second figure to produce a conclusion, the premises must differ in quality: one affirmative, the other negative.

Quality means affirmation or negation; it is not the distinction between universal and particular. If the minor is affirmative, the major must be negative; if the minor is negative, the major must be affirmative.

Two affirmative premises, or two negative premises, leave the relation between the subjects undetermined even though both have the same middle predicate.

4. Why two affirmative premises fail

Minor: All students are humans.
Major: All teachers are humans.

Both premises allow overlap between the two classes, but they also allow them to be separate. The premises do not determine how students and teachers relate to one another.

Thus they entail neither “All students are teachers,” nor “No students are teachers,” nor even “Some students are teachers.” Two affirmative premises are unproductive in the second figure.

5. Why two negative premises fail

Minor: No cat is a bird.
Major: No dog is a bird.

Both merely place cats and dogs outside the class of birds. Classes excluded from the same class may overlap or be entirely separate.

The premises do not establish a relation between cats and dogs. Two negative premises cannot determine how the minor and major terms relate.

6. Why the major premise must be universal

وضروبه الناتجة أيضا أربعة
This figure, too, has four productive moods.

In each productive mood the major premise is universal. If it tells us only that some major-term instances stand in a certain relation to the middle, those may not be the instances relevant to the minor term. Opposite premise qualities alone are therefore insufficient.

For example:
All cats are animals.
Some trees are not animals.

These premises do not, by form alone, establish the required universal relation between cats and trees. A universal major is required for the standard second-figure inference.

7. Universal major versus particular major

Valid argument:
All sparrows are birds.
No fish is a bird.
Therefore no sparrow is a fish.

Invalid arrangement:
All sparrows are birds.
Some pets are not birds.
Therefore no sparrow is a pet — this does not follow.

The second major premise excludes only some pets from birds. Other pets might themselves be birds. It lacks the universality required to rule out the entire major class.

8. Why every conclusion is negative

ونتيجة هذا الشكل لا يكون إلا سالبة.
The conclusion of this figure can only be negative.

One premise affirms and the other denies a relation to the middle. If all minor instances fall in the middle while all major instances fall outside it, minor and major are distinct. The same separation follows when the minor is outside the middle and the major is wholly inside it.

Thus every valid second-figure conclusion denies a relation. It may be universal or particular, depending on the minor premise, but its quality is negative.

9. Eliminating the twelve invalid combinations

Each premise has four possible categorical forms, so the minor and major yield 16 combinations.

First condition: qualities must differ. This eliminates eight pairs in which both are affirmative or both negative.
Second condition: the major must be universal. Of the eight remaining, four have a particular major and are eliminated.

Exactly four productive moods remain. It is important to identify not merely which moods work but which condition each invalid mood violates.

10. The four productive moods

Listing the minor premise before the major, the four moods are:
1. Universal affirmative + universal negative → universal negative.
2. Universal negative + universal affirmative → universal negative.
3. Particular affirmative + universal negative → particular negative.
4. Particular negative + universal affirmative → particular negative.

In every case the middle term is predicate of both premises, the major is universal, and the qualities differ. These are the key structural facts.

11. Mood 1: the source example

كل ج ب، ولا شيء من ب أ
All C are B, and no B is A.

The book illustrates the first productive mood with an affirmative universal minor and a negative universal major. The conclusion is “No C is A.”

The minor includes every C in B, while the major excludes A from B, so no C can be A. B is the shared predicate in the original premises. Converting the universal negative major then supplies a first-figure proof.

12. Mood 1: proof by converting the major

The book proves this mood through conversion of the major premise. A universal negative converts to another universal negative.

Minor: All C are B.
Major: No A is B.
Converted major: No B is A.
Now in the first figure: All C are B; no B is A.
Conclusion: No C is A.

Conversion makes B the subject of the major, while it remains predicate of the minor. This gives the first-figure structure and demonstrates the required conclusion.

13. Mood 1: a concrete example

Minor: Every fly is an insect.
Major: No bird is an insect.
Conclusion: No fly is a bird.

Minor term: fly.
Major term: bird.
Middle: insect.

“Insect” is predicate in both premises. The minor is universal affirmative, the major universal negative, and the conclusion universal negative. Valid entailment is distinct from, and in sound reasoning combined with, true premises.

14. Mood 2: negative minor and affirmative major

لا شيء من ج ب، وكل أ ب
No C is B, and every A is B.

The minor is universal negative and the major universal affirmative. The conclusion is “No C is A.”

The minor keeps C outside B, whereas the major puts every A inside B. No C can therefore be A: otherwise it would both belong to B and be outside B, a contradiction.

15. Mood 2: converting the minor and the result

The commentary also uses conversion of the minor followed by conversion of the conclusion to show this mood's validity.

Original minor: No C is B.
Its converse: No B is C.
Original major: All A are B.
First-figure result: No A is C.
Converse of that result: No C is A.

Converting the negative minor produces an appropriate first-figure argument. Converting its universal negative conclusion gives exactly the original intended conclusion.

16. Mood 2: a concrete example

Minor: No stone is alive.
Major: Every deer is alive.
Conclusion: No stone is a deer.

“Alive” is the predicate of both premises. The minor is universal negative, the major universal affirmative, so the qualities differ and the major is universal.

The negative conclusion follows from the necessary relation of the premises, not from an additional empirical observation about stones and deer.

17. Mood 3: particular affirmative minor

بعض ج ب، ولا شيء من أ ب، فليس بعض ج أ
Some C are B; no A is B; therefore some C are not A.

The minor is particular affirmative, the major universal negative. At least one C lies in B, but A is excluded from B. That C cannot be an A.

The conclusion is particular negative, because the premises establish exclusion for at least one C, not necessarily every C.

18. Mood 3: an existence-bearing example

Minor: Some birds are flying animals.
Major: No fish is a flying animal.
Conclusion: Some birds are not fish.

The shared predicate “flying animal” is the middle. The minor asserts the existence of at least some birds in that class, while the major excludes all fish from it.

A particular minor does not justify upgrading the conclusion to a universal. This syllogism proves a claim about some birds only.

19. Mood 4: particular negative minor

بعض ج ليس ب، وكل أ ب، فبعض ج ليس أ.
Some C are not B; all A are B; therefore some C are not A.

The minor is particular negative, the major universal affirmative. Some C lie outside B; all A are contained within B. Those C outside B cannot be A.

The conclusion is particular negative. This argument does not convert a particular negative into a universal claim; it uses the universal major to infer the required exclusion.

20. Mood 4: a worked example

Minor: Some objects are not metals.
Major: Every copper object is a metal.
Conclusion: Some objects are not copper objects.

Middle: metal.
Minor: object.
Major: copper object.

The minor identifies some objects outside the metal class. The major includes all copper objects within metals. Objects outside metals cannot be copper objects, which is the relevant inference.

21. Why reductio can prove validity

The commentary uses reductio ad absurdum (khalf). Assume both premises true and the intended conclusion false. If this forces the contradiction of a premise, the assumed falsity of the conclusion must be rejected.

Example:
All cats are animals.
No stone is an animal.
Target: No cat is a stone.

If that target were false, some cats would be stones. Those cats would be animals by the minor and non-animals by the major—an outright contradiction. Thus the negative conclusion follows.

22. Conversion and reductio are different proofs

Simple conversion exchanges subject and predicate under rules that preserve truth and quality. The first two second-figure moods can be related to first-figure reasoning by this method.

Reductio temporarily assumes the contradictory of the conclusion and shows that it cannot coexist with the premises. This need not proceed by directly rearranging the premises.

Both establish the same point—necessary consequence—but the inferential routes differ.

23. Universal and particular negative conclusions

The first two productive moods yield universal negative conclusions; the third and fourth yield particular negative conclusions.

First: all minor instances are within the middle, all major instances outside → no minor is major.
Second: all minor instances are outside, all major inside → no minor is major.
Third: some minor instances are inside, all major outside → some minor are not major.
Fourth: some minor instances are outside, all major inside → some minor are not major.

The conclusion's quantity cannot be stronger than the minor premise warrants.

24. Changing quantity changes the conclusion

Compare two arguments:
Case 1: All students are humans; no stone is human; therefore no student is a stone.
Case 2: Some students are humans; no stone is human; therefore some students are not stones.

The major remains universal negative, but the minor changes from universal to particular. The first includes all students under the middle; the second asserts this only of some.

The second argument cannot be strengthened to a universal conclusion on the strength of these premises alone, even if additional facts might establish one.

25. True conclusion does not rescue invalid form

Consider:
All humans are living beings.
All birds are living beings.

“No human is a bird” may be factually true, but it does not logically follow from these two premises, since both are affirmative. Their shared predicate does not establish exclusion between the subjects.

A valid inference differs from merely arriving at a true statement. Validity asks whether the conclusion could be false while all premises remain true.

26. A reliable classification method

For any candidate second-figure argument:
1. Identify minor and major terms from the proposed conclusion.
2. Locate the middle; it must be predicate in both premises.
3. Check quantity: is the major universal?
4. Check quality: is one premise affirmative and the other negative?
5. Determine the conclusion's quantity from the minor and its negative quality from the premises.

If the condition in step 3 or 4 fails, the proposed conclusion is not warranted by the standard second-figure form.

27. Worked case: right figure, wrong mood

Minor: Some trees are fruit-bearing.
Major: Some plants are not fruit-bearing.
Proposed conclusion: Some trees are not plants.

The common predicate is “fruit-bearing,” so the arrangement is second figure. The minor is particular affirmative and the major particular negative; qualities differ. But the major is particular, so the necessary productive condition fails.

The fact that some plants bear no fruit does not determine whether any fruit-bearing trees are plants. The conclusion is not entailed.

28. A four-mood comparative exercise

With minor before major, the four patterns are:
1. All C are B; no A is B → no C is A.
2. No C is B; all A are B → no C is A.
3. Some C are B; no A is B → some C are not A.
4. Some C are not B; all A are B → some C are not A.

In each, B is the middle and is predicate of both premises. The conclusion is always negative; its universality or particularity follows the minor premise.

29. A comparison with the first figure

First figure: Middle predicate of minor and subject of major; affirmative minor and universal major. It can yield either positive or negative conclusions.
Second figure: Middle predicate of both; premises of opposite qualities and a universal major. It can yield negative conclusions only.

Both figures have four productive moods among sixteen initial combinations, but their validity conditions differ. The first figure requires an affirmative minor; the second permits a negative minor as long as the major is universal affirmative.

30. What this lesson establishes

The second figure is identified by the middle term serving as predicate in both premises. Its two productive conditions are:
1. The premises differ in quality.
2. The major premise is universal.

The first two moods yield universal negatives; the next two particular negatives. Conversion can reduce certain moods to the first figure, while reductio shows that denying their conclusions conflicts with the premises.

The next section treats the third figure, in which the middle is subject of both premises and different rules govern the quantity of the conclusion.