The Third Figure: Conditions and Six Productive Moods

1. How the third figure differs

In the first figure, the middle term is predicate in the minor and subject in the major. In the second, it is predicate in both. In the third figure, the middle term is the subject of both premises.

We make two claims about one class and then infer a relation between the minor and major terms occurring as predicates.

Example:
All horses are animals.
All horses are living beings.
Therefore some animals are living beings.

“Horse” is the middle and serves as subject twice. It disappears from the conclusion. The particular conclusion here presupposes that horses exist.

2. Minor, major and conclusion in figure three

Let B represent the middle term, C the minor term and A the major term. The third-figure arrangement is:

Minor premise: B — C
Major premise: B — A
Conclusion: Some C — A

The predicate C of the minor premise becomes the subject of the conclusion; predicate A of the major becomes its predicate.

In “All horses are animals,” animal is the minor term. In “All horses are living beings,” living being is the major term. The conclusion joins these predicates. A term’s role depends on the conclusion, not simply the order in which premises are written.

3. The common subject does not establish validity

Consider:
No horse is a stone.
No horse is a tree.

The middle, horse, is the subject of both premises, so the arrangement is third figure. But the premises establish no definite relation between stones and trees. Two classes excluded from the same class need not stand in any specific relation to each other.

The figure is present but the argument is not productive. Locating the middle is only the first step; quantity and quality conditions must also be checked.

4. The book’s stated conditions of production

شرط إنتاج الشكل الثالث كون الصغرى موجبة وكون إحدى المقدمتين كلية
The third figure requires an affirmative minor premise and at least one universal premise.

The first is a condition of quality: the minor affirms C of the middle B. The second concerns quantity: either the minor or major must cover all members of its subject.

We must not import the rules of the first two figures unchanged. In the third figure, the major may be particular if the minor is universal affirmative.

5. Why the minor premise must affirm

An affirmative minor tells us that the middle has instances falling under the minor term. The major can then describe these same middle-term instances with respect to the major predicate.

A negative minor excludes the minor predicate from the middle and does not locate any minor-term instances to which the major can be applied.

Example:
No horse is a bird.
All horses are animals.

Neither “Some birds are animals” nor “Some birds are not animals” follows from these premises. They describe horses, not the required bird instances.

6. A negative minor: an instructive counterexample

Consider:
No cat is a fish.
All cats are animals.

The premises do not establish either “All fish are animals” or “No fish are animals.” There is no affirmative link placing any fish under the middle term cat and thus within the major premise.

The error is to treat a shared subject as if it supplied an instance of the minor predicate. The first premise excludes fish from cats; it does not provide fish to which the second premise may be applied.

7. Why at least one premise must be universal

If both premises are particular, they may concern different members of the middle class. Nothing forces the instances selected by the two premises to coincide.

Example:
Some students are Qur'an memorizers.
Some students are calligraphers.

Both claims can be true even if the two groups of students are disjoint. “Some memorizers are calligraphers” therefore does not follow.

A universal premise includes every relevant middle-term instance, allowing it to apply to the particular witnesses provided by the other premise.

8. How a universal premise secures the link

Minor: Some students are athletes.
Major: All students are human.
Conclusion: Some athletes are human.

The minor identifies some students who are athletes. The universal major includes those very students among humans, so the conclusion follows.

By contrast, “Some students are athletes; some students are human” could refer to separate individuals. The universal premise closes that inferential gap.

9. Why there are six productive moods

فضروبه الناتجة ستة
Thus, the figure has six productive moods.

The minor must be affirmative, so it can be universal affirmative or particular affirmative. The major can vary in quality and quantity, but both premises must not be particular.

Universal affirmative minor: any of the four major forms can be paired with it—universal affirmative, universal negative, particular affirmative, or particular negative.
Particular affirmative minor: the major must be universal—either affirmative or negative.

The count is therefore 4 + 2 = 6 productive moods.

10. Why the conclusion is always particular

Both premises in figure three tell us something about the same middle class. They establish a relation between the predicates for middle-term instances, but not for all instances of the minor predicate.

Example:
All horses are animals.
All horses are four-legged.

The conclusion is “Some animals are four-legged,” not “All animals are four-legged,” because animals extend beyond horses.

Important qualification: Classical syllogistic inferences from two universals to a particular conclusion presuppose relevant instances. In modern predicate logic, existence must be established separately when an empty subject class is possible.

11. Mood 1: two universal affirmatives

كل ب ج، وكل ب أ، فبعض ج أ
All B are C, and all B are A; therefore some C are A.

This is the first third-figure mood. Both premises are universal affirmative, so the affirmative-minor and universal-premise conditions hold. The conclusion is particular affirmative.

Any existing B instance belongs to both C and A. Hence, among C instances, there are some that are A.

12. Mood 1 with a concrete example

Minor: All horses are animals.
Major: All horses are living beings.
Conclusion: Some animals are living beings.

Middle: horse, subject of both premises.
Minor term: animal, predicate of the minor and subject of the conclusion.
Major term: living being, predicate of the major and conclusion.

“All animals are living beings” may be independently true, but it is not established as a universal conclusion by this syllogistic form. This form proves the claim about some animals.

13. Reducing mood 1 to the first figure

From the original minor, “All B are C,” conversion by limitation gives “Some C are B”, assuming B has instances.

Use this as a new minor:
Some C are B.
All B are A.
Therefore some C are A.

B is now predicate of the minor and subject of the major, exactly the first figure. This reduction explains the first mood’s conclusion by means of a previously established valid figure.

14. Mood 2: universal affirmative with universal negative

كل ب ج، ولا شيء من ب أ، فبعض ج ليس أ
All B are C, and no B is A; therefore some C are not A.

The minor is universal affirmative, the major universal negative, and the conclusion particular negative.

For instance, all horses are animals and no horse is a bird. Given the existence of horses, some animals are not birds. The stronger claim “No animal is a bird” does not follow.

15. Why the second mood follows

Minor: All B are C.
Major: No B is A.

Convert the minor by limitation to “Some C are B.” Since no B belongs to A, those C instances cannot belong to A. Hence “Some C are not A.”

Subtle point: The inference depends on actual B instances. If B denotes an empty class, a particular conclusion cannot be inferred from merely universal statements without an additional existence premise in modern logic.

16. Mood 3: particular affirmative and universal affirmative

بعض ب ج، وكل ب أ، فبعض ج أ
Some B are C, and all B are A; therefore some C are A.

The minor is particular affirmative; the major is universal affirmative. The minor identifies certain existing B instances as C, while the universal major makes every B an A. The resulting claim is particular affirmative.

Example:
Some students are poets.
All students are human.
Therefore some poets are human.

17. Proof by tracking the same individuals

“Some students are poets” identifies actual students who are poets. “All students are human” includes those same individuals, so some poets must be human.

In symbols, convert the particular affirmative minor:
Some C are B.
All B are A.
Therefore some C are A.

This is a first-figure syllogism with particular affirmative minor and universal affirmative major. Conversion of a particular affirmative is valid, so the reduction preserves truth.

18. Mood 4: particular affirmative with universal negative

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

The minor is particular affirmative, the major universal negative, and the conclusion particular negative.

Example:
Some students are athletes.
No student is an inanimate stone.
Therefore some athletes are not inanimate stones.

The very students identified by the minor supply the relevant athlete instances in the conclusion.

19. Why mood 4 requires no existential assumption

The minor states “Some B are C,” so B instances are explicitly provided. The major excludes every B from A. Those same B instances that are C therefore cannot be A.

Convert the minor:
Some C are B.
No B is A.
Therefore some C are not A.

This is a valid first-figure inference. Unlike the two moods with universal minors, the existence needed for the particular conclusion is already guaranteed by the minor premise.

20. Mood 5: universal affirmative with particular affirmative

كل ب ج، وبعض ب أ، فبعض ج أ
All B are C, and some B are A; therefore some C are A.

Because the minor is universal affirmative, the major may be particular affirmative. The major gives B instances that are A; the universal minor makes all of them C as well.

Example:
All students are human.
Some students are calligraphers.
Therefore some humans are calligraphers.

21. Reducing mood 5 by converting the major

The premises are:
All B are C.
Some B are A.

Convert the particular affirmative major to “Some A are B.” Then use:
Some A are B.
All B are C.
Therefore some A are C.

Converting that particular affirmative conclusion yields “Some C are A.” The proof uses conversion of the major, a first-figure syllogism, and conversion of its conclusion.

22. Mood 6: universal affirmative with particular negative

كل ب ج، وبعض ب ليس أ، فبعض ج ليس أ
All B are C, and some B are not A; therefore some C are not A.

The minor is universal affirmative and the major particular negative. Their conclusion is particular negative.

Example:
All students are human.
Some students are not poets.
Therefore some humans are not poets.

The non-poet students identified by the major are humans by the universal minor.

23. A reductio proof of mood 6

Premises:
All B are C.
Some B are not A.

Suppose the intended conclusion “Some C are not A” is false. Its contradictory would be “All C are A.” Then:
All B are C.
All C are A.
Therefore all B are A.

This contradicts the original major “Some B are not A.” Thus the intended particular negative conclusion stands.

A particular negative cannot simply be converted; that is why the reductio method is especially useful here.

24. Six moods at a glance

The sequence below is always minor premise, major premise, and conclusion:

1. Universal affirmative + universal affirmative → particular affirmative.
2. Universal affirmative + universal negative → particular negative.
3. Particular affirmative + universal affirmative → particular affirmative.
4. Particular affirmative + universal negative → particular negative.
5. Universal affirmative + particular affirmative → particular affirmative.
6. Universal affirmative + particular negative → particular negative.

In every mood the minor is affirmative, one premise universal, and the conclusion particular. Affirmative majors give affirmative conclusions; negative majors give negative conclusions.

25. The temptation to overstate a conclusion

Take:
All students are human.
Some students are poets.

Valid conclusion: Some humans are poets.
Unjustified strengthening: All humans are poets.

The premises identify poet instances within students, who are humans; they say nothing about every human beyond that group.

Rule: A third-figure conclusion is always particular, even when both premises are universal. Replacing a particular conclusion with a universal one asserts more than the premises warrant.

26. Worked case: two particular premises

Minor: Some teachers are authors.
Major: Some teachers are orators.
Proposed conclusion: Some authors are orators.

This is arranged as figure three and has an affirmative minor, but both premises are particular. The author-teachers and orator-teachers could be distinct people, so the conclusion is not entailed.

Repair: Replace the major with “All teachers are orators.” The author-teachers identified by the minor would then be orators, validating the particular affirmative conclusion.

27. Worked case: the negative-minor error

Minor: No bird is a fish.
Major: All birds are animals.
Proposed conclusion: Some fish are animals.

Bird is the subject of both premises, but the negative minor violates the productive condition. “Some fish are animals” may be independently true, yet it is not entailed by these premises.

Change the minor to “Some birds are flyers” while retaining “All birds are animals.” The conclusion “Some flyers are animals” now follows. Validity concerns what the stated premises necessitate.

28. Compare the first three figures

First figure: Middle is minor predicate and major subject. Conditions: affirmative minor, universal major. Four productive moods.

Second figure: Middle is predicate of both. Conditions: opposite premise qualities, universal major. Four moods; all conclusions negative.

Third figure: Middle is subject of both. Conditions: affirmative minor, at least one universal premise. Six moods; all conclusions particular.

The number of premises does not decide the result. The position of the middle and the productive conditions determine validity.

29. A five-step validity check

1. Identify the conclusion’s subject and predicate, thereby fixing the minor and major terms.
2. Find the middle term, repeated as subject in both premises.
3. Check the minor’s quality: it must be affirmative.
4. Check quantity: at least one premise must be universal.
5. Infer a particular conclusion whose quality corresponds to the major premise, and test it with a worked example.

This method also keeps “right figure” distinct from “productive mood.”

30. Summary and transition

The third figure has the middle as subject in both premises. Its productive conditions are an affirmative minor and at least one universal premise. They give six valid moods.

Affirmative major: particular affirmative conclusion.
Negative major: particular negative conclusion.

Conversion, reduction to the first figure, and reductio explain the results. Where a particular conclusion is drawn from universal premises alone, the existence of relevant instances must be attended to.

Next comes the fourth figure, in which the middle is subject of the minor and predicate of the major.