The Fourth Figure: Productive Conditions, Moods and Proofs

1. Identifying the fourth figure

In the fourth figure, the middle term is the subject of the minor premise and the predicate of the major premise. Relative to figure one, its two positions are reversed.

Let C denote the minor term, B the middle, and A the major:
Minor: B — C
Major: A — B
Intended conclusion: C — A

B appears in both premises but disappears from the conclusion. The conclusion must concern C and A, not merely repeat the statements about B.

2. Why figure four is not figure one

Figure one uses:
Minor: C — B
Major: B — A
Conclusion: C — A

Figure four uses the reversed placement of the middle: B — C and A — B. Having the same three terms does not make two syllogisms the same figure; the positions of the middle term decide the figure.

For example, “Every physician is human” and “Every surgeon is a physician” directly establish that every surgeon is human. The fourth figure examines what may follow when the result is expressed in the reverse subject–predicate direction.

3. Premise order versus logical roles

The first written premise is not automatically the major premise. The subject of the conclusion identifies the minor term and its predicate identifies the major term. We then assign the two premises accordingly.

Example:
Every teacher is human.
Every calligrapher is a teacher.
Therefore, some humans are calligraphers.

Here “human” is C, “calligrapher” A, and “teacher” B. The first statement is the minor and the second the major in this chosen conclusion, regardless of their order on the page.

4. The source’s two routes to production

فشرط إنتاج الشكل الرابع أحد الأمرين: وهو إما إيجاب المقدمتين مع كلية الصغرى، أو اختلافهما بالكيف مع كلية إحداهما.
The fourth figure is said to be productive in either of two ways: both premises are affirmative with a universal minor, or the premises differ in quality with at least one universal premise.

Ījāb means affirmation; difference in quality means one affirmative and one negative; and universality means covering all the members of the relevant class. These terms matter more than memorizing a list of moods.

5. Route one: both premises affirmative

Under the first route, the minor premise is universal affirmative; the major is affirmative and may be universal or particular. Every B belongs to C, and the major establishes that A has a positive connection with B.

Example:
All teachers are human.
Some authors are teachers.
Therefore some humans are authors.

The authors witnessed by the particular major are teachers, and every teacher is human. The conclusion follows even though the major is particular.

6. Two universal affirmative premises

Minor: All teachers are human.
Major: All literary scholars are teachers.
Conclusion: Some humans are literary scholars.

The major places scholars among teachers; the minor places all teachers among humans. Hence all scholars are human. If such scholars exist, at least some humans are scholars.

Nothing establishes that all humans are scholars. Reversing a universal affirmative without changing its quantity would be invalid.

7. Existence behind a particular conclusion

The move from “All scholars are teachers; all teachers are humans” to “Some humans are scholars” presupposes that scholars exist. Classical categorical logic commonly treats relevant classes as having instances.

If a purely hypothetical kind has no members, two universal statements may be true without yielding a true existential (“some”) conclusion in modern predicate logic.

Diagnostic question: whenever a particular conclusion follows from universals alone, identify the existential commitment that licenses “some.”

8. Particular affirmative major

Minor: All reciters are human.
Major: Some teachers are reciters.
Conclusion: Some humans are teachers.

Existence is explicitly provided by “some teachers” in the major. Every such teacher who is a reciter is human by the minor, establishing the particular conclusion.

We cannot replace “some” with “all”: the premises identify some teachers as reciters, not every teacher.

9. Route two: differing quality

The second route uses one affirmative and one negative premise, with at least one universal. The negative premise ordinarily contributes an exclusion relation.

But spotting opposite qualities and a universal is not enough by itself to license any desired conclusion. One must identify the relevant instances and check the direction of the two terms in the proposed conclusion.

This distinction prevents a syntactically similar but invalid argument from being taken as a productive mood.

10. A negative major and particular minor

Minor: Some reciters are teachers.
Major: No poet is a reciter.
Conclusion: Some teachers are not poets.

The minor supplies reciters who are teachers. The universal negative major excludes every reciter from being a poet, so those particular teachers are not poets.

This is a particular negative conclusion. It cannot be strengthened to “No teacher is a poet,” since the premises say nothing about all teachers.

11. Universal negative minor and affirmative major

Minor: No reciter is an inanimate object.
Major: Every Qur'an memorizer is a reciter.
Conclusion: No inanimate object is a memorizer.

All memorizers lie inside the class of reciters, and every reciter is excluded from inanimate objects. Consequently the two endpoint classes are disjoint.

This establishes a universal negative, not merely a particular one. Replacing the universal major with “some memorizers are reciters” would not support the same universal conclusion.

12. Universal affirmative minor and universal negative major

Minor: All judges are human.
Major: No stone is a judge.
Conclusion: Some humans are not stones, provided judges exist.

Every judge is human and the universal negative excludes judges from stones. Existing judges therefore witness humans who are not stones.

The proposed universal “No human is a stone” is stronger than the stated premises alone establish. The distinction illustrates why a particular conclusion must not be silently promoted to a universal.

13. A negative-particular trap

Consider:
All teachers are human.
Some calligraphers are not teachers.

It does not follow that some humans are not calligraphers. All humans might be calligraphers, with the particular non-teacher calligraphers being other individuals.

As a countermodel, take every teacher to be human and every human to be a calligrapher, while some non-teachers are calligraphers as well. Both premises are true and the proposed conclusion false. The mere words “universal” and “negative” do not prove validity.

14. Why the main text is brief and the notes elaborate

وشرائط إنتاج الشكل الرابع مع كثرتها وقلة حدودها مذكورة في المبسوطات
The conditions for producing results in the fourth figure, being numerous and having fewer limits, are discussed in extended treatises.

The main text deliberately does not set out every condition in this concise manual. In the same printed edition, the notes expand the topic with two broad requirements and a table marking eight productive cases.

It is useful to distinguish the short original exposition from the pedagogical elaboration in the notes, and then assess each proposed result by its own valid proof.

15. Reading the source’s eight marked cases

وضروبه الناتجة ثمانية بالخلف، وبعكس الترتيب، ثم النتيجة، أو بعكس المقدمتين.
The text lists eight productive arrangements and refers to proof by reductio, reversing the order and converting the result, or converting the premises.

The table on printed page 79 combines four possible quantities/qualities for each premise into 16 cells. Eight cells contain a displayed result; the others are marked as failures of production.

To interpret the table critically, always identify which terms occur in the desired conclusion and in which direction. A result written with its subject and predicate interchanged is not automatically the same claim.

16. Checking the direction of a tabulated conclusion

“Some humans are teachers” and “Some teachers are human” are legitimate particular-affirmative converses. But “Some humans are not teachers” does not entail “Some teachers are not human.”

The first pair follows the valid conversion of a particular affirmative. A particular negative cannot be converted in this way.

When a fourth-figure table displays a particular negative, its subject and predicate must not be reversed without a separate valid argument.

17. Source fidelity and logical verification

Some printed table entries require particular care about the direction of the conclusion: the bare surface form of the premises does not always establish the displayed orientation. Faithfully recording a table and establishing logical consequence are distinct tasks.

Test case:
All teachers are human.
Some calligraphers are not teachers.
It does not follow that some humans are not calligraphers, despite having a universal affirmative and a particular negative.

The sound method is to preserve the source's reported classification while testing each proposed conclusion by a valid conversion, reductio, or explicit witness-based argument.

18. Reduction through conversion

Minor: All reciters are human.
Major: Some teachers are reciters.
Conclusion: Some humans are teachers.

Convert the particular affirmative major: “Some reciters are teachers.” Because all reciters are human, these teachers are human; then convert the particular affirmative result to obtain “Some humans are teachers.”

At every step, conversion is applied only where its quantity and quality rules permit. In particular, a particular negative cannot be converted in the same way.

19. Establishing a result by reductio

Minor: Some reciters are teachers.
Major: No poet is a reciter.
Target: Some teachers are not poets.

Assume the contradictory of the target: “All teachers are poets.” The minor supplies reciters who are teachers; under the assumption, they are also poets. But the major excludes every poet from reciters. This is a contradiction.

The assumption is false; hence some teachers are not poets. Reductio succeeds only when the true contradictory, not merely another negative sentence, is assumed.

20. Why conclusion quality follows the exclusion

With two affirmative premises, the proof establishes a positive overlap: “Some teachers are reciters” together with “All reciters are human” gives a particular affirmative conclusion.

With one negative premise, the proof may instead establish exclusion: “No poet is a reciter” with “Some reciters are teachers” entails “Some teachers are not poets.”

This is a distinction of quality. Whether the result is universal or particular is a separate issue of quantity and must follow from what the premises cover.

21. Two particulars do not force a connection

Premises:
Some teachers are poets.
Some calligraphers are teachers.

Both can be true while the poet-teachers and calligrapher-teachers are distinct people. “Some calligraphers are poets” then need not follow.

Two particular premises do not force their witnesses to be the same individuals. Hence the proposed endpoint relation may fail regardless of what figure the two premises superficially resemble.

22. Why two negatives are insufficient

Premises:
No reciter is a stone.
No judge is a stone.

These do not determine whether reciters and judges overlap or are disjoint. Both classes merely exclude stones. Their common exclusion does not establish a relation between them.

Exclusion from a common class differs from establishing a relation between two classes. Two negative premises are not enough to give the intended syllogistic conclusion.

23. The middle term disappears

A complete syllogism uses the middle to connect the premises, but the middle itself is absent from the intended conclusion:
All teachers are human.
Some writers are teachers.
Therefore some humans are writers.

“Teacher” occurs in both premises, but only “human” and “writer” remain in the conclusion.

Repeating “Some teachers are writers” merely restates a premise. The desired inferential work is to establish a relation between the two endpoint terms.

24. Must the conclusion always be weaker?

أن النتيجة في القياس تتبع أدون المقدمتين في الكيف والكم
In a syllogism, the conclusion is said to follow the weaker premise in quality and quantity.

As a general working rule, a negative premise leads to a negative conclusion, while a particular premise often limits the result to a particular.

The note on printed page 80, however, discusses qualifications to such general formulations and recognizes cases yielding a universal conclusion. Hence the quantity of a conclusion should be proved from its exact mood, not decided only by a slogan.

25. How a universal negative can arise

Minor: No reciter is an inanimate object.
Major: Every memorizer is a reciter.
Conclusion: No inanimate object is a memorizer.

The two premises are universal, one affirmative and one negative. Every memorizer is a reciter and no reciter is inanimate, so the endpoint classes cannot overlap.

The conclusion is universal negative. This shows why a blanket claim that the fourth figure can yield only particular conclusions would be mistaken.

26. Limits on converting premises

Recall the rules of simple conversion:
Universal affirmative → particular affirmative, with existence.
Universal negative → universal negative.
Particular affirmative → particular affirmative.
Particular negative → no generally valid simple converse.

Thus “converting the premises” is not permission to reverse every sentence at will. Directly converting a particular negative can turn an apparently convincing reduction into an invalid proof.

27. A reliable procedure for testing figure four

For a new example, work in this order:
1. Mark the subject and predicate of the proposed conclusion.
2. Identify the minor and major terms.
3. Locate the middle term shared by the premises.
4. Check that it is subject of the minor and predicate of the major.
5. Identify each premise's quantity and quality.
6. Prove the conclusion by conversion, reductio, or instances; if it fails, construct a counterexample.

This gives logical consequence, not memorized shape labels, the decisive role.

28. Worked example with term labels

Minor: Every exegete is a scholar.
Major: Some authors are exegetes.
Conclusion: Some scholars are authors.

Middle: exegete — shared by both premises.
Minor term: scholar — subject of the conclusion.
Major term: author — predicate of the conclusion.

The middle is subject in the minor and predicate in the major: figure four. The universal affirmative minor plus particular affirmative major satisfies the first route. The authors who are exegetes are scholars, so the result follows.

29. Worked counterexample for a failed mood

Premises:
All teachers are human.
Some calligraphers are not teachers.
Target conclusion: Some humans are not calligraphers.

Build a countermodel: all teachers are human, all humans are calligraphers, and some additional non-teachers are calligraphers. Both premises hold, but the conclusion is false because every human is a calligrapher.

The failure of entailment is established. One explicit counterexample teaches more than merely labeling the mood unproductive.

30. The transition to conditional syllogisms

In figure four, the middle is the subject of the minor and predicate of the major. The text states two routes to productivity: two affirmatives with universal minor, or differing quality with at least one universal. The printed table marks eight arrangements; each proposed conclusion still needs a sound inference.

The core skill is identifying the middle, checking quantity and quality, and either proving the result or providing a counterexample.

The next section, al-iqtirāniyyāt min al-sharṭiyyāt, extends conjunction-based syllogisms to premises containing connected and disjunctive conditionals.