The First Figure: Conditions of Validity and Four Productive Moods

1. The task of studying the first figure

The previous lesson identified four figures of conjunctive syllogism. In the first figure, the middle term is the predicate of the minor premise and the subject of the major premise. This arrangement is important, but not every combination in that arrangement yields a valid inference.

We will learn the two productive conditions and explain why only four out of sixteen possible combinations of quantity and quality yield a conclusion.

For example, why does “All humans are animals” together with “All animals are bodies” entail “All humans are bodies”? The aim is to understand the connection, not merely memorize a conclusion.

2. Why the first figure is considered primary

وأشرف الأشكال من الأربعة «الشكل الأول»
Of the four figures, the first figure is regarded as the most distinguished.

It is privileged because the passage to the conclusion is especially transparent: the minor term falls under the middle, and what is universally established or denied of the middle then applies to the minor term.

This does not mean that every imaginable first-figure combination is valid. Its productive moods require particular conditions; without them even the first figure can fail.

3. Locating the three terms

The standard arrangement of the first figure is:
Minor premise: minor term — middle term
Major premise: middle term — major term
Conclusion: minor term — major term

Example:
All students are humans.
All humans are living beings.
Therefore all students are living beings.

“Student” is the minor term, “living being” the major, and “human” the middle. The middle connects the premises but does not appear in the conclusion.

4. The source gives two conditions

أما الشرائط فاثنان: أحدهما: إيجاب الصغرى، وثانيهما: كلية الكبرى.
There are two conditions: first, the minor premise must be affirmative; second, the major premise must be universal.

Affirmative minor means that the minor term is placed within the middle term rather than excluded from it. The minor may be universal affirmative or particular affirmative.

Universal major means that the predicate of the major is asserted or denied of every instance of the middle term. The major may be universal affirmative or universal negative.

5. Why the minor premise must be affirmative

Productive example: Every cat is an animal. Every animal has a body. So every cat has a body. The minor premise places cats within animals.

Unproductive example: No cat is a bird. Every bird is an animal. We cannot infer that no cat is an animal: a cat can be an animal without being a bird.

A negative minor separates the minor term from the middle term. Thus the major premise's judgment about the middle does not transfer to the minor in the way required for first-figure inference.

6. Why the major premise must be universal

If the major premise is particular, the individual or group classified under the middle by the minor premise need not belong to the “some” instances mentioned by the major.

Example: All students are humans. Some humans are poets. Neither “All students are poets” nor “Some students are poets” follows. The poets may be entirely different humans.

A universal major makes a judgment about every middle-term instance, allowing it to transfer to those included under the middle by the minor premise.

7. How the two conditions work together

An affirmative minor establishes inclusion of the minor term under the middle. A universal major states what holds, or fails to hold, for every member of that middle. Together these yield a relation between minor and major.

Affirmative major: the major term is affirmed of all the middle, so the conclusion is affirmative.
Negative major: the major term is denied of all the middle, so the conclusion is negative.

The quantity of the conclusion tracks the minor: a universal minor gives a universal conclusion; a particular minor gives a particular one.

8. Why there are sixteen initial combinations

Each categorical premise may have four basic quantity-and-quality forms:
Universal affirmative (A) — All A are B.
Universal negative (E) — No A are B.
Particular affirmative (I) — Some A are B.
Particular negative (O) — Some A are not B.

There are four possibilities for the minor and four for the major, giving 4 × 4 = 16 initial combinations. That counts arrangements, not valid inferences.

9. The affirmative-minor condition removes eight

If we eliminate the two negative forms of the minor premise, only two forms remain:
Universal affirmative minor
Particular affirmative minor

Each excluded negative minor could be combined with four major forms, so 2 × 4 = 8 arrangements are eliminated by the first condition.

Eight combinations remain with an affirmative minor, but not every major premise is universal. The second condition removes more.

10. The universal-major condition removes four more

Among the remaining eight combinations there are two affirmative minors. The particular affirmative and particular negative major forms must both be excluded because the major must be universal.

The count is:
2 affirmative minor forms × 2 particular major forms = 4 more invalid arrangements.

After removing eight and then four from the original sixteen, four productive moods remain. Each has an affirmative minor and a universal major.

11. The four productive first-figure moods

Mood 1: Universal affirmative minor + universal affirmative major → universal affirmative conclusion (AAA).
Mood 2: Universal affirmative minor + universal negative major → universal negative conclusion.
Mood 3: Particular affirmative minor + universal affirmative major → particular affirmative conclusion.
Mood 4: Particular affirmative minor + universal negative major → particular negative conclusion (O in conclusion; minor I, major E).

These descriptions list the minor premise before the major. The following examples show how each conclusion follows and why its quantity and quality are fixed.

12. Mood 1: the textbook example

كل ج ب، وكل ب د، ينتج كل ج د.
All C are B, and all B are D; therefore all C are D.

Both premises are universal affirmatives. The first places every C within B; the second places every B within D.

Consequently no C falls outside the scope of D, and the conclusion is also universal affirmative. This is the AAA form.

13. Mood 1: analyzing a concrete argument

Minor: All date palms are trees.
Major: All trees are plants.
Conclusion: All date palms are plants.

Minor term: date palm
Middle term: tree
Major term: plant

Both premises are universal affirmative. If every date palm is inside the set of trees, and every tree is inside the set of plants, then all date palms must be plants. The inference rests on nested inclusion, not a superficial verbal resemblance.

14. Mood 2: the textbook example

كل إنسان حيوان، ولا شيء من الحيوان بحجر
Every human is an animal, and no animal is a stone.

Conclusion: No human is a stone. The minor places all humans under animal, while the major excludes stones from every animal.

The result is universal negative: stonehood is denied of all humans. The negative quality comes from the major, while the affirmative minor and universal major satisfy the production conditions.

15. Mood 2: where the negation applies

Minor: All roses are flowers.
Major: No flowers are metals.
Conclusion: No roses are metals.

Unlike the first mood, the major premise here is universal negative. If metalhood is denied of every flower, and every rose is a flower, metalhood must be denied of every rose.

Incorrect conclusion: Some roses are metals. Correct conclusion: No roses are metals. The valid conclusion excludes the predicate from the entire minor class.

16. Mood 3: the textbook example

بعض الحيوان فرس، وكل فرس صهال
Some animals are horses, and every horse neighs.

بعض الحيوان صهال
Some animals neigh.

The minor does not say that every animal is a horse; it identifies only some animals as horses. The major applies neighing to all horses. Hence neighing is established for some animals, not every animal, and the conclusion is particular affirmative.

17. Why mood 3 cannot yield a universal conclusion

Minor: Some students have memorized the Qur’an.
Major: Everyone who has memorized the Qur’an knows its entirety by heart.
Conclusion: Some students know the entire Qur’an by heart.

The minor tells us about some students, not all students. Although the major is universal, it applies to whoever has memorized the Qur’an; the minor does not establish that every student has done so.

Therefore “All students know the whole Qur’an by heart” is unjustified. A universal major does not enlarge the group identified by the minor.

18. Mood 4: the textbook example

بعض الحيوان ناطق، ولا شيء من الناطق بناهق
Some animals are rational, and no rational being brays.

بعض الحيوان ليس بناهق
Some animals do not bray.

The minor establishes that certain animals are rational; the major denies braying of every rational being. Hence braying is denied of those particular animals. The conclusion is particular negative.

19. Mood 4: another worked example

Minor: Some books are dictionaries.
Major: No dictionaries are novels.
Conclusion: Some books are not novels.

Minor term: books
Middle term: dictionaries
Major term: novels

The minor is particular affirmative and the major universal negative, so the result is particular negative. “No books are novels” would be invalid, because other books may be novels.

20. Reading all four conclusions together

Read the resulting quantity and quality of each productive mood:
A minor + A major: All minor-term instances are major-term instances.
A minor + E major: No minor-term instances are major-term instances.
I minor + A major: Some minor-term instances are major-term instances.
I minor + E major: Some minor-term instances are not major-term instances.

The minor premise is written first. For these valid first-figure moods, the conclusion’s quantity follows the minor and its quality follows the major. Do not use the shortcut outside this specific context without checking its conditions.

21. An invalid combination with a negative minor

Minor: No fish are birds.
Major: All birds are living things.
Invalid conclusion: No fish are living things.

The first premise separates fish from birds. The second says birds are living things. Being outside birds does not imply being outside all living things.

Fish can be living things while both premises remain true. Therefore the stated universal negative conclusion does not follow. This illustrates the affirmative-minor requirement.

22. An invalid combination with a particular major

Minor: All pens are writing instruments.
Major: Some writing instruments are blue.
Invalid conclusion: Some pens are blue.

The blue writing instruments mentioned in the major might all be instruments of some other kind. Nothing guarantees that any blue one is a pen.

The minor is universal affirmative, but the major is particular affirmative. Meeting only one of the two conditions does not make a productive first-figure argument.

23. A common mistake about the conclusion’s quantity

Minor: Some birds are waterbirds.
Major: All waterbirds are living things.

Valid conclusion: Some birds are living things.
Conclusion not established by these premises: All birds are living things.

That general statement is true in fact, but these two premises do not establish it for every bird. A statement being true and its being entailed by these specific premises are separate issues. A universal first-figure conclusion requires a universal affirmative minor.

24. A common mistake about the conclusion’s quality

Minor: Every square is a shape.
Major: No shape is a sound.
Valid conclusion: No square is a sound.

The claim “Every square is a sound” contradicts the universal exclusion stated by the major premise.

In the first figure, the minor must be affirmative to establish connection through the middle. If the major is universally negative, the conclusion is negative, whether the minor is universal or particular.

25. Check the figure before applying its rules

Minor: All roses are flowers.
Major: All jasmines are flowers.
Invalid conclusion: All roses are jasmines.

“Flower” is the predicate in both premises. This is therefore arranged in the second figure, not the first. Belonging to the same broader class does not make roses and jasmines identical.

The first-figure conditions apply only when the middle is the minor’s predicate and major’s subject. First identify the figure, then apply its rules.

26. A qualification about a merely possible minor

The note on printed page 76 adds that a merely possible minor premise, considered as such, is not productive. Here the minor only states that belonging to the middle is possible, not that it actually holds.

For example, “Every A can be B” does not establish that every A actually is B. A major about all actual Bs cannot automatically transfer its result to As merely on the strength of possible membership.

This is a modal qualification. The four standard moods treated here use an affirmative relation strong enough to establish the required middle-term connection.

27. Five steps for checking an argument

For a new argument, proceed in this order:
1. Identify the minor and major terms from the conclusion’s subject and predicate.
2. Find the middle term shared by the premises and absent from the conclusion.
3. Check whether the middle predicates the minor and is the subject of the major.
4. Verify an affirmative minor and a universal major.
5. Derive the correctly quantified and qualified conclusion.

For example: “Some fruits are apples; all apples are food; therefore some fruits are food.” Here “apple” is the middle, the minor is particular affirmative, and the major universal affirmative; the conclusion is therefore particular affirmative.

28. Consolidated summary of the first figure

In the first figure the middle term is the minor premise’s predicate and the major premise’s subject. The two production conditions are an affirmative minor and a universal major.

Of sixteen possible combinations, twelve are unproductive and four productive:
1. A minor + A major → A conclusion.
2. A minor + E major → E conclusion.
3. I minor + A major → I conclusion.
4. I minor + E major → O conclusion.

The goal is not just to remember their names, but to test term position, premise conditions and entailment in unfamiliar arguments. The next lesson begins the conditions and productive moods of the second figure.