Proof: Syllogism, Induction, Analogy and the Four Figures

1. From propositions to argument

Earlier lessons explained propositions, truth and falsity, contradiction, and conversion. Those discussions usually concerned the form of a proposition or its relation to another proposition.

We now turn to the argument (ḥujjah): connected statements from which another statement is inferred. Two claims about a person, for instance, may jointly support a further claim.

The crucial logical question is not merely what conclusion is stated, but whether it genuinely follows from the premises.

2. The three divisions of argument

الحجة على ثلاثة أقسام: أحدها القياس، وثانيها الاستقراء، وثالثها التمثيل.
Argument has three divisions: syllogism, induction, and analogy.

The division concerns the route of reasoning. Syllogism derives a necessary consequence from structured premises; induction moves from particular observations toward a general claim; analogy transfers a judgment from one particular case to another through a relevant shared feature.

These three kinds are named at this point in the source. Here we establish their basic differences; their extended treatments belong to later parts of the book.

3. Necessary consequence in syllogism

In a syllogism (qiyās), premises are arranged so that accepting them entails a determinate conclusion. The conclusion is not a mere guess; it is required by the logical relationship between the premises.

Example:
Every human is mortal.
Zayd is human.
Therefore Zayd is mortal.

If the premises are true, denying the conclusion would conflict with them. The distinctive feature is consequence by necessity, not simply placing several sentences together.

4. How induction proceeds

Induction (istiqrāʾ) examines particular cases and arrives at a more general judgment. For example, repeated observations that samples of a type of metal expand when heated may motivate a general rule about that metal.

The samples are particular instances; the proposed rule is general. If not every relevant case has been examined, a counterexample may still be possible.

Thus moving from particulars to a generalization differs from a syllogism in which the conclusion follows from the stated structure of the premises.

5. How analogical reasoning proceeds

Analogy (tamthīl) extends a judgment from one particular case to another on the basis of a relevant shared characteristic.

Suppose two devices have the same tested internal mechanism. Knowledge of one device's behavior may support a claim about the other. The reasoning moves from one particular device to another.

Superficial resemblance is insufficient. Unless the similarity concerns the feature relevant to the judgment, the analogy is weak.

6. Comparing the three routes

Syllogism: a necessary result from suitable general and particular premises.
Every bird is living; a pigeon is a bird; therefore a pigeon is living.

Induction: a generalization from several observed cases.
A property appears in many plants of a kind; a general rule is proposed.

Analogy: a judgment about one particular extended to another through a relevant shared basis.
One device's known feature supports an inference about a comparable device.

The principal differences are necessity, generalization, and particular-to-particular transfer, respectively.

7. The textbook definition of qiyās

وهو قول مؤلف من قضايا يلزم عنها قول آخر بعد تسليم تلك القضايا.
A syllogism is a discourse composed of propositions from which another statement necessarily follows once those propositions are granted.

Composed of propositions points to connected premises. Another statement necessarily follows requires a logical connection, not a merely accidental or temporal association.

The phrase once the propositions are granted means that we provisionally accept the premises and examine what follows from them.

8. Granting premises versus verifying them

Granting premises does not mean that every proposed premise has been established as fact. Logic can first assess the form of the inference and whether the conclusion follows.

Example:
All trees are made of iron.
All date palms are trees.
Therefore all date palms are made of iron.

The inferential form is valid: the conclusion follows if the premises are granted. But the first premise is false, so the argument does not establish a true fact about the world. Distinguish valid form from true premises.

9. The conclusion is not mere repetition

The conclusion of a syllogism often combines a relationship presented separately across the premises. Its full propositional form need not have appeared explicitly before.

Example:
Zayd is human.
Every human is an animal.
Therefore Zayd is an animal.

The first premise connects Zayd with humanity; the second connects humanity with animality. Together they yield the relation between Zayd and animality.

10. Two principal forms of qiyās

A basic division of qiyās is:
1. Exceptional syllogism (istithnāʾī): the conclusion or its contradictory is explicitly present in the premises.
2. Conjunctive syllogism (iqtirānī): neither the complete conclusion nor its contradictory is explicitly present as such in the premises.

“If Zayd is human he is an animal; but he is human” draws the consequent as a result. By contrast, “Zayd is human; every human is an animal” yields its conclusion by linking the premises.

What matters is how the conclusion occurs in the premises, not merely how many sentences are written.

11. How exceptional syllogism works

Exceptional syllogism does not mean exempting something from a rule in the everyday sense. It is a logical form in which a conditional or related premise is followed by another premise affirming or denying a part.

For example:
If Zayd is human, he is an animal.
But Zayd is human.
Therefore Zayd is an animal.

The inference succeeds because affirming the antecedent allows the consequent to follow, assuming the conditional is true.

12. The source’s affirmative example

إن كان زيد إنسانا كان حيوانا لكنه إنسان.
If Zayd is human, he is an animal; but he is human.

The antecedent is “Zayd is human”; the consequent is “Zayd is an animal.” Affirming the antecedent licenses the conclusion that he is an animal.

This differs from the invalid move: “If Zayd is human he is an animal; Zayd is an animal; therefore he must be human.” Other kinds of animal exist, so that conclusion does not follow.

13. Denying the consequent

وإن كان زيد حمارا كان ناهقا لكنه ليس بناهق.
And if Zayd were a donkey he would bray, but he does not bray.

Here the conditional is granted and the consequent is denied. It follows that Zayd is not a donkey: if he were, the assumed conditional would require braying.

This is the valid pattern “If A then B; not B; therefore not A.” Simply denying A, by contrast, does not establish not B.

14. A common conditional fallacy

The following inference is invalid:
If it rains, the ground gets wet.
The ground is wet.
Therefore it rained.

Washing can also make the ground wet. Thus affirming the consequent does not necessarily affirm the antecedent.

Likewise, “If it does not rain, the ground is dry” need not hold in every circumstance. In an exceptional syllogism, check both the conditional itself and the form of the exception.

15. Recognizing conjunctive syllogism

وإن لم تكن النتيجة ونقيضها مذكورا يسمى اقترانيا.
If neither the conclusion nor its contradictory is mentioned, the syllogism is called conjunctive.

Example:
Zayd is human.
Every human is an animal.
Conclusion: Zayd is an animal.

Neither the complete proposition “Zayd is an animal” nor its negation appears as a premise. The conclusion is obtained by joining the premises.

16. Categorical and conditional conjunctive syllogism

فصل في القياس الاقتراني: وهو قسمان: حملي وشرطي.
Chapter on conjunctive syllogism: it has two kinds, categorical and conditional.

A categorical conjunctive syllogism draws a conclusion by relating categorical premises: every human is an animal; Zayd is human; therefore Zayd is an animal.

A conditional conjunctive syllogism may employ conditional propositions or combined conditional structures. Their detailed treatment follows later. Here we focus on the basic components and four figures of categorical syllogism.

17. The two terms of a conclusion

To analyze a categorical syllogism, first identify its conclusion. The thing being spoken about is its subject, and what is predicated of it is its predicate.

Example: Every human is an animal.
Subject: human.
Predicate: animal.

The textbook calls the subject of the conclusion the minor term and its predicate the major term. Their names express their logical roles, not their word lengths.

18. Why “minor” and “major”?

The source calls the conclusion's subject minor and predicate major, noting that the former usually has fewer instances and the latter usually more.

In “Every human is an animal,” humans constitute a narrower class than animals. That explains the terminology.

For formal analysis, however, the decisive criterion is: the subject of the conclusion is minor and its predicate is major. We do not determine them by counting real individuals in every case.

19. Minor and major premises

والمقدمة التي فيها الأصغر تسمى صغرى، والتي فيها الأكبر كبرى.
The premise containing the minor term is called minor; the one containing the major term is called major.

Example:
Minor premise: Zayd is human.
Major premise: Every human is an animal.
Conclusion: Zayd is an animal.

The conclusion's subject, Zayd, occurs in the minor premise; its predicate, animal, occurs in the major. The premise names depend on their roles relative to the conclusion.

20. The middle term connects the premises

والجزء الذي تكرر بينهما يسمى حدا أوسط.
The term repeated between the two premises is called the middle term.

Example:
Zayd is human.
Every human is an animal.
Therefore Zayd is an animal.

Human is the middle term. It is linked to Zayd in the minor premise and to animal in the major, but does not occur in the conclusion. It supplies the bridge between the other terms.

21. The source example: world, changing, originated

العالم متغير، وكل متغير حادث.
The world is changing, and everything changing is originated.

Minor premise: The world is changing.
Major premise: Every changing thing is originated.
Conclusion: The world is originated.

The world is the minor term, originated the major term, and changing the middle term. This is the book's illustrative inference. The present lesson analyzes its syllogistic structure, rather than independently defending its philosophical premises.

22. Pairing, mood and figure

واقتران الصغرى بالكبرى يسمى قرينة وضربا.
The pairing of minor and major premises is called a connection and a mood.

A mood (ḍarb) concerns the combination of affirmative/negative quality and universal/particular quantity in the premises. A figure (shakl), by contrast, concerns whether the middle term is the subject or predicate of each premise.

These answer distinct questions: Where does the middle occur?—figure. What are the quantity and quality of the premises?—the basis of mood.

23. A reliable method to identify the figure

To find the figure of a categorical syllogism, first identify the minor and major terms in the conclusion. Next find the term shared by the two premises: the middle term.

Now note only its positions:
Predicate of minor, subject of major — first figure.
Predicate of both — second figure.
Subject of both — third figure.
Subject of minor, predicate of major — fourth figure.

These four exhaust the possibilities because the middle can occupy either of two positions in each of two premises.

24. First figure: middle as minor predicate

العالم متغير، وكل متغير حادث.
The world is changing, and everything changing is originated.

Minor: world — changing.
Major: every changing thing — originated.
Conclusion: the world is originated.

The middle term, changing, is the predicate of the minor and the subject of the major. This is the first figure.

Its further conditions and productive moods are treated separately in the next lesson.

25. Second figure: middle as predicate of both

كل إنسان حيوان، ولا شيء من الحجر بحيوان.
Every human is an animal, and no stone is an animal.

Minor: every human — animal.
Major: no stone — animal.
Conclusion: no human is a stone.

The middle, animal, is the predicate in both premises, so this is the second figure. Middle-term position identifies the figure; valid inference also requires suitable quantity and quality.

26. Third figure: middle as subject of both

كل إنسان حيوان، وبعض الإنسان كاتب.
Every human is an animal, and some humans are writers.

Major: every human — animal.
Minor: some humans — writers.
Conclusion: some animals are writers.

The middle term, human, is the subject of both premises, which makes this the third figure. The humans known to be writers are also animals, yielding a particular conclusion.

27. Fourth figure: middle minor subject

كل إنسان حيوان، وبعض الكاتب إنسان.
Every human is an animal, and some writers are human.

Minor premise: Every human is an animal.
Major premise: Some writers are human.
Conclusion: Some animals are writers.

In the conclusion, animal is the minor term and writer the major term. Hence the premise about animals is minor and the premise about writers is major. The middle, human, is subject of the minor and predicate of the major: exactly the fourth figure.

If one instead concludes “Some writers are animals,” the minor and major terms are re-assigned and the same material can be arranged as a first-figure syllogism. The stated conclusion matters when assigning figure.

28. The four figures at a glance

All four figures use a shared middle term; what differs is its location in the premises.

First: predicate of minor, subject of major.
Second: predicate of both.
Third: subject of both.
Fourth: subject of minor, predicate of major.

First mark the minor, major and middle terms. Identify figure from middle-term position, not from whether you already believe the conclusion.

29. Integrated worked classification

Consider:
Every scholar is human.
Every human is living.
Therefore every scholar is living.

Minor term: scholar, the conclusion's subject.
Major term: living, the conclusion's predicate.
Middle term: human, occurring in both premises and absent from the conclusion.

The middle is the predicate of the minor and the subject of the major. This is therefore a categorical conjunctive syllogism of the first figure. The full conclusion is inferred, not simply repeated from a premise.

30. Recap and preparation

The three broad types of argument are syllogism, induction and analogy. In a syllogism, a conclusion necessarily follows from granted premises; induction generalizes from particulars, and analogy reasons from one particular to another.

A syllogism may be exceptional or conjunctive. Categorical conjunctive syllogisms require identifying the minor and major premises, minor and major terms, and middle term. The middle term's location determines the four figures.

The next lesson studies the first figure in detail, including its conditions for valid inference and productive moods.