Lesson 12 · The syllogism I: proof, its parts, truth and validity

Sources: Book pp. 38–41; S5 slides 2–8, 16–18.

1. Proof and its three kinds

The logician's highest goal is the study of proof (ḥujja, also dalīl): known taṣdīqs arranged so that they lead to an unknown one. The arranging itself is inference (istidlāl). And as Lesson 2 showed, every chain of proof must end in axioms.

Everyone reasons this way already. Who does not infer fire from smoke? The book says 99% of people are logicians by nature, without knowing it, and yet make many mistakes. Hence the need for rules.

(The direct inferences of Unit 4, contradiction and conversion, take one proposition to another and are not counted here.)

Terminology trap: S5 is titled "Analogy (القياس)", but in logic qiyās means the syllogism. The book keeps "analogy" for tamthīl. The jurists' qiyās is closer to tamthīl (Lesson 17).

2. The definition, phrase by phrase

Qiyās: a statement composed of (two or more) statements such that, if they are granted, another statement follows from them by itself, necessarily.

S5 slide 2: "Two or more propositions, given true, necessarily implies a third proposition."

• "a statement composed of statements": it is made of propositions (premises). This excludes the direct inferences, which start from a single proposition.
• "if they are granted": the premises need not actually be true for the argument to count as a syllogism. But the conclusion is only guaranteed true if they are. Truth flows forward, falsehood flows back:
  • true premises → true conclusion;
  • false conclusion → at least one false premise;
  • but false premises do not make the conclusion false.
• "follows necessarily": this excludes induction and analogy, which only make a conclusion likely.
• "by itself": this excludes the syllogism of equality (qiyās al-musāwāt), which needs an outside premise. S5 slides 4–5: "Ten plus ten equals twenty; four times five equals twenty; therefore ten plus ten equals four times five" works only through the outside rule "if A and B both equal C, then A equals B". (More in Lesson 16.)

3. The parts

S5 slide 8's example:

Every human being is an animal. Every animal is a body (jism). ∴ Every human being is a body.

The way the premises are laid out is the form (ṣūra) or figure (shakl); S5 calls it "type". The middle term is what carries the proof: the book calls it the intermediary for confirmation (wāsiṭa fi'l-ithbāt).

Book slip (p. 41 onward): the translation calls the minor and major terms "minor premise" and "major premise". Keep them apart: terms are ḥudūd, premises are muqaddimāt.

Two fiqh examples in the same shape

• S5 slide 3: "Janāba is a kind of ḥadath. Every ḥadath prevents ṣalāh. ∴ Janāba prevents ṣalāh."
• Book p. 40: "The drinker of wine is a sinner. The testimony of every sinner is rejected. ∴ The testimony of the wine drinker is rejected."

4. Truth and validity (S5 slides 16–18)

• Truth (ṣidq): are the premises true? This is about the argument's matter (mādda).
• Validity (ṣiḥḥa): does the conclusion follow from the premises? This is about its form (ṣūra).

The two are independent, so an argument can be valid with false premises, or invalid with true premises. Only a valid argument with true premises (a sound one) proves anything.

S5 slide 17:

Jesus is the son of God. The son of God is God. ∴ Jesus is God.

The form is valid: it is the same shape as "Zayd is a man; every man is mortal; ∴ Zayd is mortal". The first premise is false: "He neither begets nor is born" (al-Ikhlāṣ 112:3); "It is not befitting for Allah to take a son" (Maryam 19:35). So the argument is valid but unsound. The lesson for debate: when the form is valid, you answer by attacking a premise, not the logic.

S5 slide 18:

খিচুড়ি আমার সবচেয়ে প্রিয় খাবার। আজ বৃষ্টি পড়ছে। তাই আজ আমি আমার সবচেয়ে প্রিয় খাবার খাচ্ছি।

The premises may well be true, and the conclusion may even happen to be true (it is a rainy day, after all), but it does not follow: the two premises share no middle term. Invalid.

S5 slides 6–7:

Human beings are not horses. Horses are not thinking animals. ∴ Human beings are not thinking animals.

Both premises are true; the conclusion is false. The form is invalid: from "A is not B" and "B is not C" nothing follows about A and C (Lesson 13, rule 2).

5. Two families: iqtirānī and istithnāʾī

• Iqtirānī (conjunctive): the conclusion is not stated in the premises; its two terms are spread across them, joined by the middle. All the examples above.
  • ḥamlī if all its premises are categorical;
  • sharṭī if a conditional is involved: "Whenever water is running, it is protected; whenever water is protected, it does not become impure; ∴ whenever water is running, it does not become impure."
• Istithnāʾī (exceptive): one premise is conditional, and the conclusion (or its contradictory) already appears in it word for word. The other premise "excepts" with but (lākin):
  • "If Muḥammad is knowledgeable, we must respect him. But he is knowledgeable. ∴ We must respect him."
  • "If a person is just, he does not disobey Allah. But this person disobeyed Allah. ∴ He is not just."

Lessons 13 and 14 cover the iqtirānī ḥamlī; Lesson 15 the conditional kinds.

ʿAqīda link. The classical kalām proof that the world had a beginning:

The world is changing. Everything changing is originated (ḥādith). ∴ The world is originated.

Minor term: the world. Major term: originated. Middle term: changing. Everything now depends on the two premises, which is exactly where the theologians argued.

Quiz 12

1. Name the three kinds of proof and what each gives (certainty or probability).
2. Why does the definition say "if granted" rather than "if true"? What follows when (a) the premises are true, (b) the conclusion is false, (c) a premise is false?
3. Why does "by itself" exclude the syllogism of equality?
4. Label the minor, major and middle terms, and the minor and major premises: "Every intoxicant is forbidden. Wine is an intoxicant. ∴ Wine is forbidden."
5. Valid or invalid? True premises or not?
   • a) "Every prophet is human. Every human eats. ∴ Every prophet eats."
   • b) "Some Muslims are Arabs. Some Arabs are Christians. ∴ Some Muslims are Christians."
   • c) "Every fish flies. Every flying thing is a bird. ∴ Every fish is a bird."
6. Explain why the slide-17 argument is valid but unsound, and how a Muslim should answer it.
7. Iqtirānī or istithnāʾī? a) "If the Qurʾān were from other than Allah, they would have found much contradiction in it; but they do not; ∴ it is from Allah." b) "The world is changing; everything changing is originated; ∴ the world is originated."
8. Review: what is the contradictory of "Every S is P"?