Lesson 16 · Syllogisms in real speech

Sources: Book pp. 67–71; S5 slides 4–5, 19.

People rarely speak in neat three-line syllogisms. This lesson is about finding the syllogism inside ordinary speech, building chains of them, and two special forms.

1. The enthymeme (qiyās muḍmar)

Qiyās muḍmar: a syllogism with one premise, or the conclusion, left unsaid. The unsaid part is the ḍamīr (the hidden one).

We leave things out when they are obvious, or when we trust the listener. The logician's job is to fill the gap and test it, because the hidden premise is often where the weakness hides.

Notice that in speech the conclusion often comes first, followed by "because".

S5 slide 19: "O mankind! Worship your Lord, who created you and those before you" (al-Baqara 2:21). Filled in:

Your Lord is the One who created you and those before you. (minor) Whoever created you deserves your worship. (major, unsaid) ∴ Your Lord deserves your worship. (conclusion, given as the command "worship your Lord")

The description "who created you" is the middle term: the reason for the command. (The uṣūl scholars have a rule for this: when a ruling is attached to a description, the description points to the reason for the ruling.)

2. Finding the premises: analysis (taḥlīl)

Lesson 2's five stages of thought, applied to proof. To prove a categorical conclusion:

1. Split the goal into its subject (your minor term) and predicate (your major term).
2. List what you know about each: everything said of it and everything it is said of, both affirmatively and negatively.
3. Look for a shared term that links the two lists and can serve as a middle term in a valid figure.

Example. Goal: "Nabīdh is forbidden."

• What we know of nabīdh: it is a drink, made from dates, intoxicating.
• What "forbidden" is said of: carrion, pork, every intoxicant ("every intoxicant is ḥarām", Muslim).
• The shared term: intoxicating. "Nabīdh intoxicates; every intoxicant is forbidden; ∴ nabīdh is forbidden" (Figure 1, mood 1).

If no shared term turns up, the conclusion cannot (yet) be proven from what you know.

3. Compound syllogisms (qiyās murakkab)

Analysis stops when the premises are clear or axiomatic. If a premise is itself in need of proof, it must be the conclusion of another syllogism, and that one may need another, until axioms are reached. A chain like this is a compound syllogism.

Mawṣūl (connected): each intermediate conclusion is stated, then reused.

Every poet is sensitive. Every sensitive person is tormented. ∴ Every poet is tormented. Every poet is tormented. Everyone tormented has strong feelings. ∴ Every poet has strong feelings.

Mafṣūl (separated): the intermediate conclusions are left out.

Every poet is sensitive. Every sensitive person is tormented. Everyone tormented has strong feelings. ∴ Every poet has strong feelings.

ʿAqīda link. The kalām argument for a Maker, as one separated chain:

The world is changing. Whatever changes is originated. Whatever is originated has an originator. ∴ The world has an originator.

Spelled out, it is two Figure 1 syllogisms: first "the world is originated", then "the world has an originator".

4. The syllogism of contradiction (qiyās al-khalf)

Sometimes a conclusion cannot be reached directly. Then you prove it by refuting its contradictory: since two contradictories cannot both be false (Lesson 4), if the contradictory is false, the conclusion is true.

Method

1. Take the contradictory of what you want to prove.
2. Join it to a premise you already know is true, and draw a conclusion.
3. That conclusion contradicts something you know is true.
4. So the contradictory was false, and your conclusion is proven.

The book counts it as a compound syllogism: a conditional iqtirānī ("if the conclusion is false, then X") plus an istithnāʾī ("but X is false; ∴ the conclusion is not false"). You have already used it for Figure 2, mood 4 (Lesson 14).

ʿAqīda link. The theologians' version of the proof from mutual hindrance is a khalf:

Suppose there were two gods. Suppose one wills that a body move and the other that it stay still. Either both wills happen (motion and rest together: impossible), or neither happens (then both are powerless, and neither is a god), or one happens (then the other is powerless, and not a god). Every branch is impossible, so the supposition is false: there are not two gods.

It combines khalf with sabr wa taqsīm (Lesson 15).

5. The syllogism of equality (qiyās al-musāwāt)

A is equal to B. B is equal to C. ∴ A is equal to C.

It looks like a syllogism, but there is no repeated middle term: the first premise ends in "equal to B", the second begins with "B". It only works through an outside premise: "things equal to the same thing are equal". When that outside premise is true, the inference works; when the matching outside premise is false, it fails:

• "A is half of B; B is half of C" does not give "A is half of C", because half of a half is a quarter, not a half.
• "Two is half of four; four is half of eight" does not give "two is half of eight".

S5 slides 4–5: "Ten plus ten equals twenty. Four times five equals twenty. ∴ Ten plus ten equals four times five", through the outside rule "if A and B are both equal to C, then A = B".

Book slip (p. 39): "Therefore C A is a quarter of B" should read "Therefore A is a quarter of C".

Quiz 16

1. Fill in the hidden part and name it (minor, major or conclusion):
   • a) "Zayd's testimony is rejected, because he drinks wine."
   • b) "Every created thing is in need, and the idols are created."
   • c) "He cannot be a god: he ate food."
2. Fill in al-Baqara 2:21 as a full syllogism. What is the middle term?
3. Use analysis to find a middle term for "Gambling is forbidden". (Hint: what does the Qurʾān say of it in al-Māʾida 5:90?)
4. What is the difference between a mawṣūl and a mafṣūl compound syllogism? Write the kalām chain for "the world has an originator" in mawṣūl form.
5. Explain the steps of a qiyās al-khalf. Why does it work?
6. Why is "A is equal to B; B is equal to C; ∴ A is equal to C" not a syllogism "by itself"? Give a case where the same pattern fails.
7. Review: "If the Qurʾān were from other than Allah, they would have found much contradiction in it. But they do not find it." Which move is this, and what follows?