Lesson 15 · Conditional reasoning: the conditional iqtirānī and the istithnāʾī
Source: Book pp. 56–67.
1. The conditional iqtirānī (in brief)
An iqtirānī syllogism can be built with conditionals as well. The book only sketches this, and so will we.
- Two muttaṣilas sharing a whole part work exactly like categorical syllogisms, with the same figures and rules: "Whenever a person is wise, he is content with what suffices him. Whenever he is content with what suffices him, he is not needy. ∴ Whenever a person is wise, he is not needy."
- A ḥamliyya and a muttaṣila: "Whenever a metal is gold, it is rare. Everything rare is valuable. ∴ Whenever a metal is gold, it is valuable."
- A ḥamliyya and a munfaṣila: "Three is a number. Every number is either even or odd. ∴ Three is either even or odd."
Two cautions from the book:
- Only luzūmiyya conditionals work (Lesson 10). A coincidence links nothing.
- Munfaṣilas cannot be combined directly. First turn each into the muttaṣilas it implies, then combine those.
Turning a munfaṣila into muttaṣilas (pp. 58–59)
| Munfaṣila | The conditionals it gives |
|---|---|
| Ḥaqīqiyya: "A number is either even or odd." | if even, not odd; if odd, not even; if not even, odd; if not odd, even |
| Māniʿat al-jamʿ: "It is either a tree or a rock." | if a tree, not a rock; if a rock, not a tree (only these two) |
| Māniʿat al-khuluww: "Zayd is either in the water or not drowning." | if not in the water, not drowning; if drowning, in the water (only these two) |
2. The istithnāʾī (exceptive) syllogism
One premise is a conditional. The other, introduced by "but" (lākin), affirms or denies one part of it. The conclusion, or its contradictory, is already written in the conditional.
Conditions
- One of the premises must be universal (the conditional should hold "always" or "whenever").
- The conditional must be luzūmiyya, not coincidental.
- A disjunctive conditional must be affirmative (convert a negative one first).
With a muttaṣila: two valid moves, two fallacies
Take: "Whenever water is running, it does not become impure."
| Move | Example | Valid? |
|---|---|---|
| Affirm the antecedent → affirm the consequent | ...but this water is running; ∴ it does not become impure. | ✅ |
| Deny the consequent → deny the antecedent | ...but this water did become impure; ∴ it is not running. | ✅ |
| Affirm the consequent → affirm the antecedent | ...but this water did not become impure; ∴ it is running. | ❌ it may be a large still body of water |
| Deny the antecedent → deny the consequent | ...but this water is not running; ∴ it became impure. | ❌ same reason |
Why. The consequent can be wider than the antecedent (Lesson 7). Whatever is running is protected, but not everything protected is running. Establishing the narrower establishes the wider; denying the wider denies the narrower; but not the other way round.
With a munfaṣila
| Kind | Affirm one → deny the other | Deny one → affirm the other |
|---|---|---|
| Ḥaqīqiyya | ✅ | ✅ |
| Māniʿat al-jamʿ (no joining) | ✅ | ❌ both might be absent |
| Māniʿat al-khuluww (no absence) | ❌ both might be present | ✅ |
With a ḥaqīqiyya of even/odd, all four work: "This number is even, so not odd"; "This number is not even, so it is odd"; and so on.
More than two options. "A word is either a noun, a verb or a particle." Affirm one ("it is a noun") and you deny the rest ("not a verb and not a particle"). Deny one ("it is not a noun") and you are left with the others as a smaller disjunction ("it is either a verb or a particle"), which you can narrow again. This step-by-step narrowing is sabr wa taqsīm ("probing and dividing"), the method of elimination used in Lesson 7 and in debates everywhere.
Book slip (p. 66): "not a verb and it is not a noun" should read "not a verb and not a particle", and "either a verb or a noun" should read "either a verb or a particle".
3. ʿAqīda link: two Qurʾānic proofs
The proof from mutual hindrance (dalīl al-tamānuʿ), al-Anbiyāʾ 21:22. "Had there been within them gods besides Allah, they would both have been ruined."
If there were gods in the heavens and earth besides Allah, they would be ruined. (muttaṣila) But they are not ruined. (denying the consequent) ∴ There are no gods in them besides Allah. (denying the antecedent)
The form is valid. The theologians' debates were about the matter: exactly why the conditional holds (why two gods would bring ruin).
The proof by elimination, al-Ṭūr 52:35–36. "Or were they created by nothing, or are they the creators [of themselves]? Or did they create the heavens and the earth? Rather, they are not certain." The options are laid out and closed one by one:
They were either created by nothing, or created themselves, or have a Creator. Not by nothing (nothing comes from nothing). Not by themselves (nothing makes itself before it exists). ∴ They have a Creator.
This is a disjunction of the ḥaqīqiyya kind followed by denying all but one option: sabr wa taqsīm.
Quiz 15
- Write out the four conditionals implied by "A number is either even or odd". Which of them can you get from "It is either a tree or a rock"?
- Valid or invalid? Name the move.
- a) "If he is a prophet, he is truthful. He is a prophet. ∴ He is truthful."
- b) "If he is a prophet, he is truthful. He is truthful. ∴ He is a prophet."
- c) "If the wuḍūʾ is valid, the water was pure. The water was not pure. ∴ The wuḍūʾ is not valid."
- d) "If it is khamr, it is forbidden. It is not khamr. ∴ It is not forbidden."
- Why are the two fallacies fallacies? Use the idea of "wider" and "narrower".
- "The thing is either a tree or a rock. It is not a tree." What follows, and why?
- "Zayd is either in the water or not drowning. He is drowning." What follows?
- Set out al-Anbiyāʾ 21:22 as an istithnāʾī syllogism and name the move.
- Set out al-Ṭūr 52:35 as sabr wa taqsīm.
- Review: name the figure and mood: "No created thing is eternal; Allah is eternal; ∴ Allah is not a created thing."
Show answers
- If even, not odd; if odd, not even; if not even, odd; if not odd, even. From "tree or rock" (no joining) only the first kind: if a tree, not a rock; if a rock, not a tree.
- a) valid, affirming the antecedent · b) invalid, affirming the consequent (many truthful people are not prophets) · c) valid, denying the consequent · d) invalid, denying the antecedent (other things are forbidden too)
- The consequent can be wider than the antecedent. Affirming the wider (truthful) does not establish the narrower (prophet), and denying the narrower (khamr) does not deny the wider (forbidden).
- Nothing follows. It is a māniʿat al-jamʿ: both can be absent (it could be a horse), so denying one does not establish the other.
- He is in the water. It is a māniʿat al-khuluww: the two sides cannot both be false, and "drowning" is the denial of "not drowning", so the other side must be true.
- "If there were gods in them besides Allah, they would be ruined; but they are not ruined; ∴ there are no gods in them besides Allah." Denying the consequent.
- "They were created either by nothing, or by themselves, or by a Creator. Not by nothing; not by themselves. ∴ By a Creator."
- Taking the minor as "Allah is eternal" (S = Allah) and the major as "No created thing is eternal" (P = created thing), the middle (eternal) is the predicate of both: Figure 2, mood 1 (A + E → E).