Lesson 7 · The four relations (al-nisab al-arbaʿ)

Source: Book pp. 27–30.

In Lesson 4 we compared words by their concepts. Here we compare two universals by their instances: can they be true of the same individuals?

1. The four relations

The bird and black example: a crow is both; a dove is a bird but not black; black cloth is black but not a bird. Each concept is wider than the other in one respect, narrower in another, hence "in one respect".

Picture them as circles: two circles exactly on top of each other (=), one circle inside a bigger one (>), two circles that partly overlap (×), two circles apart (//).

A note on "capable of laughter": the book writes "human" and "laughter", but it means ḍāḥik, "capable of laughing" (it explains this itself on p. 35, Lesson 9).

2. The relations between their contradictories

Take any two universals and form their contradictories: "not-A" and "not-B". How do those relate? The book proves four rules.

Partly separated (tabāyun juzʾī) simply means: they are not true together of everything. That covers two possibilities, either × or //, depending on the example.

Examples

• Equal: human = rational, so non-human = non-rational.
• Reversed: animal > human, so non-animal < non-human. Every non-animal (a rock) is a non-human, but not every non-human is a non-animal (a horse is a non-human and an animal).
• × gives sometimes ×, sometimes //:
  • bird × black → non-bird × non-black (paper is both; a black shirt is a non-bird but not non-black; a dove is non-black but not a non-bird);
  • animal × non-human → non-animal // human (no human is a non-animal).
• // gives sometimes ×, sometimes //:
  • human // rock → non-human × non-rock (a horse is both; a rock is non-human only; a human is non-rock only);
  • existent // non-existent → non-existent // existent (their contradictories are each other).

3. How the book proves them: elimination

The method is elimination (istiqṣāʾ, also called sabr wa taqsīm, "probing and dividing"; Lesson 15). List every possible relation, refute all but one, and the one left standing is proven.

Proof that equals have equal contradictories. Suppose some not-A is not a not-B. Then, since there is no middle between contradictories, that thing is a B. So it is a B that is not an A. But A = B means every B is an A. Contradiction. So every not-A is a not-B; by the same reasoning every not-B is a not-A. They are equal.

Book slip (p. 28): in the proof of the second rule, "contradicts our original supposition that A=B" should read "A > B".

4. Why this matters

Every one of the four relations is a bundle of propositions you will meet in Unit 4:

• A = B: "every A is B" and "every B is A";
• A > B: "every B is A" and "not every A is B";
• A × B: "some A is B", "not every A is B", "not every B is A";
• A // B: "no A is B" (which also gives "no B is A").

And in Unit 5 the whole question "does the conclusion follow?" becomes a question about how three circles can sit relative to each other.

ʿAqīda link: īmān and islām. On the reading of al-Ḥujurāt 49:14 ("The Bedouins say: 'We believe.' Say: 'You have not believed; rather say, We have submitted'"), islām is the outward submission and īmān the inner conviction with it. On that reading, every muʾmin is a muslim but not every muslim is a muʾmin: muslim > muʾmin. By the reversal rule, non-muslim < non-muʾmin: every non-muslim is a non-muʾmin, but not the other way round. (Scholars discuss how the two words relate when each is used on its own; the exercise here is only to see the logic.)

Quiz 7

1. Name the relation and draw it with the book's sign:
   • a) human / animal
   • b) human / rational
   • c) white / bird
   • d) believer / disbeliever
   • e) prophet / messenger (taking every messenger to be a prophet, but not every prophet a messenger)
2. What is the relation between "non-animal" and "non-human"? Give an example that shows it is not equality.
3. What is the relation between "non-white" and "non-bird"? Give one thing that is both, and one that is only one of them.
4. Why does "partly separated" have to cover two possibilities?
5. Prove, by elimination, that if A = B then not-A = not-B.
6. On the reading of al-Ḥujurāt 49:14, what is the relation between "non-muslim" and "non-muʾmin"?
7. Review: is "existence" univocal (mutawāṭiʾ) or graded (mushakkik)? Why?