Qiyās al-Khalf: Compound Reductio, Impossibility and Proof of the Claim

1. From analogy to reductio

The previous lesson examined transfer of a judgment by analogy. Qiyās al-khalf (reductio) offers a different method: rather than establishing a claim directly, assume its contradictory and show that this assumption entails an impossibility.

If denying a proposed conclusion produces a consequence that conflicts with an accepted logical principle, the denial must be rejected, thereby supporting the original claim.

The method depends on a valid entailment from the contradictory assumption to an impossibility, not merely on finding the assumption inconvenient or surprising.

2. The text introduces qiyās al-khalf

ومن الأقيسة المركبة قياس يسمى قياس الخلف، ومرجعه إلى قياسين
Among compound syllogisms is one called qiyās al-khalf, which reduces to two syllogisms.

Compound syllogisms are arguments in which a conclusion reached in one stage functions as a premise in a subsequent stage. In this form of reductio, a conditional conclusion is obtained first and then used to establish the sought claim.

It is not enough merely to declare the opposing position false: both inferential stages must be shown.

3. Direct and indirect proof

Direct proof: derive the target from the accepted premises without assuming its contradictory.
Proof by reductio: assume the contradictory of the target, derive an impossibility, and reject the assumption.

For example, the claim that every square has four sides can be grounded directly in the definition of a square. A reductio instead assumes a square lacking four sides and derives a conflict with that definition.

This is an additional instructional example. The text itself uses “Every human is an animal” to illustrate the technique.

4. Claim and contradictory

The claim (muddaʿā) is the proposition to be established; its contradictory has the opposite truth value in every case. They cannot both be true or both be false.

The contradictory of “Every human is an animal” is “Some human is not an animal,” not “No human is an animal.” The former correctly denies the universal assertion.

Reductio must begin with the genuine contradictory. Refuting a different, stronger or weaker statement may leave the original claim unproved.

5. Why two syllogisms?

The book analyzes reductio through two syllogisms. The first connects two hypothetical conditionals and concludes that if the contradictory of the claim holds, an impossibility follows.

The second is an exceptive syllogism. It takes that conditional conclusion and adds that the impossible consequence does not hold; denying the consequent then rejects the contradictory assumption.

The progression is:
1. Assume the contradictory.
2. Establish that it entails an impossibility.
3. Deny the impossibility, thereby rejecting the assumption.
4. Affirm the original claim.

6. First stage: two connected conditionals

أحدهما اقتراني شرطي مركب من المتصلتين
The first is a connected hypothetical syllogism composed of two conditional propositions.

Suppose the original claim is A. One condition links the denial of A to a contradictory state; another says that state entails an impossibility.

Connecting the conditionals yields: If not-A holds, an impossibility follows. This conditional relation is the conclusion of the first syllogism.

7. Chaining conditionals correctly

To chain two conditionals, the consequent of the first must properly connect with the antecedent of the second:

1. If A, then B.
2. If B, then C.
Therefore: If A, then C.

In a reductio, the starting condition is the contradictory of the claim and the end is an impossibility. Intermediate steps may vary, but each entailment must be established.

Unrelated consequents and antecedents cannot be linked to obtain the desired conclusion.

8. What denying the claim means

The text uses the expression “the claim is not established.” Within a formal reductio it represents assuming the denial of the claim, not merely lacking evidence for it.

“I do not know whether this claim is proved” reports an epistemic state.
“The claim is false” denies the proposition.

Reductio proceeds from the second kind of assumption. Not yet possessing a proof is not the same as proving a proposition false.

9. The source conditional conclusion

لو لم يثبت المدعى ثبت المحال
If the claim is not established, the impossible will be established.

This is the result of the first syllogism. It must be supported by premises from which the unacceptable consequence validly follows.

Simply saying “If my assertion were false, something impossible would follow” is not proof. The argument must show which impossibility follows and by what entailment.

10. What counts as an impossibility?

An impossibility (muḥāl) is a consequence incompatible with an accepted necessary or logical principle. For instance, the same proposition cannot be both affirmed and denied in the same respect and at the same time.

The surprising is not necessarily impossible. Nor does disliking a result establish a contradiction.

Since the impossibility of the consequent is a central premise in the second syllogism, its status must be adequately justified.

11. Second stage: an exceptive syllogism

The second syllogism takes the conditional established in the first and then denies its consequent.

First premise: If the claim is denied, an impossibility follows.
Second premise: But that impossibility does not hold.
Conclusion: Therefore the denial of the claim does not hold.

This applies the valid hypothetical rule denying the consequent entails denying the antecedent, studied earlier.

12. The text denies the impossible consequence

لكن المحال ليس بثابت، فبالضرورة ثبت المدعى
But the impossible is not established; necessarily, therefore, the claim is established.

This states the final inference. If denial of the claim entails something impossible, and that impossible consequence cannot hold, the denial is rejected.

The original claim follows because a proposition and its genuine contradictory cannot both be false.

13. The valid modus tollens pattern

In a conditional inference, this pattern is valid:
1. If the antecedent holds, the consequent holds.
2. The consequent does not hold.
3. Therefore the antecedent does not hold.

Symbolically: P ⇒ Q; ¬Q; therefore ¬P.

In reductio, the antecedent is the contradictory of the claim and the consequent is an impossibility. Denying the impossible rejects the contradictory.

Affirming an antecedent merely from the truth of its consequent would be a different, invalid form.

14. Double negation restores the claim

Let the claim be A, and its contradictory not-A.

Reductio rejects not-A, leaving not-not-A. In standard classical two-valued logic, eliminating this double negation gives A.

The step requires the assumption to be the actual contradictory. Refuting a merely contrary or loosely opposing statement does not necessarily prove the initial claim.

15. Why the two contradictories cannot both fail

The source appeals to the impossibility of both contradictories being false (irtifāʿ al-naqīḍayn).

If “Every human is an animal” and “Some human is not an animal” were both false, the universal affirmative and its genuine contradictory would both have been denied. Classical logic excludes that possibility.

Thus, refuting the true contradictory establishes the original proposition. Do not confuse contradictories with propositions that are merely contraries.

16. The textbook example: identify the claim

كل إنسان حيوان
Every human is an animal.

This is the claim in the source example. “Animal” is used in the classical logical sense as a genus including humans, not as an insult.

The proposition is universal in quantity and affirmative in quality. To defend it indirectly, the source assumes its contradictory.

17. The correct contradictory in the example

بعض الإنسان ليس بحيوان
Some human is not an animal.

This is the contradictory of “Every human is an animal.” A universal affirmative is contradicted by a particular negative, asserting at least one exception.

“No human is an animal” is a universal negative, not the formal contradictory of the original universal affirmative. Confusing these forms spoils the first step of reductio.

18. How does the assumed exception entail an absurdity?

The classical argument presupposes that humanity falls within the genus animal. If some human were not an animal, an individual would have to be human while lacking what is held to belong to the logical definition of human.

This would jointly affirm and deny the relevant genus of the same individual in the same respect: a contradiction.

The reasoning depends on the substantive logical account of “animal” as part of the concept of human; it cannot be obtained from a superficial resemblance of words.

19. Completing the textbook inference

First stage: If “Every human is an animal” were false, “Some human is not an animal” would be true. Under the assumed genus relation, that supposed exception would contradict an established feature of being human.

Second stage: Such a contradiction cannot hold. Hence “Some human is not an animal” is false. That proposition is the contradictory of the original claim.

Conclusion: Every human is an animal.

The argument first derives the unacceptable consequence and then denies it to recover the claim.

20. Worked supplementary example: square sides

Claim: Every square has four sides.
Contradictory: Some square does not have four sides.

By definition, a square is a four-sided figure with equal sides and right angles. Assuming a square that does not have four sides conflicts with the very definition used to identify it as a square.

The assumed contradictory is therefore rejected. Here having four sides belongs to the definition of the square, not to a merely observed regularity.

21. An unlikely outcome is not a contradiction

Consider the claim “Every intelligent student finishes first in an examination.” Its contradictory is “Some intelligent student does not finish first.”

That exception is entirely possible and may occur for many reasons. It does not entail a logical contradiction. Saying that intelligent students “ought” to finish first is not a reductio proof.

Reductio requires a genuine impossibility entailed by the contradictory assumption, not an expectation or preference.

22. Verifying the purported impossibility

Two checks are central to reductio:
1. Entailment: does the contradictory really lead to the alleged consequence?
2. Impossibility: is that consequence genuinely impossible, rather than merely unexpected?

“If the road is blocked we arrive late; we arrived late, so the road was blocked” is an invalid inference. Many other causes could explain lateness, and arriving late is not impossible.

Both failure of entailment and failure to establish an impossibility would undermine a purported reductio.

23. Relation to contradiction

The chapter on contradiction established that a proposition and its contradictory necessarily differ in truth value. Reductio applies that principle in an inferential process.

The earlier chapter identified the relation between two propositions. Reductio goes further: it assumes the contradictory, derives an impossibility, rejects the contradictory, and establishes the desired proposition.

The earlier rules governing contradictories of quantified propositions are therefore essential here.

24. Relation to exceptive syllogisms

An earlier lesson established the valid exceptive form: denying the consequent of a conditional denies its antecedent. The final step of qiyās al-khalf uses that pattern.

Conditional: If the contradictory is true, impossibility follows.
Exception: But impossibility does not hold.
Conclusion: Therefore the contradictory is not true.

The original claim then follows by the impossibility of both contradictories being false. Thus reductio combines previously studied forms of reasoning.

25. Common mistakes in reductio

Mistake 1: assume an opposing statement that is not the true contradictory.
Mistake 2: fail to establish the necessary implication from the contradictory to the alleged impossibility.
Mistake 3: substitute an undesirable or unlikely result for an impossibility.
Mistake 4: affirm the antecedent from the consequent of a conditional.

A sound reductio avoids all four. Clearly identifying the first syllogism’s conclusion and the second syllogism’s exception makes the reasoning auditable.

26. Full worked proof pattern

Let A be the claim and M an impossible consequence. The pattern is:
1. Assume not-A.
2. Establish from premises: If not-A, then M.
3. M is impossible, so M is false.
4. Therefore not-A is false.
5. By the rule of contradictories, A is true.

Step 2 belongs to the connected hypothetical argument, steps 3–4 to the exceptive inference, and step 5 applies the principle of contradiction.

27. Conclusion and next topic

Qiyās al-khalf is a compound proof: assume the contradictory of a claim, demonstrate that it entails an impossibility, reject the contradictory, and establish the claim.

For soundness, the opposing assumption must be the actual contradictory, the entailment must be valid, the consequent genuinely impossible, and the second inference must correctly deny the consequent.

The source uses “Every human is an animal” and rejects the corresponding particular negative. The following section begins a new discussion of the form and matter of syllogism.