Contraposition: Classical Rules, Validity and the Later Convention

1. A new operation after simple conversion

In simple conversion, the subject and predicate exchange positions while truth and affirmative/negative quality are preserved. Thus “Every human is an animal” converts to “Some animals are humans.”

Contraposition (ʿaks al-naqīḍ) involves complementary concepts as well as a change of position. Merely reversing subject and predicate is insufficient.

We first examine the earlier logicians' convention and then the distinct later convention discussed in the commentary.

2. Complement of a term versus denial of a proposition

The complement of the concept “human” is nonhuman; the complement of “animal” is nonanimal. These are terms, not complete propositions.

“Zayd is not human” is a negative proposition; “nonhuman” is a concept applicable to whatever is not human.

Contraposition works with complementary terms. The presence of “non-” within a term must not be confused with the denial of an entire proposition.

3. The earlier definition of contraposition

«هو جعل نقيض الجزء الأول من القضية ثانيا، ونقيض الجزء الثاني أولا مع بقاء الصدق والكيف»
Make the complement of the first part the second, and the complement of the second part the first, while preserving truth and quality.

This is the earlier logicians' formulation. If the original subject is “human” and predicate “animal,” the resulting subject is “nonanimal” and predicate “nonhuman.”

Preserving truth means truth of the premise supports truth of the result. Preserving quality means that an affirmative stays affirmative and a negative stays negative.

4. The earlier transformation pattern

Let the original subject be S and predicate P. The earlier form is:

Original subject S → new predicate non-S.
Original predicate P → new subject non-P.

Example:
Original: Every human is an animal.
Contrapositive: Every nonanimal is nonhuman.

The pattern alone does not guarantee a valid inference for every quantity. The four categorical cases must be checked separately.

5. Why truth and quality matter

Rearranging words does not establish a valid contraposition. Truth preservation matters; otherwise a true premise could yield a false conclusion.

Quality means affirmation or negation and differs from quantity, which is universal or particular. A valid transformation may keep quality while altering quantity.

For example, under this traditional account a universal negative yields a particular negative: quality stays negative while quantity changes.

6. Universal affirmative: A → A

«فتنعكس الموجبة الكلية بهذا العكس كنفسها»
A universal affirmative contraposes to another universal affirmative of the same form.

Original: Every human is an animal.
Contrapositive: Every nonanimal is nonhuman.

If humans are included within animals, anything outside the class of animals must also be outside the class of humans. Both statements are universal affirmatives.

7. Why A → A is valid

Assume that every human is an animal. Choose something that is not an animal. If it were human, the universal premise would make it an animal, contradicting our choice.

It therefore cannot be human. That is precisely what “Every nonanimal is nonhuman” says.

The converse is not a word trick; it follows from the inclusion expressed by the original universal proposition.

8. Comparing two kinds of conversion

One universal affirmative gives two distinct results:

Original: Every human is an animal.
Simple converse: Some animals are humans.
Contrapositive: Every nonanimal is nonhuman.

Simple conversion retains the original concepts and exchanges positions. Contraposition introduces complementary concepts in reversed order.

Confusing the operations may change the quantity or meaning of the resulting proposition.

9. Particular affirmative: no guaranteed result

«والموجبة الجزئية لا تنعكس بهذا العكس»
A particular affirmative does not validly contrapose by this method.

Consider “Some animals are nonhumans,” true of horses. The mechanical result “Some humans are nonanimals” is false.

A true particular affirmative can therefore produce a false result. There is no general A-like contraposition rule for I-propositions.

10. A negative-looking predicate can be affirmative

“Some animals are nonhuman” is affirmative in form: “nonhuman” is a complementary predicate, while that predicate is affirmed of some animals.

“Some animals are not human” uses a negative copula. Both can describe the same cases, but an affirmative with an infinite/complementary term differs in form from a negative proposition.

This distinction applies the earlier lesson on the position of negation.

11. Why the particular case fails

“Some humans are learned” asserts only that at least one object is both human and learned. It does not entail “Some nonlearned things are nonhuman.” The chosen universe might contain only humans, some learned.

A particular premise gives information about at least one overlapping instance, not about every complementary class.

Thus the new existential claim about both complements does not follow.

12. Universal negative: E → O

«والسالبة الكلية تنعكس إلى سالبة جزئية»
A universal negative yields a particular negative in this traditional scheme.

Original: No human is a horse.
Result: Some nonhorses are not nonhumans.

“Nonhorse” is the new subject, “nonhuman” the new predicate, and the copula remains negative. The particular conclusion also reflects the traditional assumption that humans exist.

The printed example has «بعض اللافرس ليس بالإنسان»
“Some nonhorses are not human.”

That assertion might be true in some circumstances, but it does not follow from the premise alone. By the stated earlier definition, the complement “nonhuman” is needed in the predicate.

13. Existential commitment in E → O

The move from “No humans are horses” to “Some nonhorses are not nonhumans” uses an existing human as a witness: that human is a nonhorse and is not a nonhuman.

If the original subject has no instances, a universal negative alone does not guarantee a particular existence claim. Modern formal logic often makes this qualification explicit.

The traditional rule must therefore be read with its existential assumption.

14. Why not a universal negative converse?

“No humans are horses” does not entail “No nonhorses are humans.”

A human is both a nonhorse and a human, so the proposed universal denial is false.

This shows why the quantity cannot simply remain universal. In the book's traditional scheme, the valid result is particular negative.

15. Particular negative: O → O

«والسالبة الجزئية تنعكس إلى سالبة جزئية»
A particular negative yields a particular negative.

Original: Some animals are not human.
Intended result: Some nonhumans are not nonanimals.

Take a horse: it is an animal and not human. The same instance is a nonhuman that is not a nonanimal. Hence the negative particular conclusion is supported.

16. Reading the printed particular example carefully

The textbook's illustration uses a horse, which is an animal but is not human. Its logically fitting transformed result is:

Original: Some animals are not human.
Contrapositive: Some nonhumans are not nonanimals.

The printed line appears to omit the negating element on the transformed subject. The accompanying horse example and the stated rule require “nonhuman.” This is an interpretive correction, not a silent change to the quoted text.

17. The four categorical outcomes

Under the earlier convention, the categorical outcomes are:

Universal affirmative → universal affirmative.
Particular affirmative → no generally valid contrapositive.
Universal negative → particular negative, assuming subject existence.
Particular negative → particular negative.

The table is not merely mnemonic. Complement both terms, exchange them, then test whether the required quantity and quality preserve truth.

18. Why later logicians use a different definition

The commentary records a different later convention, motivated by logical concerns about unrestricted use of the earlier pattern.

«جعل نقيض الثاني أولا وعين الأول ثانيا مع بقاء الصدق ومخالفة الكيف»
Make the complement of the second part the first, keep the original first part as the second, preserve truth, and reverse quality.

Unlike the earlier approach, the original subject is not complemented; rather, the proposition changes from affirmative to negative or vice versa.

19. A positive example in the later convention

Under the later convention:

Original: Every human is an animal.
New subject: nonanimal.
New predicate: human.
Result: No nonanimal is human.

The original is universally affirmative; the result is universally negative. Thus quality is reversed, while the inference preserves truth.

20. Same premise, two valid formulations

Compare the two styles using one premise:

Original: Every human is an animal.
Earlier: Every nonanimal is nonhuman.
Later: No nonanimal is human.

The earlier result is affirmative with a complemented predicate. The later result is negative with the original subject as predicate.

Their meaning is closely related, but their logical form and the relevant definition differ.

21. Contraposition is not contradiction

“Every human is an animal” and “Every nonanimal is nonhuman” can both be true. They are therefore not contradictories.

In contradiction, the truth of a proposition requires the falsity of its contradictory. In contraposition, a true premise supports a true transformed proposition.

The word “negation” in the name refers to complementary terms, not to constructing the logical contradictory of the whole proposition.

22. Keep compound predicates intact

A compound predicate must be taken as a complete concept before it is complemented.

“Every cat is an animal” gives “Every nonanimal is a noncat.” If the predicate were “a person living in the city,” negating only “city” would not complement that whole predicate.

Logical transformation concerns complete meanings, not arbitrary fragments of wording.

23. Why modal contraposition needs care

The commentary on printed page 71 also discusses modal converses. A proposition qualified by necessity, duration, possibility, or a particular time cannot always be transformed like a bare categorical proposition.

“Necessarily” is not equivalent to “sometimes.” Mishandling the modality may destroy truth preservation.

The text points to larger treatises for the full rules. The key lesson here is that a valid contraposition must preserve truth with its modal qualifications.

24. Worked example: animals and cats

Original: Every cat is an animal.

Step 1: Complement the subject “cat” to “noncat.”
Step 2: Complement the predicate “animal” to “nonanimal.”
Step 3: Put “nonanimal” first and “noncat” second.

Result: Every nonanimal is a noncat. No cat can be a nonanimal, so the result follows. This illustrates valid contraposition of a universal affirmative.

25. Worked example: a particular negative

Original: Some birds are not birds of prey.

Take a bird that is not predatory. It is a non-predatory thing but is not a nonbird, because it is a bird.

Result: Some non-predatory things are not nonbirds.

The same existing instance witnesses the new proposition, explaining why a particular negative contraposes to a particular negative.

26. A method for checking proposed results

Check a proposed contrapositive with four questions:

1. Have the complete subject and predicate concepts been identified?
2. Are their complements formed correctly?
3. Has one convention—earlier or later—been used consistently?
4. Must the new proposition be true whenever the original is true?

A true premise with a false proposed result is enough to expose an invalid general rule. This tests logical validity, not merely memorized names.

27. Consolidation and the next chapter

Under the earlier convention, contraposition reverses the complementary terms while preserving truth and quality. The later convention complements the second term, retains the first term, and reverses quality.

Earlier categorical outcomes:
Universal affirmative → universal affirmative.
Particular affirmative → no general contrapositive.
Universal negative → particular negative, assuming subject existence.
Particular negative → particular negative.

Above all, term complementation, propositional negation, and contradiction must remain distinct. The next independent chapter begins the treatment of arguments and syllogism.