Simple Conversion: Definition, Categorical Rules and Reductio

1. From contradiction to simple conversion

The previous level studied contradiction, where one proposition precisely denies another and their truth values are opposed. Simple conversion (ʿaks mustawī) has a different aim: interchange the terms of a true proposition to obtain another true proposition, without negating the original judgment.

Example:
“All humans are animals” — original.
“Some animals are humans” — valid converse.

The second statement supports rather than denies the first. Forming a contradictory and producing a converse are therefore different logical operations.

2. Textbook definition of simple conversion

«العكس المستوي، ويقال له: العكس المستقيم أيضاً، وهو عبارة عن جعل الجزء الأول من القضية ثانياً، والجزء الثاني أولاً مع بقاء الصدق والكيف»
Simple conversion, also called direct conversion, places the first part of a proposition second and the second first, while preserving truth and quality.

In a categorical proposition, interchanging the parts means exchanging subject and predicate. Preserving truth means that a true original must yield a true converse. Preserving quality means affirmatives remain affirmative and negatives remain negative.

Preserving quantity is not required: a universal proposition may validly convert into a particular one.

3. Interchanging subject and predicate

In a categorical proposition the subject is what the claim concerns; the predicate is what is affirmed or denied of it. Simple conversion exchanges their positions, keeping the complete meaning of each term intact.

“Some students are writers”:
Original subject — students.
Original predicate — writers.
Converse — “Some writers are students.”

The same overlapping individuals make both statements true. Conversion reverses the direction of attribution; it does not introduce a new property.

4. Why truth must be preserved

Reversing words is not sufficient. A proper converse must be necessarily true whenever the original is true.

“All humans are animals” is true, but “All animals are humans” is false; a horse is an animal but not human. That reversal of wording is therefore not a valid simple converse.

The valid converse is “Some animals are humans.” The actual human members belong to both classes, making the particular converse true.

5. Quality stays; quantity may change

Quality means affirmation or negation. Quantity means universal or particular scope. Simple conversion preserves quality, but may change quantity where required.

“No human is a stone” — universal negative.
“No stone is a human” — also a universal negative.
“All humans are animals” — universal affirmative.
“Some animals are humans” — particular affirmative.

Thus conversion does not preserve every quantifier. It preserves truth and affirmative or negative quality.

6. The four categorical conversion rules

The four quantified categorical forms behave differently:
Universal affirmative (A) — converts to a particular affirmative (I).
Universal negative (E) — converts to a universal negative (E).
Particular affirmative (I) — converts to a particular affirmative (I).
Particular negative (O) — has no generally valid converse.

“No guaranteed converse” does not mean every reversed statement must be false. It means truth of the original does not ensure truth of the reversal in every case.

7. Universal negative converts universally

«فالسالبة الكلية تنعكس كنفسها»
A universal negative converts into another universal negative.

Textbook example:
«لا شيء من الإنسان بحجر»
No human is a stone.

Its converse:
«لا شيء من الحجر بإنسان»
No stone is a human.

If two classes share no members, no member of the first is in the second and no member of the second is in the first. This symmetric exclusion supports universal-negative conversion.

8. Indirect proof for the universal negative

The textbook does not rely only on an example; it also gives an indirect proof (burhān bil-khalf). We temporarily suppose the intended converse false, then combine that supposition with the known true original and derive an impossibility.

Original: “No human is a stone.”
Proposed converse: “No stone is a human.”

If the proposed converse were false, its contradictory would be true: “Some stones are humans.” We then combine this claim with the original universal negative.

9. The impossible result in the reductio

The assumed proposition “Some stones are humans” identifies an object as both a stone and a human. But “No human is a stone” then excludes that same object from the class of stones.

Reasoning:
1. Some stones are humans — hypothetical negation of the converse.
2. No humans are stones — true original.
3. Therefore some stones are not stones — denying of the very thing already asserted.

This is the impossibility of denying a thing of itself. Therefore the initial assumption fails and the universal-negative converse stands.

10. What the reductio establishes

The indirect proof shows that conversion is not just a memorized pattern; it is a necessary consequence.

When a universal negative is true, supposing its converse false creates a contradiction with the original. Therefore the converse does not merely happen to be true in the human-and-stone example; its validity follows from complete disjointness between the two classes.

This reasoning generalizes to other true universal-negative propositions.

11. Why the particular negative does not convert

«والسالبة الجزئية لا تنعكس لزوماً»
A particular negative does not have a necessarily valid converse.

Textbook example:
«بعض الحيوان ليس بإنسان»
Some animals are not humans.

Simply reversing the terms gives “Some humans are not animals,” which is false because every human is an animal. The original was true, but the reversed statement failed.

One valid counterexample is sufficient to defeat the claim that every particular negative converts.

12. A wider subject explains failure

In “Some animals are not humans,” animals form a wider class while humans form a narrower subclass. Horses and birds make the original true.

The reversal “Some humans are not animals” demands a human outside the wider animal class. But every human belongs to that class.

Thus switching terms in a particular negative does not preserve exclusion in the way it does for two completely disjoint classes.

13. Not guaranteed is not never true

A particular negative and its term-reversal may both happen to be true. For example, a population might contain both non-poet students and poets who are not students.

“Some students are not poets.”
“Some poets are not students.”

Yet the first statement alone does not entail the second. Thus “the particular negative does not convert” means there is no universally reliable conversion rule, not that a reversed claim can never be true.

14. Universal affirmative converts particularly

«والموجبة الكلية تنعكس إلى موجبة جزئية»
A universal affirmative converts to a particular affirmative.

Textbook example:
«كل إنسان حيوان»
Every human is an animal.

Correct converse:
«بعض الحيوان إنسان»
Some animals are humans.

The original places all humans within the class of animals. Given the existing human members, the converse correctly says that some animals are humans, but it does not say all animals are humans.

15. Why universal scope is reduced

If we converted “All humans are animals” into “All animals are humans,” the statement would remain universal but become false. The predicate animal is broader than the subject human.

Original: every member of the human class is an animal.
Invalid reversal: every animal is human.
Valid converse: some animals are human.

The broader class has members outside the narrower one. Hence universal affirmation must be restricted to particular affirmation during conversion.

16. Existential basis of the particular converse

Deriving “Some animals are humans” from “All humans are animals” assumes the relevant human class has members. The textbook develops the traditional rule through classes that do have instances.

A universal affirmative includes every subject-instance under its predicate. A particular affirmative claims at least one shared instance.

Modern formal systems can interpret universal statements over empty classes differently. Here we retain the classical rule taught in *Al-Mirqat*, while identifying the existential assumption rather than concealing it.

17. Particular affirmative converts particularly

«والموجبة الجزئية تنعكس إلى موجبة جزئية»
A particular affirmative converts to another particular affirmative.

Textbook example:
«بعض الحيوان إنسان»
Some animals are humans.

Its converse:
«بعض الإنسان حيوان»
Some humans are animals.

A shared member belongs to both classes regardless of which is mentioned first. This is why particular affirmation is preserved under conversion.

18. Shared membership in a fresh example

Suppose some teachers are also authors:
“Some teachers are authors.”
“Some authors are teachers.”

The first has teachers as subject and authors as predicate. The second exchanges them. The very same person who makes the original true makes its converse true.

No new shared member is required. Particular affirmative conversion relies on the overlap already established by the original claim.

19. Converting twice

A correctly converted particular affirmative or universal negative can be converted once more to recover its original form.

“Some students are memorizers.”
First converse: “Some memorizers are students.”
Second converse: “Some students are memorizers.”

Likewise:
“No tree is a stone.”
Converse: “No stone is a tree.”

But converting a universal affirmative first reduces it to a particular, and converting that back does not recover universality. Conversion may therefore lose information about quantity.

20. Why some qualifications may change

«حفظ النسبة ليس بضروري في العكس»
Maintaining every relational qualification unchanged is not necessary in conversion.

The book illustrates the issue through youth and old age: a young person may become old at another time, which does not imply simultaneous youth and old age.

The primary requirements are preservation of truth and quality. Yet when time or another qualification affects the meaning, the valid converse must be interpreted carefully. A simple word reversal does not settle the temporal scope.

21. Keeping the complete predicate intact

Some predicates are compound descriptions, not single nouns. The whole predicate must become the new subject, rather than a fragment of it.

Example:
Original: “Some students are in the library.”
Invalid word reversal: “Some libraries are students.”
Meaning-preserving converse: “Some people in the library are students.”

“People in the library” captures the complete predicative description. This clarifies the source’s discussion of objections involving a composite predicate.

22. The source response to a conversion objection

The textbook considers an objection: why does reversing a sentence with a locative predicate sometimes yield a false statement?

The answer is that the predicate is not merely the name of a place; it is being located in that place. Reversing the bare place-name changes the original predicate and thus fails to produce a valid converse.

From “Some people are in the garden,” we cannot derive “Some gardens are people.” The valid converse is “Some individuals in the garden are people.” Word-order reversal is not the same as conversion of the whole predicate.

23. Testing the rules by class relationships

The rules can be understood through relations between classes.

Universal negative: no overlap; exclusion holds in both directions.
Universal affirmative: the first class is wholly included in the second; reverse inclusion need not be total.
Particular affirmative: at least one member is shared; overlap is symmetric.
Particular negative: some members of the first class are outside the second; nothing follows about members of the second being outside the first.

These four relations explain the rules rather than merely listing them.

24. Caution with conditional propositions

The general definition refers to exchanging the first and second parts of a proposition, and the commentary also mentions the generality of a conditional antecedent alongside a categorical subject.

This does not permit mechanically swapping antecedent and consequent in every conditional. “If something is human, it is an animal” does not imply “If something is an animal, it is human.”

The four categorical rules are the focus here. Any conditional conversion must likewise preserve truth, and its special conditions require separate care.

25. Conversion of modal propositions

The commentary also addresses conversion of modal propositions. Necessity, permanence, actuality and possibility change a proposition’s truth conditions. Therefore a modal qualification cannot automatically be carried over unchanged when the terms are exchanged.

Being true at all times is a stronger claim than being true at some time. A valid converse of a modal original may have a weaker modality.

The source gives conversions of several modal forms under general absoluteness (muṭlaqa ʿāmma) and notes exceptions. Understanding each modality’s scope is essential before attempting detailed modal conversions.

26. Original, contradictory, and converse

Different operations can be performed on the same proposition.

Original: “All humans are animals.”
Contradictory: “Some humans are not animals” — opposed in truth value.
Simple converse: “Some animals are humans” — preserves the original’s truth.

Contradiction changes quantity and quality as needed to ensure opposite truth values. Conversion swaps subject and predicate while preserving truth and quality. These are not interchangeable operations.

27. Four worked conversions

1. Universal affirmative: “All parrots are birds.”
Valid converse: “Some birds are parrots.”
Reason: the bird class contains other species too.
2. Universal negative: “No triangle is a circle.”
Valid converse: “No circle is a triangle.”
Reason: complete disjointness.
3. Particular affirmative: “Some teachers are authors.”
Valid converse: “Some authors are teachers.”
Reason: shared individuals.
4. Particular negative: “Some birds are not pigeons.”
No guaranteed converse: “Some pigeons are not birds” is false.

28. A reliable conversion procedure

To construct a categorical converse:
1. Identify the original subject and the complete predicate.
2. Determine affirmative versus negative and universal versus particular.
3. Apply the appropriate rule; do not force a converse for a particular negative.
4. Exchange terms and select the correct quantifier.
5. Verify that whenever the original is true, the result must be true.

For compound predicates, temporal restrictions, and modal qualifications, a merely verbal reversal is insufficient.

29. Summary and transition

Simple conversion, also called direct conversion, exchanges the two parts of a proposition while preserving truth and quality.

The four rules:
Universal affirmative → particular affirmative.
Universal negative → universal negative.
Particular affirmative → particular affirmative.
Particular negative → no guaranteed converse.

The universal negative is supported by reductio. Particular affirmation rests on shared membership. Universal affirmation requires reduced quantity, while particular negation lacks a generally valid converse.

The next topic is ʿaks al-naqīḍ (conversion by contraposition), another operation involving terms and negation.