Contradiction: Definition, Conditions and Application to Propositions

1. Why contradiction matters

The preceding lessons covered categorical and conditional propositions. We now ask when two propositions differ so that one must be true and the other false. In classical logic this relation is called tanāquḍ (contradiction).

“Zayd is standing” and “Zayd is not standing” appear opposed. Yet if the first concerns morning and the second evening, both may be true. The words “is” and “is not” alone do not establish contradiction.

We will define contradiction, study its conditions, and then apply the principle to quantified, modal and conditional propositions.

2. The source definition of contradiction

«التناقض هو اختلاف القضيتين بالإيجاب والسلب بحيث يقتضي لذاته صدق إحداهما كذب الأخرى أو بالعكس»
Contradiction is a difference between two propositions in affirmation and negation such that, in themselves, the truth of one entails the falsehood of the other, and conversely.

Affirmation and negation must concern the very same predication. “In themselves” means that the exclusion follows from their logical relationship, not merely from accidental external circumstances.

The textbook's example is:
«زيد قائم، وزيد ليس بقائم»
Zayd is standing; Zayd is not standing.

With the same subject, predicate and relevant circumstances, the two cannot both be true or both be false.

3. Contradiction versus other opposition

“This object is white” and “This object is black” cannot both hold in the same respect, but both may be false if the object is red. That is not contradiction as defined here.

“This object is white” and “This object is not white” negate precisely the same property. With the relevant conditions held fixed, exactly one is true.

Distinction: opposed positive predicates may both fail, whereas an affirmation and its exact negation cannot. This comparison clarifies the textbook’s definition.

4. Eight required identities

«وشرط لتحقق التناقض بين القضيتين المخصوصتين وحدات ثمانية»
Eight kinds of identity are conditions for establishing contradiction between two determinate propositions.

They are:
1. Identity of subject
2. Identity of predicate
3. Identity of place
4. Identity of time
5. Identity of potentiality and actuality
6. Identity of condition
7. Identity of part and whole
8. Identity of relation

The list is not merely terminological. Every condition tests whether the statements actually affirm and deny the very same thing.

5. Identity of subject

«زيد قائم، وعمرو ليس بقائم»
Zayd is standing, and Amr is not standing.

These are not contradictory. The first is about Zayd, the second about Amr. Both could easily be true together.

Compare:
“Zayd is standing.”
“Zayd is not standing.”

Now the subject is identical. Yet identity of subject alone is insufficient: time, place and all relevant qualifications must also match. Contradiction negates the very same complete claim.

6. Identity of predicate

«زيد قاعد، وزيد ليس بقائم»
Zayd is sitting, and Zayd is not standing.

The subject is identical, but one proposition affirms sitting while the other denies standing. Both may be true; they are not contradictory.

The exact contradictory of “Zayd is sitting” is “Zayd is not sitting”, not “Zayd is not standing”.

The predicate affirmed must be the very predicate denied. Substituting another predicate changes the claim under discussion.

7. Identity of place

«زيد موجود أي في الدار، وزيد ليس بموجود أي في السوق»
Zayd is in the house, and Zayd is not in the market.

Both concern Zayd's presence, but the locations differ: house versus market. Both claims may be true together.

The exact negation of “Zayd is in the house” is “Zayd is not in the house”, with all other relevant conditions fixed.

A different place changes the spatial qualification of the assertion, defeating contradiction.

8. Identity of time

The book contrasts Zayd's state at night with his state by day:
“Zayd sleeps at night.”
“Zayd does not sleep in the daytime.”

Both can be true because they concern different times. A person may sleep at night and remain awake by day.

A genuine pair keeps the time fixed:
“Zayd is sleeping now.”
“Zayd is not sleeping now.”

When time qualifies the judgment, changing that time changes the proposition. That is why temporal identity matters.

9. Potentiality and actuality must match

The text distinguishes a thing's capacity (potentiality) to produce an effect from actually producing that effect, using the capacity for intoxication as an example.

Compare:
“This seed can develop into a tree” — a potentiality claim.
“This seed is actually a tree now” — an actuality claim.

The first can be true while the second is false, without their being exact negations.

For contradiction, compare potential with potential, or actual with actual. The contradictory of the capacity claim is that this seed lacks that capacity.

10. Identity of condition

The source discusses the movement of a writer's fingers under two different conditions: while writing and while not writing.

“Zayd moves his fingers while writing” and “Zayd does not move his fingers when he is not writing” may both be true; the conditions differ.

A proper comparison is:
“When Zayd is writing, his fingers move.”
“When Zayd is writing, his fingers do not move.”

When subject, time and other qualifications also coincide, affirmation and negation concern the very same conditional claim.

11. Identity of part and whole

A claim about a whole and a claim about one of its parts need not concern the same predication. The text illustrates this with a dark body and the color of its teeth.

Compare:
“Most of the animal’s body is black.”
“The animal's teeth are not black.”

These relate to the whole and one part. Both can be true because the predominant color of the body does not entail that the teeth share it. The question would change if the first statement explicitly included every part.

To judge contradiction accurately, identify which exact part or whole receives the predicate.

12. Identity of relation

«زيد أب أي لبكر، وزيد ليس بأب أي لخالد»
Zayd is Bakr's father, and Zayd is not Khalid's father.

Both may be true. Fatherhood is a relational predicate: we must ask, “father of whom?” The first relation is to Bakr, the second to Khalid.

A contradictory pair would be:
“Zayd is Bakr's father.”
“Zayd is not Bakr's father.”

Here the other term of the relation is identical. An identical relational word is insufficient unless its object is identical too.

13. An eight-point comparison

To test whether two propositions contradict, first express their full meanings and compare all relevant qualifications.

Checklist:
Same subject? Same predicate?
Same place? Same time?
Same potential/actual respect? Same condition?
Same part or whole? Same relational object?

A difference in a relevant qualification means the apparent negation no longer denies precisely the same claim.

“Ali was present in Lahore on Monday” and “Ali was not present in Karachi on Tuesday” contain affirmative and negative language, but both place and time differ. Contradiction has not been established.

14. Why some commentators reduce the eight

The commentary notes that some logicians subsume the eight identities under identity of subject and predicate. On that analysis, time, place, condition or relational object is already contained in a fully specified subject or predicate.

In “Zayd is in the house now” and “Zayd is not in the house now”, the fully specified predicate being in the house now already includes time and place.

This does not make time or location irrelevant. Both approaches demand an affirmation and denial of one precisely determined relation. The difference is how its qualifications are organized.

15. Quantity must change in quantified contradictories

«لا بد في التناقض في المحصورتين من كون القضيتين مختلفتين في الكم»
In contradiction between two quantified propositions, their quantities must also differ.

The quantifiers “all”, “none”, “some” and “some not” determine the scope of the subject. Merely changing affirmation into negation without the correct universal/particular shift may allow both propositions to have the same truth value.

The pairings are:
Universal affirmative ↔ particular negative
Universal negative ↔ particular affirmative

Both quantity and quality change, ensuring opposite truth-values under the standard assumptions of this traditional discussion.

16. Negating a universal affirmative

The source example:
«كل حيوان إنسان»
Every animal is human.

Its correct contradictory:
«بعض الحيوان ليس بإنسان»
Some animal is not human.

If all animals are human, there can be no counterexample, so the particular negative is false. If the universal is false, at least one counterexample exists, making the particular negative true.

“No human is an animal” is not its exact contradictory; it is a universal negative, a stronger claim than what is required for denial of “every”.

17. Negating a universal negative

The source example:
«لا شيء من الحيوان بإنسان»
No animal is human.

Its exact contradictory is:
«بعض الحيوان إنسان»
Some animal is human.

One counterexample is enough to refute a universal denial, so its contradictory is particular affirmation.

“All animals are human” and “No animals are human” differ in quality but are both universal. The true contradictory of a universal negative is a particular affirmative.

18. Why two universals or two particulars fail

The textbook uses the following examples:
“All animals are human” — false.
“No animal is human” — also false, since some animals are human.

Two universal propositions can thus both be false; they are not contradictories.

Likewise:
“Some animals are human” — true.
“Some animals are not human” — also true.

Two particular propositions can thus both be true.

Consequently both quantity and quality must be reversed: universal affirmative pairs with particular negative, and universal negative with particular affirmative.

19. Singular and quantified propositions

“This book is new” and “This book is not new” concern a specified individual. There is no universal or particular quantifier; contradiction requires the same identified subject and predicate.

By contrast, “Every book is new” quantifies over a class. Its contradictory is “Some book is not new.”

Thus the tests differ:
Singular propositions — exact affirmation and denial about one individual.
Quantified propositions — the proper reversal of quantity as well as quality.

20. Modal propositions require attention to modality

The book explicitly requires appropriate modal opposition in the contradiction of modal propositions. Necessity, possibility and permanence qualify a claim differently.

For example:
“Every human is necessarily a writer.”
“No human is necessarily a writer.”

These do not automatically form a contradictory pair: both may be false. Simply reversing affirmation and negation while leaving modal scope unexamined is insufficient.

A correct modal contradictory must negate the full modal assertion.

21. Negating necessity by possibility

The source pairs the appropriate contradictories of necessity with general possibility:
الضرورية ↔ الممكنة العامة
A necessity claim is negated by the appropriate possibility of its contrary.

For example:
“Every human is necessarily a writer” asserts necessity.
“It is possible for some human not to be a writer” expresses the relevant counter-possibility.

Denying that a property is necessary does not mean the property is absent from everyone in actuality.

Keep denial of necessity distinct from denial of actual occurrence.

22. A compact map of modal oppositions

The text summarizes the relevant modal pairings as:
Necessary ↔ general possible
Permanent ↔ general absolute/actual
General conditional ↔ temporal possible
General customary ↔ temporal absolute

These are not mere verbal substitutions. Quality, quantity and modal scope must all be matched correctly for each pair.

To negate a modal claim, determine precisely what would make the original claim false, and express that failure at the proper scope.

23. Contradiction of compound modalities

The commentary explains that compound modal propositions have several components, so negating the whole requires careful treatment of their partial negations and the logical connection among them.

For instance, a claim that a property is “both necessary and permanent” makes two demands. To falsify the combined claim, it is enough that one required component fails; both need not fail simultaneously.

The object is to negate the proposition as a whole, not mechanically insert a negative word into every part.

24. Contradiction among conditionals

«ويشترط في أخذ تناقض الشرطيات الاتفاق في الجنس والنوع، والمخالفة في الكيف»
To construct the contradictory of conditionals, they must agree in genus and species, while differing in quality.

Same genus: a connected conditional should be compared with the negation of that same connected relation, or a disjunctive with the same disjunctive relation.
Same species: the relation involved, such as an entailment rather than a coincidence, must not be silently changed.
Different quality: the whole judgment is affirmed in one and denied in the other.

Merely denying a component is not necessarily denying the conditional as a whole.

25. Negating a connected conditional

Suppose we assert:
“Whenever fire burns normally, heat is present.”

The direct denial of the whole connected conditional is:
“It is not true that heat must be present in every such case of normal burning fire.”

The negation targets neither the antecedent as a separate fact nor the consequent alone, but the claimed general entailment between them.

Saying merely “The fire is not burning” does not contradict the conditional. The truth of a conditional differs from the actual occurrence of its antecedent now.

26. Negating a disjunctive conditional

For a particular integer consider:
“This number is either even or odd.”

A denial of the whole disjunctive judgment is:
“It is not true that precisely one of these alternatives applies under the stipulated division.”

“This number is not even” alone does not contradict the disjunction: the number could be odd, leaving the original disjunction true.

A genuine negation must address the whole disjunctive relation, whether it is an exclusive, exclusion-of-joint-truth, or exclusion-of-joint-falsehood form.

27. Checking quantity and quality in conditionals

Connected conditionals can be quantified by expressions such as “always” and “sometimes”. The negation of “on every occasion” is “on at least one occasion, not so”, not necessarily “never”.

For example:
“Whenever it rains, the ground becomes wet.”
“On at least one rainy occasion, the ground did not become wet.”

The second rejects the claimed universal connection.

The two concern the same conditional relation, with properly differing quantity and opposite quality. This applies the quantified-proposition principle to conditionals.

28. Worked practice: forming exact contradictories

Identify the exact assertion before forming its contradictory.

1. “Every student is present” → “Some student is not present.” One absence refutes the universal.
2. “No bird flies” → “Some bird flies.” One flying bird is sufficient.
3. “Zayd is at home today” → “Zayd is not at home today,” provided the same relevant time within the day is intended.
4. “This door is always closed” → “At some relevant time this door is not closed.”

In each case the full scope of the original claim is preserved.

29. Synthesis and transition to conversion

Contradiction is an opposition in which one proposition's truth requires the other's falsehood and conversely. Relevant identities must hold so the negation targets precisely the same judgment.

Singular proposition: exact denial about the same individual and predicate.
Quantified proposition: appropriate change of quantity and quality.
Modal proposition: correct treatment of the scope of necessity, permanence and possibility.
Conditional proposition: agreement of genus and species with negation of the whole conditional judgment.

The next chapter is al-ʿaks al-mustawī (simple conversion). Instead of negating a proposition, it reverses the relevant terms and asks when truth is preserved.