Conjunctive Syllogisms with Conditional Propositions

1. From categorical to conditional syllogisms

Earlier figures mostly used categorical propositions: a predicate is affirmed or denied of a subject. Here a premise can instead be a connected conditional or a disjunction.

Categorical: Every human is an animal.
Connected conditional: Whenever Zayd is human, he is an animal.
Disjunction: This integer is either even or odd.

The difference is not merely grammatical. The kind of relation changes. The task remains to connect what is given and derive only what follows.

2. The source section and its scope

فصل في الاقترانيات من الشرطيات
Section on conjunctive syllogisms involving conditional propositions.

An iqtirani syllogism places premises together so that their common relation allows a conclusion. At least one premise may be conditional rather than both being categorical.

This section sketches the relevant figures and forms. The following section introduces exceptional syllogisms (istithna'i) separately; the two methods must not be conflated.

3. The four figures extend to conditionals

وحالها في الأشكال الأربعة
Their arrangement is discussed through the four figures.

The common component again plays the role of a middle term. Its position in the two premises distinguishes the four figures, just as in categorical syllogisms.

The book likens the figures, productive moods, and required conditions of these syllogisms to their categorical counterparts. This does not license an inference from any arbitrarily paired conditionals.

4. The weaker-premise principle recalled

النتيجة في القياس تتبع أدون المقدمتين في الكيف والكم
A syllogistic conclusion follows the weaker premise in quality and quantity.

In quality, negation is weaker than affirmation: a valid syllogism with one negative premise has a negative conclusion. In quantity, particularity is weaker than universality: when one premise is particular, a valid conclusion cannot be stronger in quantity.

The source recalls this principle at the transition to conditional syllogisms. It describes the conclusion of a valid inference, not a substitute for proving validity.

5. Figure one: the book’s connected example

كلما كان زيد إنسانا كان حيوانا، وكلما كان حيوانا كان جسما
Whenever Zayd is human, he is an animal; and whenever he is an animal, he is a body.

The first premise connects being human to being animal. The second connects being animal to being a body. “Being animal” is the shared middle condition linking both premises.

First link: human → animal
Second link: animal → body
Composing the links gives human → body.

6. Why the first-figure conclusion follows

كلما كان زيد إنسانا كان جسما
Whenever Zayd is human, he is a body.

This is the conclusion stated in the book. Suppose Zayd is human. The first premise then makes him an animal; the second makes whatever is an animal, in this context, a body. Hence being a body follows.

The middle condition “animal” disappears from the conclusion but remains the bridge between the premises. The proof uses two consecutive implication links.

7. The abstract structure of the first figure

Represent the first example abstractly:
Premise 1: If A, then B.
Premise 2: If B, then C.
Conclusion: If A, then C.

B is the shared intermediate component. The consequent of the first premise matches the antecedent of the second, producing one directed chain.

The reverse “If C, then A” does not follow. Reversing a conditional needs a separate justification and cannot be smuggled into the conclusion.

8. A common error: reversing implication

Suppose:
If Zayd is human, he is an animal.
If Zayd is an animal, he is a body.

It does not follow that “If Zayd is a body, he is human.” Being human is sufficient for being a body, but being a body need not be sufficient for being human.

Following a sufficient condition forward differs from reversing it. A correct chain keeps the direction of implication intact.

9. Figure two: the source premises

كلما كان زيد إنسانا كان حيوانا، وليس ألبتة إذا كان حجرا كان حيوانا
Whenever Zayd is human, he is an animal; and it is never the case that, if he is a stone, he is an animal.

The first premise affirms a link from humanity to animality. The second denies such a link from being a stone to animality. “Being an animal” occurs as a common side of the stated relations.

Unlike figure one, the premises do not form a simple forward chain: one is affirmative and the other negative.

10. Figure two: the negative conclusion

ليس ألبتة إذا كان زيد إنسانا كان حجرا
It is never the case that if Zayd is human, he is a stone.

The source concludes negatively from affirmation of animality under humanity and the denial of animality under the stone condition.

Where the same property is affirmed on one side and denied on the other, their joint satisfaction is excluded, provided the relevant subject, respect, and conditions remain fixed.

11. The middle in figure two

In figure two the middle occupies the predicate or consequent side of both premises. The source example uses animality as the common notion.

Human ⇒ animal.
Stone ⇒ animal is denied as a conditional relationship.
Conclusion: the human and stone descriptions cannot jointly hold in the intended respect.

Merely counting occurrences of “human,” “stone,” and “animal” misses the logical positions and the affirmative/negative contrast.

12. Why differing quality matters

Two conditionals affirming the same consequent do not by themselves make their antecedents incompatible.

For example:
If Zayd is a student, he is human.
If Zayd is a teacher, he is human.

These statements do not imply that a student cannot be a teacher. A person can be both. A negative separation requires the appropriate contrast—affirmation on one side and denial on the other—in the same relevant respect.

13. Figure three: a common antecedent

كلما كان زيد إنسانا كان حيوانا، وكلما كان زيد إنسانا كان كاتبا
Whenever Zayd is human, he is an animal; and whenever Zayd is human, he is a writer.

Both conditionals share the same antecedent: Zayd being human. Two different consequences are associated with that condition.

The common component is not the consequent of one and antecedent of the other. It appears at the start of both, which distinguishes this arrangement from figure one.

14. Figure three: understanding the particular result

قد يكون إذا كان زيد حيوانا كان كاتبا
It may be the case that, when Zayd is an animal, he is a writer.

The source marks its conclusion with qad yakun, not a universal and necessary claim that every animal is a writer. The common case of being human is where the two descriptions are brought together.

A restricted or particular relationship can be contemplated; the example does not warrant the universal proposition “All animals are writers.”

15. Why the third-figure conclusion is not universal

Consider:
Every student is human.
Every student can read.

It does not follow that every human can read. Some humans may be outside the student class and unable to read. The premises tell us only that the two predicates meet in the student class.

In the third figure, a shared subject connects two predicates. With relevant instances, it can establish a particular overlap, not a universal rule for either wider class.

16. Compare the three displayed figures

Put the core patterns together:
Figure one: A ⇒ B and B ⇒ C; therefore A ⇒ C.
Figure two: the same consequent is approached affirmatively and negatively; the extremes are separated.
Figure three: a common antecedent has two different consequents; a restricted overlap is expressed.

The word “if” alone never identifies the figure. The position and relation of the common component across the premises do.

17. Where the fourth figure fits

The book says these syllogisms fall within the four figures, but this brief passage explicitly illustrates figures one, two, and three. It does not supply a separate fourth-figure example here.

The fourth figure can still be understood through placement of the common component; it should not be attributed to the source through an invented quotation.

This distinction separates the generality of a rule from the examples actually printed.

18. Five ways to combine conditional premises

The commentary distinguishes five main combinations by premise type:
1. Two connected conditionals.
2. Two disjunctions.
3. A categorical and a connected conditional.
4. A categorical and a disjunction.
5. A connected conditional and a disjunction.

The first uses chains or relations of implication; the second requires care with alternatives. Mixed forms require a bridge between different propositional structures. Classifying the premises does not by itself establish a valid conclusion.

19. Recognizing connected and disjunctive propositions

A connected conditional states a relationship between an antecedent and a consequent:
If it rains, the ground will be wet.

A disjunction places alternatives in a defined relationship:
This integer is either even or odd.

When examining a syllogism, classify each whole premise. A categorical clause can occur inside a larger conditional without making that whole premise categorical. Its overall logical structure decides its type.

20. A conjunctive syllogism from disjunctions

إما كل أ ب، أو كل ج د
Either every A is B, or every C is D.

This is a disjunction whose alternatives are categorical propositions. The operative question is not how often “all” occurs but how the two alternative assertions are related.

When a second disjunctive premise further constrains one alternative, any new alternative in the conclusion must be established by an appropriate valid inference.

21. The source’s disjunctive conclusion

دائما إما كل أ ب، أو كل ج ه، أو كل د ز
Always, one of the alternatives obtains: every A is B, or every C is H, or every D is Z.

The book offers this form as a conclusion from disjunctive premises. Da'iman qualifies the overall disjunctive statement; it does not assert that every listed alternative is simultaneously true.

Each symbolic component is itself a whole categorical proposition. One must still examine which alternatives are licensed and under what relation between the original premises.

22. What “either/or” does in a disjunction

“Or” joins clauses into one compound proposition. It does not establish the truth of each clause separately.

For example:
This integer is either even or odd.

The compound claim can be true without identifying which alternative holds. More information—such as the number itself or exclusion of one alternative—is required to affirm a particular side.

In disjunctive syllogisms, a valid conclusion preserves the appropriate alternative structure instead of asserting an arbitrary disjunct.

23. A mixed categorical–connected syllogism

كلما كان ب ج، فكل ج أ، وكل أ د
Whenever B is C, every C is A; and every A is D.

The first whole premise is connected: its consequent contains the categorical “Every C is A.” The other premise, “Every A is D,” is categorical in its own right.

We must distinguish the whole conditional from the categorical proposition embedded inside it. The common term A lets the categorical conclusions be chained under the condition.

24. Deriving the mixed-form conclusion

كلما كان ب ج، فكل ج د
Whenever B is C, every C is D.

Under the antecedent condition, the first premise supplies “Every C is A.” The second premise states “Every A is D” without that condition. Chaining the two yields “Every C is D” under the original condition.

It would be mistaken to discard the first condition and claim unconditionally that every C is D.

25. The remaining mixed combinations

وعلى هذا القياس باقي التركيبات
The remaining combinations follow the corresponding method.

The remaining forms in the fivefold classification are:
1. One categorical premise and one disjunction.
2. One connected conditional and one disjunction.

For either, the figure, alternatives, shared part, and production conditions must be checked. These pages do not provide separate worked original examples for both forms; inventing a supposed source quotation would misrepresent the text.

26. Worked classification of three arguments

Example 1: If A then B; if B then C; hence if A then C.
This is the connected chain of figure one.

Example 2: If A then C; under B the C relationship is denied; the extremes are separated.
This resembles the source's second-figure pattern, provided the negation has the appropriate scope and reference.

Example 3: If A then B; if A then C; the two predicates meet in the restricted common A case.
This illustrates the particular aspect of figure three, not “Every B is C.”

27. Four steps for analyzing a conditional syllogism

Analyze any conjunctive conditional syllogism in four steps:
1. Classify each whole premise as categorical, connected, or disjunctive.
2. Locate the common part as antecedent, consequent, subject, or predicate.
3. Check quality, quantity, and the exact conditional or alternative relationship.
4. State only the conclusion actually entailed by those premises.

A useful validity test is whether the premises could all be true while the proposed conclusion is false. If so, the inference needs correction.

28. Lesson synthesis and next section

Conjunctive syllogisms involving conditionals still work with premises, a shared component, and an entailed conclusion. What changes is the possible mixture of connected, disjunctive, and categorical premises.

The first figure gives a directed chain, the second contrasts opposite relations to a common side, and the third connects two predicates through a shared antecedent in a restricted case. The text then gives disjunctive and mixed categorical–connected forms.

The next section on al-qiyas al-istithna'i considers conclusions drawn through explicit exceptions using “but.”