Conditional Propositions: Connectives, Disjunctions and Their Main Kinds

1. From categorical to conditional propositions

«باب الشرطيات»
The chapter of conditional propositions.

A categorical proposition predicates something of a subject or denies it: for example, “Zayd is learned.” A conditional proposition, by contrast, concerns a relation between two complete propositions. We must identify the two claims rather than merely count words.

“If Zayd is human, he is an animal” connects one claim conditionally to another. “This number is either even or odd” expresses an exclusion between alternatives. These are the starting points of the connective and disjunctive forms.

2. A conditional resolves into two propositions

«وهي التي تنحل إلى قضيتين»
It is a proposition that resolves into two propositions.

“If it rains, the ground becomes wet” has two propositional components:
1. It rains.
2. The ground becomes wet.

Joining the two by a conditional construction does more than place two sentences together; it advances a judgment about their connection. The result is one conditional proposition. This is a structural illustration, not a claim that every rainfall wets every patch of ground.

3. The two main kinds of conditional proposition

«وهي قسمان: أحدهما المتصلة وثانيهما المنفصلة»
It has two kinds: the connective and the disjunctive.

A connective conditional expresses a link between one supposition and a further proposition; it often takes the form “if … then …”.

A disjunctive conditional asserts, or denies, incompatibility between alternatives; it often takes the form “either … or …”. The word “or” alone does not settle the precise logical relationship.

4. Identifying the antecedent and consequent

The antecedent (muqaddam) is the proposition introduced as the condition; the consequent (tālī) is what is connected to it. These names reflect their logical roles, not simply the number of words before and after a pause.

«إن كان زيد إنسانا كان حيوانا»
If Zayd is human, he is an animal.

Antecedent: Zayd is human.
Consequent: Zayd is an animal.

The claim concerns the connection between humanity and animality. Affirming the antecedent and affirming the connection must not be confused.

5. The affirmative connective conditional

«أما المتصلة: فهي التي حكم فيها بثبوت نسبة على تقدير ثبوت نسبة أخرى في الإيجاب»
In an affirmative connective conditional, one relation is affirmed on the supposition that another holds.

The antecedent is assumed, and a consequent is asserted in relation to that supposition. Humanity implies animality because human is a species of animal in the classical classification.

An affirmative conditional does not, by its mere wording, separately affirm that its antecedent actually obtains. It asserts the relevant conditional connection.

6. What a negative connective denies

«ليس البتة إذا كان زيد إنسانا كان فرسا»
It is not at all the case that if Zayd is human, he is a horse.

This is the textbook’s negative connective conditional. It does not deny that Zayd is human; it denies the supposed conditional relation making horsehood follow from humanity.

Do not equate it without qualification with “If Zayd is human, he is not a horse.” The first denies a conditional link; the second states a different affirmative conditional whose consequent is negative.

7. Negating a conditional versus a negative consequent

Compare the structures:
1. “It is not true that if Zayd is human, he is a horse.” — denial of the conditional claim.
2. “If Zayd is human, he is not a horse.” — affirmation of a conditional with a negative consequent.

Both concern horsehood, but the scope of negation changes their logical operation. Identifying a negative word is insufficient; we must ask whether it denies the whole conditional connection or belongs inside the consequent.

This becomes important when we later examine quantifiers and truth conditions for conditional propositions.

8. The necessary or entailing connective

«إن كان ذلك الحكم لعلاقة بين المقدم والتالي سميت لزومية»
If that judgment arises from a relation between antecedent and consequent, the connective is called necessary (luzūmiyyah).

In a necessary connective, the two clauses are not merely side-by-side facts. A genuine relation makes the consequent relevant to the supposition of the antecedent. Such a relation can be conceptual, causal, grounded in a shared cause, or tied to correlatives.

“If this figure is a square, it has four sides” illustrates a conceptual link: four-sidedness is included in the concept of a square.

9. The cause–effect relation

The commentary treats a cause–effect relation as an important source of genuine linkage. Where one condition causes another under the relevant qualifications, a conditional may state a real connection.

For example, “If water is heated to its boiling point at the specified pressure, it begins to boil.” Here the consequent follows the stipulated physical conditions.

By contrast, “If the sun has risen, Zayd must have gone to the market” lacks such entailment. Sunrise alone does not establish Zayd’s journey. A causal relationship is different from merely occurring around the same time.

10. Two effects of a common cause

The relation may also arise where antecedent and consequent are two effects of a third cause. In such a case the antecedent need not itself cause the consequent; both may be tied to the same underlying cause.

Imagine a specified alarm system in which one malfunction both illuminates a warning lamp and triggers an alarm. The two effects share a cause; the light does not necessarily cause the alarm.

This is an explanatory illustration. A common cause does not make every pair of its effects invariably simultaneous; the relevant conditions of the particular system matter.

11. Correlatives: father and son

Correlativity (taḍāyuf) concerns paired concepts understood in relation to each other, such as fatherhood and sonship.

If Zayd is Amr’s father, Amr is Zayd’s son. The link is not coincidental; fatherhood inherently refers to a corresponding relationship.

But the persons and the relation must be correctly specified. From “Zayd is a father,” one cannot infer that any arbitrary person is his son. The identification of the relata is essential.

12. The coincidental connective

«وإن كان ذلك الحكم بدون العلاقة سميت اتفاقية»
If that judgment is made without such a relation, it is called coincidental.

«إذا كان الإنسان ناطقا فالحمار ناهق»
If a human is rational, a donkey brays.

The two clauses are stated together without the previously described relation of entailment. Accepting both claims does not show that human rationality causes, or essentially entails, a donkey’s braying. This lack of a necessary link is what distinguishes the coincidental connective.

13. Do not confuse entailment with coincidence

Compare the two connections:
Necessary: “If Zayd is human, he is an animal.” Animality is included in humanity.
Coincidental: “If Zayd is writing, Amr is in the garden.” Both may happen to be true, but Zayd’s writing does not entail Amr’s location.

Thus the wording “if … then …” does not itself prove a necessary link. One must identify the genuine relation between antecedent and consequent.

14. Simultaneous truth does not establish necessity

Two events may occur at the same time, tempting us to treat them as logically linked. “Prayer is taking place in the mosque” and “a shop is open in the market” may both be true, without either following from the other.

Co-occurrence is an observation. Entailment requires an actual relation connecting the propositions. The former does not establish the latter.

Classical logic examines the relationship within a conditional precisely so that coincidence is not mistaken for an argument.

15. A worked analysis of a connective

Consider: “If every square has four sides, this particular square also has four sides.”

Antecedent: Every square has four sides.
Consequent: This particular square has four sides.
Relation: applying a universal fact to an instance known to fall under it.
Kind: an affirmative connective with an intelligible conceptual relation, on the specified assumption.

Four questions organize the analysis: What is the antecedent? What is the consequent? Is the link affirmed or denied? Is it necessary or coincidental? Only then can the proposition be classified reliably.

16. Definition of the disjunctive conditional

«وأما المنفصلة: فهي التي حكم فيها بالتنافي بين شيئين في موجبة، وسلب التنافي بينهما في سالبة»
A disjunctive conditional affirms incompatibility between two things in its affirmative form, and denies that incompatibility in its negative form.

Incompatibility (tanāfī) concerns whether two propositions may both be true, both false, or neither, under a specified respect. Different restrictions on these possibilities produce different kinds of disjunction.

For “This number is either even or odd,” two alternative claims are presented. The word “or” alone does not establish the precise truth–falsity relation.

17. The three kinds of disjunction

«الشرطية المنفصلة على ثلاثة أضرب»
A disjunctive conditional is of three kinds.

1. Genuine (ḥaqīqiyyah): the alternatives can neither both be true nor both be false.
2. Exclusive of conjunction (māniʿat al-jamʿ): they cannot both be true, but can both be false.
3. Exclusive of omission (māniʿat al-khulū): they cannot both be false, but can both be true.

Truth means that a component proposition holds, and falsehood that it fails. These relationships must be tested under the same time and respect.

18. Genuine disjunction: exclusion on both sides

«هذا العدد إما زوج أو فرد»
This number is either even or odd.

This is a genuine disjunction, provided the subject is an integer. An integer cannot be both even and odd; nor can an integer be neither.

Exclusion of conjunction: even and odd cannot be jointly true.
Exclusion of omission: neither can be jointly false of an integer.

When both restrictions apply, the disjunction is genuine. The key is understanding both prohibitions, not merely repeating “either … or …”.

19. Possible and impossible cases in genuine disjunction

Let A mean “this integer is even” and B mean “it is odd.” Only two combinations are allowed:
A true, B false: the integer is even.
A false, B true: the integer is odd.

Two combinations are excluded:
Both true: an integer is both even and odd — impossible.
Both false: an integer is neither even nor odd — impossible.

Thus a genuine disjunction allows exactly one true side; this follows from the exclusion of joint truth and joint falsehood, not just from the word “or”.

20. Excluding joint truth: māniʿat al-jamʿ

«هذا الشيء إما شجر أو حجر»
This thing is either a tree or a stone.

One and the same thing cannot be both a tree and a stone in the same relevant respect. Therefore the two alternatives cannot both be true.

Yet the thing might be neither a tree nor a stone—for example, it could be a bird. Both component claims would then be false without violating the rule of māniʿat al-jamʿ. This form excludes joint truth, not joint falsehood.

21. Excluding joint falsehood: māniʿat al-khulū

Māniʿat al-khulū is a disjunction whose component claims cannot both be false: at least one must hold. They may, however, both be true.

Consider: “This thing is not human, or it is an animal.” If something is human, it necessarily is an animal; the two clauses cannot both fail.

For a nonhuman animal, such as a horse, “not human” and “animal” are both true. This illustrates why joint truth is permitted in this kind.

22. Book example: at sea or not drowning

«إما أن يكون زيد في البحر أو لا يغرق»
Either Zayd is in the sea, or he does not drown.

The book uses this as an example of exclusion of joint falsehood. Its intended point is that both clauses can be true: someone may be at sea without drowning.

The book treats their joint failure as excluded under the intended supposition. Outside that restricted setting, a person can drown in a river; the example is therefore not a universal physical law. We are studying the logical form under its stated or understood assumptions.

23. A clear additional example of māniʿat al-khulū

Use the rule every human is an animal. For any object define:
A: It is not human.
B: It is an animal.

For a human: A false, B true.
For a horse: A true, B true.
For a stone: A true, B false.

Both false would mean something is human but not an animal, contradicting the stated rule. This gives an especially transparent additional example of exclusion of joint falsehood.

24. Comparing the three disjunctive relations

Test all three by the same standard:
Genuine: both true impossible; both false impossible. Example: an integer is even or odd.
Exclusive of conjunction: both true impossible; both false possible. Example: an object is a tree or a stone.
Exclusive of omission: both false impossible; both true possible. Example: not human or animal.

The genuine form combines both exclusions. The remaining kinds exclude only one of the two combinations. Ask “Can both be true?” and “Can both be false?” before deciding.

25. Negative disjunction: denying the exclusion

«وسلب التنافي بينهما في سالبة»
In the negative form, their incompatibility is denied.

Thus the negative disjunctive denies the alleged incompatibility between the alternatives.

Suppose someone claims, “A person is either a poet or an orator; they cannot be both.” One may deny that exclusion: a person can be both a poet and an orator.

What is denied is the alleged incompatibility, not that every individual actually has both attributes. Denying an exclusion does not by itself prove the two predicates of any particular subject.

26. Worked recognition exercise

Classify each by the kind of judgment being made:
1. If Zayd is human, he is an animal. — affirmative necessary connective: a conceptual genus–species link.
2. If Zayd writes, Amr is in the garden. — coincidental if both just happen to be true without entailment.
3. This integer is even or odd. — genuine disjunction: neither both true nor both false.
4. This thing is a tree or a stone. — exclusive of conjunction: both can be false.
5. This thing is not human or is an animal. — exclusive of omission: both can be true.

Giving the reason makes classification a test of understanding, not vocabulary recall.

27. Summary: conditional relations and exclusion

The connective concerns the conditional link between antecedent and consequent. It may affirm or deny a connection; an affirmed connection may be necessary or coincidental.

The disjunctive form addresses exclusion between alternatives, or denial of that exclusion. A genuine disjunction excludes joint truth and joint falsehood; exclusive of conjunction excludes only joint truth; exclusive of omission excludes only joint falsehood.

The next stage is to study the book’s further divisions of conditional propositions, including opposition, coincidence, singular and quantified forms, and their characteristic quantifier words.