The discussion concerns the connected conditionals, namely the two types known as entailment and disjunctive division.

They also state that the first figure of the categorical syllogism yields all four kinds of conclusions—affirmative, negative, particular, and universal; that the second figure yields only the universal and particular negatives; and that the third yields only the particulars, whether affirmative or negative.

In the entailment-type conditional, affirming the very antecedent produces the very consequent, and denying the consequent produces the negation of the antecedent. Muslim dialecticians express this with the maxim: “The existence of the implicant entails the existence of the implied, and the absence of the implied entails the absence of the implicant.”

Concise as it is, this principle in fact embraces every form of proof—whether one calls it a demonstration, a syllogism, or anything else—for whatever serves as evidence for something else necessarily implies it. Thus, whenever the implicant (the proof) is realized, the implied (the sought conclusion) must also be realized, and whenever the implied is absent, the implicant must likewise be absent.

Hence, whoever knows a claim to be false knows that no sound proof can support it, for it is impossible for a valid proof to uphold falsehood. Conversely, whoever knows a proof to be sound knows that its consequent—the desired conclusion—is true; for if the proof is true, the point it indicates must be true, whatever variety may exist in the outward forms, premises, or arrangements of proofs.

Accordingly, a proof that can be cast in the form of a categorical syllogism can equally be cast in the form of a conditional syllogism, or in other formulations beyond the precise wording and order they describe. Our present concern is with what they themselves have mentioned, just as we earlier treated the connected conditional.

As for division (taqsīm), it may preclude both combination and absence, as when one says, “A number is either even or odd”; these two contradictories can neither coexist nor both be absent. It may also preclude combination yet not absence, as with two contraries that cannot coexist yet may both be lacking, such as saying, “This is either black or red,” while it may in fact be neither.