The sense of this passage is to perform a reductio ad absurdum proof on the claim that two formulas, (A → B) and (~A → B), can be true simultaneously.
The proof is: Aristotle then declares the last line to be impossible, completing the reductio.
For example, the contrariety of A and E statements, "All S are P," and "No S are P," follows by a reductio ad absurdum argument similar to the one given by Aristotle.
Later logicians, notably Chrysippus, are also thought to have endorsed connexive principles.
By 100 BCE logicians had divided into four or five distinct schools concerning the correct understanding of conditional ("if...then...") statements.
Sextus Empiricus described one school as follows: And those who introduce the notion of connexion say that a conditional is sound when the contradictory of its consequent is incompatible with its antecedent.
That this school accepted Aristotle's thesis seems clear because the definition of the conditional, requires that Aristotle's thesis be a logical truth, provided we assume that every statement is compatible with itself, which seems fairly fundamental to the concept of compatibility.
In De Syllogismo Hypothetico, he argues that from, "If A, then if B then C," and "If B then not-C," we may infer "not-A," by modus tollens.
However, in the 19th century Boolean syllogisms, and a propositional logic based on truth functions, became the standard.
The objection that is made to the truth-functional definition of conditionals is that there is no requirement that the consequent actually follow from the antecedent.