The third concerns opposed realities that are mutually incompatible, whether in existence or nonexistence—in which case they are contradictories—or in existence alone, which is broader than contradiction, or in nonexistence alone, which is narrower than contradiction.

Thus, categorical syllogisms concern two or more similar realities, while disjunctive conditional syllogisms concern two or more opposed realities. Such reasoning is called division, testing, the posing of alternatives, and expository reasoning. Conjunctive conditional syllogisms concern two differing realities that are not opposites; this is called mutual entailment.