1. In Aristotelian logic, with respect to a syllogism, the conjunction of the two premises of that syllogism with the denial of its conclusion. For example, with respect to the syllogism:

• Some $A$s are $B$s
• All $B$s are $C$s
• Hence, some $A$s are $C$s.

Its corresponding antilogism is:

• Some $A$s are $B$s
• All $B$s are $C$s
• No $A$s are $C$s

In general, the validity of a syllogism corresponds to the necessary falsehood of its corresponding antilogism.

2. In a deductive system $\text{L}$, the classically valid inference:

• • In many cases, antilogism is considered an extension of contraposition. In the context of relevant logics, the term is frequently used to describe the collection of formulae of the form: $\left(\left(\phi \wedge \psi \right)\to \xi \right)\to \left(\left(\phi \wedge \neg \xi \right)\to \neg \psi \right)$