Disjunctive Syllogism

In classical logic disjunctive syllogism (historically known as modus tollendo ponens) is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.

Either the breach is a safety violation, or it is not subject to fines.

The breach is not a safety violation.

Therefore, it is not subject to fines.

In propositional logic, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), is a valid rule of inference. If we are told that at least one of two statements is true; and also told that it is not the former that is true; we can infer that it has to be the latter that is true. If either P or Q is true and P is false, then Q is true. The reason this is called "disjunctive syllogism" is that, first, it is a syllogism, a three-step argument, and second, it contains a logical disjunction, which simply means an "or" statement. "Either P or Q" is a disjunction; P and Q are called the statement's disjuncts. The rule makes it possible to eliminate a disjunction from a logical proof. It is the rule that:

where the rule is that whenever instances of "", and "" appear on lines of a proof, "" can be placed on a subsequent line.

Disjunctive syllogism is closely related and similar to hypothetical syllogism, in that it is also type of syllogism, and also the name of a rule of inference.