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.

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 "${\displaystyle P\lor Q}$", and "${\displaystyle \neg P}$" appear on lines of a proof, "${\displaystyle Q}$" can be placed on a subsequent line.

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