Disjunctive syllogism

In classical logic, disjunctive syllogism (historically known as modus tollendo ponens (MTP), Latin for "mode that affirms by denying") is a valid argument form which is a syllogism having a disjunctive statement for one of its premises. An example in English: 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 it is known that at least one of two statements is true, and that it is not the former that is true; we can infer that it has to be the latter that is true. Equivalently, if P is true or Q is true and P is false, then Q is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where 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, which is another rule of inference involving a syllogism. It is also related to the law of noncontradiction, one of the three traditional laws of thought. For a logical system that validates it, the disjunctive syllogism may be written in sequent notation as where is a metalogical symbol meaning that is a syntactic consequence of , and . It may be expressed as a truth-functional tautology or theorem in the object language of propositional logic as where , and are propositions expressed in some formal system. Here is an example: I will choose soup or I will choose salad. I will not choose soup. Therefore, I will choose salad. Here is another example: It is red or it is blue. It is not blue. Therefore, it is red.

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