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. 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: 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 P is true 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. '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 ' P ∨ Q {displaystyle Plor Q} ', and ' ¬ P {displaystyle eg P} ' appear on lines of a proof, ' Q {displaystyle Q} ' 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. It is also related to the law of noncontradiction, one of the three traditional laws of thought. The disjunctive syllogism rule may be written in sequent notation: where ⊢ {displaystyle vdash } is a metalogical symbol meaning that Q {displaystyle Q} is a syntactic consequence of P ∨ Q {displaystyle Plor Q} , and ¬ P {displaystyle lnot P} in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: where P {displaystyle P} , and Q {displaystyle Q} are propositions expressed in some formal system.