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: Propositional logic. 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 "formula_2", and "formula_3" appear on lines of  a proof, "formula_4" can be placed on a subsequent line. Disjunctive syllogism is closely related and similar to hypothetical syllogism, in that it is also a type of syllogism, and also the name of a rule of inference. It is also related to the law of noncontradiction, one of the . Formal notation. The "disjunctive syllogism" rule may be written in sequent notation: where formula_6 is a metalogical symbol meaning that formula_4 is a syntactic consequence of formula_2, and formula_9 in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: where formula_11, and formula_4 are  propositions expressed in some formal system. Natural language examples. Here is an example: Here is another example: Inclusive and exclusive disjunction. Please observe that the disjunctive syllogism works whether 'or' is considered 'exclusive' or 'inclusive' disjunction. See below for the definitions of these terms. There are two kinds of logical disjunction: The widely used English language concept of "or" is often ambiguous between these two meanings, but the difference is pivotal in evaluating disjunctive arguments. This argument: is valid and indifferent between both meanings. However, only in the "exclusive" meaning is the following form valid: With the "inclusive" meaning you  could draw no conclusion from the first two premises of that argument. See affirming a disjunct. Related argument forms. Unlike "modus ponens" and "modus ponendo tollens", with which it should not be confused, disjunctive syllogism is often not made an explicit rule or axiom of logical systems, as the above arguments can be proven with a (slightly devious) combination of reductio ad absurdum and disjunction elimination. Other forms of syllogism include: Disjunctive syllogism holds in classical propositional logic and intuitionistic logic, but not in some paraconsistent logics.