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 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 "P \lor Q", and "\neg P" appear on lines of a proof, "Q" 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.

Formal notation

For a logical system that validates it, the disjunctive syllogism may be written in sequent notation as where \vdash is a metalogical symbol meaning that Q is a syntactic consequence of P \lor Q, and \lnot P. It may be expressed as a truth-functional tautology or theorem in the object language of propositional logic as where P, and Q are propositions expressed in some formal system.

Natural language examples

Here is an example: Here is another example:

Strong form

Modus tollendo ponens can be made stronger by using exclusive disjunction instead of inclusive disjunction as a premise:

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 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.