Type | Rule of inference |
---|---|

Field | Propositional calculus |

Statement | If is true or is true and is false, then is true. |

Symbolic statement |

In classical logic, **disjunctive syllogism**^{[1]}^{[2]} (historically known as * modus tollendo ponens* (

An example in English:

- I will choose soup or I will choose salad.
- I will not choose soup.
- Therefore, I will choose salad.

In propositional logic, **disjunctive syllogism** (also known as **disjunction elimination** and **or elimination**, or abbreviated **∨E**),^{[7]}^{[8]}^{[9]}^{[10]} 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:

- It is red or it is blue.
- It is not blue.
- Therefore, it is red.

Here is another example:

- 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.

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

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.^{[11]}