A disjunctive syllogism, also known as disjunction-elimination and or-elimination (∨E), and historically known as modus tollendo ponens,, is a classically

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

 valid

Validity

In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

, simple argument form:

A is B or C

A is not C

Therefore, A is B

In logical operator notation:
where represents the logical assertion

Logical assertion

A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....

.

Roughly speaking, we are told that at least one of two statements is true; then we are told that it is not the former that is true; so we infer that it has to be the latter that is true. The reason this is called "disjunctive syllogism" is that, first, it is a syllogism

Syllogism

A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

--a three-step argument

Argument

In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...

--and second, it contains a disjunction, which means simply an "or" statement. "Either P or Q" is a disjunction; P and Q are called the statement's disjuncts.

Note that the disjunctive syllogism works whether 'or' is considered 'exclusive' or 'inclusive' disjunction. See below for the definitions of these terms.

Here is an example:

Either 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 either red or blue.

It is not blue.

Therefore, it is red.

Inclusive versus exclusive disjunction

There are two kinds of logical disjunction:

• inclusive
  Logical disjunction
  In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...
  
  means "and/or" - at least one of them is true, or maybe both.
• exclusive ("xor") means exactly one must be true, but they cannot both be.

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:

Either P or Q.

Not P.

Therefore, Q.

is valid and indifferent between both meanings. However, only in the exclusive meaning is the following form valid:

Either P or Q (exclusive).

P.

Therefore, not Q.

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 ponendo ponens and modus ponendo tollens

Modus ponendo tollens

Modus ponendo tollens is a valid rule of inference, sometimes abbreviated MPT. It is closely related to Modus ponens and modus 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

Reductio ad absurdum

In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

 and disjunction elimination

Disjunction elimination

In propositional logic disjunction elimination, or proof by cases, is the inference that, if "A or B" is true, and A entails C, and B entails C, then we may justifiably infer C...

.

Other forms of syllogism:

• hypothetical syllogism
  Hypothetical syllogism
  In logic, a hypothetical syllogism has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic, it is a short-hand for the theory of consequence.-Propositional logic:...
  
  
• categorical syllogism

Disjunctive syllogism holds in classical propositional logic and intuitionistic logic

Intuitionistic logic

Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...

, but not in some paraconsistent logic

Paraconsistent logic

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...

s.