# Online Encyclopedia

# Disjunctive syllogism

A **disjunctive syllogism** is one valid, simple argument form:

- Either P or Q.
- Not P.
- Therefore, Q.

In logical operator notation:

- ¬

where represents the logical assertion.

Roughly, we are told that it has to be one or the other that is true; then we are told that it is not the one that is true; so we infer that it has to be the other that 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 disjunction, which means simply an "or" statement. "Either P or Q" is a disjunction; P and Q are called the statement's *disjuncts*.

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:

- Either the Browns win or the Bengals win.
- The Browns do not win.
- Therefore, the Bengals win.

*Inclusive versus exclusive:*

It should be noted with importance that there are two kinds of logical disjunction:

*inclusive*means "and/or" where at least one term must be true or they can both be true.*exclusive*("xor") means one must be true and the other must be false. Both terms cannot be true and both cannot be false.

The popular English language concept of *or* is often ambiguated 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.

*Other forms of syllogism:* hypothetical syllogism, categorical syllogism.