The Disjunctive syllogism reference article from the English Wikipedia on 24-Jul-2004
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Disjunctive syllogism

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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:

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).
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.