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