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Disjunctive syllogism

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A disjunctive syllogism is one valid, simple argument form: A or B If not A Then B

In logical operator notation:

<math> p \lor q, <math>
<math> p \quad<math>
<math> \vdash q <math>

where <math>\vdash<math> 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.

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