## Disjunctive Syllogisms

The components of a disjunctive proposition--p and q--are called disjuncts. Such a statement does not actually assert that p is true, or that q is, but it does say that one or the other of them is true.

Example:

If we know independently that one of the disjuncts is not true, we can infer that the other must be true. If you know that the meeting will be either in room 302 or 306, and you find that it is not in 302, you can infer that it is in 306.

A disjunctive syllogism has the following structure:

 Either the meeting is in room 302, or it is in room 306. It is not in room 302. Therefore, it is in room 306.

So long as we eliminate all the disjuncts but one, that one must be true--assuming, of course, that the disjunctive premise is true to begin with.

The disjunctive syllogism proceeds by denying one of the disjuncts. Is it equally valid to argue by affirming a disjunct? Is the following inference valid?

 Either p or q. p. Therefore, Not-q.

The answer depends on how we are using the conjunction "or." We sometimes use it in what is called the exclusive sense to mean, "p or q but not both," as in, "Tom is either asleep or reading." We also use "or" in the inclusive sense to mean, "p or q or both," as in, "If she's tired or busy, she won't call back." An argument that denies a disjunct is valid in either case, but an argument that affirms a disjunct is valid only if "or" is used in the exclusive sense.

The problem is that nothing in the logical form of the argument tells us which sense is being used. To make it clear that p and q are exclusive alternatives, people sometimes say, "p, or else q." But, in most cases, we have to decide from the context which sense is intended. For logical purposes, therefore, we assume that "or" is used inclusively, so that affirming a disjunct is fallacious.

In cases where such an argument seems valid intuitively, it is easy to translate the argument into a different form that makes the validity clear.

Comprehension Questions
 1. Determine whether the following disjunctive syllogism is valid or invalid. Either I will graduate in 1998 or in 1999I will graduate in 1999I will not graduate in 1998 a) It is valid. b) It is is invalid. 2. Determine whether the following disjunctive syllogism is valid or invalid. Either the next Olympics will be held in Atlanta, Georgia or in Athens, Greece.It won't be held in Athens, Greece.It will be held in Atlanta, Georgia. a) It is valid. b) It is is invalid.

Hypothetical Propositions