Dilemmas and Disjunctions

We should look at the differences between a dilemma and a disjunction. We should also examine some ways to identify and challenge each.

Some have mentioned “if p, then q,” is equivalent to “not-p or q.” However, others immediately object this is the fallacy of denying the antecedent. That is to say, it does not follow from p’s negation that q is also false. But we must take note that’s not what is being said. “Regular” language, such as English, often fails us with respect to logical expression.

“Not-p or q” is actually a disjunction. Only a dilemma demands one side is true to the exclusion of the other (ex. “if p, then q,” as a dilemma is equivalent to “either p or not-q,” since if p, then not-not-q [i.e. q], and if not-q, then by modus tollens not-p). With a dilemma, affirming either side as true provides warrant for the falsity of the other; denying either side as true provides warrant for the truth of the other. In a valid disjunction, one or both sides may be true. Because of this, affirming one side of the disjunction as true tells us nothing of the truth of the other side. As an example:

1.      Either evolution is true or design is.

2.      Evolution is true.

3.      Therefore, design is false [they may both be true, logically speaking].

What one needs to do in a disjunction is to eliminate, if possible, one of the sides, regarding it to be false. In this way, the other side is regarded to be true. Here’s where the tricky stuff starts, however. Suppose I put forth a disjunctive argument (copied from William Lane Craig):

1.      Either God exists or the moon is made of green cheese.

2.      The moon is not made of green cheese.

3.      Therefore, God exists.

I think one is safe in asserting (2). So, it follows God exists, right? As far as validity goes, this is correct. But one should not expect many atheists or agnostics to come to belief in God from this argument. Why not? Because one may object that (1) is a false disjunction. A disjunction is false if none of the sides (or “disjuncts”) are true. It seems to me “God exists” is indeed true. However, one may demand to see reasons we should think that is true (and even if we do, the argument becomes superfluous, since we already accept the conclusion!).

Another way to tackle a disjunction, besides targeting its validity or truth, is to deny the minor premise. Here is an example:

1.      Either actions are determined or they are free.

2.      They are not determined.

3.      Therefore, they are free.

By denying (2), one would claim the disjunct “actions are determined” is true. I must stress that this does not prove, formally, that actions are not free (one could raise issues of compatibilism, for instance). But it does allow the objector to avoid accepting the conclusion.

The Christian may face this or a dilemma when encountering atheist objections. How is one to identify the difference? First, examine the language used. This is especially helpful if the argument is not formally laid out. If the objector says, “either A or B, but not both,” you know you are dealing with a dilemma. If that is the case one should seek to demonstrate either the dilemma is false (by providing a third option) or that one of the horns of the dilemma does not entail much of an objection after all. Second, look at the claim of the either-or statement (or form if applicable). Does it affirm one as true, and thus conclude the other as false? If it does, it is probably a dilemma. Third, can both sides be true? If yes, then you are dealing with a disjunction.

Identifying a dilemma or disjunction as such can be extremely helpful for the Christian to decide what tactic to take to overcome the argument or objection. Have any questions, examples, or comments? Post one here!