Affirming a Disjunct

Jason Iuliano

You have two choices with Obama. You either believe that he is a man of Christ, or you think he’s a liar.

Penn Jillette

Can you spot the logical error in Penn Jillette’s statement? If so, you’ve iden­tified the fallacy known as affirming a disjunct (AAD).

Either Madison likes cats or she likes dogs. I know she likes cats. Therefore, she doesn’t like dogs.

This argument is flawed because it fails to account for the possibility that Madison likes both cats and dogs. Here’s the example mapped out in logical form.

As you can infer from the table above, this fallacy is caused by an ambiguity in the English word or. Because context makes the meaning clear in every­day usage, you might never have thought about this ambiguity before, but you actually use the word or in two very distinct ways. One type of or is known as “inclusive,” and the other type is known as “exclusive.” Since this difference is the key to understanding the fallacy of AAD, let’s take a moment to explore it.

When you were a child, your mother probably had a hard time convinc­ing you to eat your vegetables. At some point, she likely said, “Either you eat your veggies or you’re not getting dessert.” Chances are, you diligently ate your vegetables and your mother brought out dessert a short while later.

Now, suppose that you ate your vegetables, but your mom failed to pro­vide dessert. You would’ve approach her and complained, “Mom you prom­ised dessert if I ate my vegetables!” But what if she responded, “No, I didn’t, Sweetie. I said, ‘Either you eat your veggies or you’re not getting dessert.’ You ate your veggies and you didn’t get dessert, so my statement is truthful. You really should listen more carefully.”

Aside from ensuring that you’ll never again eat your vegetables in exchange for dessert, what has your mother taught you? Well, she has given you a valuable lesson in the difference between two types of or.

Let’s break them down, so you’ll never be fooled again. On the one side, there is the inclusive or. This is the or used in classical logic. Take the basic proposition “Either A or B.” If this proposition is using an inclusive or, it is true in three circumstances:

(1) A is True.

(2) B is True.

(3) Both A and B are True.

Therefore, if your mother intended to use this inclusive or, her statement would have been true in three situations:

(1) You eat your veggies.

(2) You don’t get dessert.

(3) You eat your veggies AND you don’t get dessert.

Unlucky for you, it was this third possibility that played out. Lest you think your mother played a terrible trick on you, let me point out that this is the type of or that online search engines use. Go try it now to see for yourself. If you type “apple or banana” into Google, it will return results that have the word “apple,” results that have the word “banana,” and results that have both “apple” and “banana” in them.

Exclusive “or” is quite different. It is true only in two circumstances:

(1) A is True.

(2) B is True.

If both A and B are true, then an exclusive or statement is false. If Google searched using this form of or, it would return results that had the word “apple” and results that had the word “banana,” but it would exclude results that had both “apple” and “banana.”

To sum up, for inclusive or, there is an implicit or both that follows the statement. However, for exclusive or, there is an implicit but not both that follows the statement (Crain and Thornton 2013).

AAD occurs when someone mistakenly believes that an inclusive or is an exclusive or. To return to the original example, this is precisely what Penn Jillette did when he said, “You have two choices with Obama. You either believe that he is a man of Christ, or you think he’s a liar.” Penn failed to realize that there is a third choice: someone can believe that Obama is both a man of Christ and a liar.

Importantly, not every or statement can fall prey to AAD. Consider the following argument:

Either George Washington is dead or alive. George Washington is dead. Therefore, he is not alive.

This argument - known as a disjunctive syllogism - is sound because the first premise correctly uses an exclusive or. In other words, that premise sets out two possibilities that are mutually exclusive. When someone is dead, s/ he is not alive, and when someone is alive, s/he is not dead.

The next time you make an or statement, think a little bit harder about which or you intend to use, and make sure you don’t fall prey to AAD.

Reference

Crain, S., and R. Thornton. 2013. “Unification in Child Language.” In From Grammar to Meaning: The Spontaneous Logicality of Language, edited by Ivano Caponigro and Carlo Cecchetto. Cambridge, UK: Cambridge University Press, 235-236.