Logic 101 · Series · Uncategorized

Parodies and Counterexamples

In a previous post we looked at reductio ad absurdum arguments. These arguments allow us to assume a premise is true, derive a contradiction, and then assert that our original assumption was false.

In that post I shared an extended example of St. Anselm’s famous ontological argument formulated as a reductio ad absurdum argument. Assuming that God is defined as “the greatest conceivable being,” that argument was the following:

  1. Suppose God exists only in the mind as a mere idea.
  2. Existence in reality is greater than existence as a mere idea in the mind.
  3. We can conceive of a God who exists in reality.
  4. If (2) and (3) are true, then we can conceive of a God that is greater than one that exists in the mind as a mere idea.
  5. Therefore, we can conceive of a God that is greater than one that exists in the mind as a mere idea.
  6. If (5) is true, then we can conceive of a being greater than the greatest conceivable being.
  7. If we can conceive of a being greater than the greatest conceivable being, then God does not exist only in the mind as a mere idea.
  8. Therefore, God does not exist only in the mind as a mere idea.
  9. Therefore, God exists only in the mind as a mere idea and God does not exist only in the mind as a mere idea.
  10. Therefore, it is false that God exists only in the mind as a mere idea.
  11. If (10) is true, then God exists in reality.
  12. Therefore, God exists in reality.

That Pesky Gaunilo

Anselm shared his argument in a text called the Proslogion, and this text got into the hands of a monk named Gaunilo. He produced a response to Anselm’s argument, and Anselm was thrilled that somebody thought through his argument carefully. He said that any time the Proslogion was copied, Gaunilo’s response should be appended to it so that readers could consider his comments on the ontological argument.

So what did Gaunilo do that impressed Anselm? He provided a parody of the ontological argument that was supposed to show that there was something obviously wrong with the argument. A parody of an argument is a complete argument that uses substituted terms in order to show that the argument is somehow unsound. So, Gaunilo provided substituted terms into Anselm’s argument to show that there had to be something wrong with it. Suppose there is a lost island that is the greatest island imaginable. (Call it Lost Island.) Here’s Gaunilo’s “Lost Island” parody:

  1. Suppose Lost Island exists only in the mind as a mere idea.
  2. Existence in reality is greater than existence as a mere idea.
  3. We can conceive of a Lost Island that exists in reality.
  4. If (2) and (3) are true, then we can conceive of a Lost Island that is greater than the one that exists in the mind as a mere idea.
  5. Therefore, we can conceive of a Lost Island that is greater than one that exists in the mind as a mere idea.
  6. If (5) is true, then we can conceive of an island greater than the greatest conceivable Island.
  7. If we can conceive of an island greater than the greatest conceivable Island, then the Lost Island does not exist only in the mind as a mere idea.
  8. Therefore, Lost Island does not exist only in the mind as a mere idea.
  9. Therefore, Lost Island exists only in the mind as a mere idea and Lost Island does not exist only in the mind as a mere idea.
  10. Therefore, it is false that Lost Island exists only in the mind as a mere idea.
  11. If (10) is true, then Lost Island exists in reality.
  12. Therefore, Lost Island exists in reality.

Thanks for ruining all the fun, Guanilo!

What’s going on in the Lost Island parody? You can see where “Lost Island” was substituted for “God” (and “island” for “being”) in many of the premises. Premise (2) is the same in both arguments. Without saying exactly which premise is false, this argument shows that the argument must have a false premise because it leads to an obviously false conclusion. Since both Anselm’s and Gaunilo’s arguments are deductively valid, Gaunilo’s argument shows that Anselm’s argument is somehow unsound. That’s how a parody argument works.

Counterexamples

From the texts that I have gone through on logic, I haven’t seen anybody explicitly distinguish a parody argument from a counterexample. From what I’ve encountered, authors sometimes use the term “parody” as a synonym for “counterexample,” and yet they use examples to show that parody arguments (like the one I just sketched) aren’t the same thing as counterexamples.

But there’s a major difference between the two. As I’ve said, a parody argument shows that the structure of an argument leads to an obviously false conclusion, and therefore there must be a false premise in the parody. But if there’s a false premise in the parody, there must be a false premise in the original, parodied argument (even though its conclusion is presumably true). A counterexample, on the other hand, is a complete argument that uses substituted terms to show that an argument with the same form is invalid or incomplete. In other words, it substitutes different terms than the original argument uses in order to show that all the premises can be true, but that the conclusion is false. And that’s just what it means for an argument to be invalid.

These arguments are great if you can’t recall a specific rule that says an argument is invalid. Suppose you have been getting terrible grades in logic class and ask a friend for advice. They say, “Look, I know you’re capable of understanding logic. When you try hard at something, you understand it. But you haven’t tried hard at all. And that’s why I think you don’t understand logic.”

Nice pep talk, right? Not so fast. This kind of reasoning might inspire you to try harder, but the argument is actually a flawed one. Your friend is essentially arguing:

  1. If you try hard to understand logic, then you will understand it.
  2. But you haven’t tried hard to understand logic.
  3. So, you won’t understand it.

You accept all three premises as true. After all, you know introspectively that whenever you do try hard that you understand what you’re studying. And you also know that you haven’t tried hard. So you also know that you won’t understand logic.

Now, since you haven’t tried hard to understand logic, you also don’t know that this argument commits a fallacy called denying the antecedent. This argument takes the form:

  1. If P, then Q.
  2. Not-P.
  3. Therefore, not-Q.

But suppose another friend comes along and tells you that your previous friend’s pep-talk was mistaken. “After all,” he says, “that’s like arguing that whenever the grass outside your window is wet, it has rained today. But the grass outside your window isn’t wet right now, is it? But that doesn’t mean that it hasn’t rained today.” In other words, your second friend has provided a counterexample:

  1. If the grass outside your window is wet, then it has rained today.
  2. But the grass outside your window isn’t wet.
  3. Therefore, it hasn’t rained today.

Premises (4) and (5) are both obviously true to you, but (6) is obviously false. And since argument (4)-(6) is invalid, argument (1)-(3) is invalid also. That’s how a counterexample works.

Being Charitable: Finding Hidden Premises

Now I have to make some qualifications to the definitions I’ve given for parody and counter-example arguments. Life just isn’t simple, is it?

I would add that a parody argument allows us to conclude that there is either a false premise in the argument that is hidden (implicit) or stated outright (explicit). By the same token, a counter-example argument allows us to conclude that the argument as it currently stands is invalid.

What I’m getting at here is that when we use parody arguments or counter-examples, we should look for hidden premises that either (1) make the argument sound or (2) make the argument valid. Let’s return to our examples above.

Suppose that Anselm assumed a premise that says “Only a morally perfect being can be a greatest conceivable being.” Also suppose he had good reasons for thinking this. Well, if this premise plays into Anselm’s original argument, that means that it makes no sense to say there could be a greatest conceivable Lost Island, since an island isn’t a moral being. Now we’ve got more work to do.

Also, suppose that your first friend who was talking to you about logic was assuming a premise that said, “You will understand logic if and only if you try hard to understand it.” Now we have a valid argument:

(HP) You will understand logic if and only if you try hard to understand it.

  1. [Therefore,] If you try hard to understand logic, then you will understand it.
  2. But you haven’t tried hard to understand logic.
  3. Therefore, you won’t understand it.

Premise (1) is now unnecessary, because the conclusion follows from (HP) and (2) alone.

If you have good reasons for thinking there aren’t any hidden premises that make the argument sound (or valid), then you should ask whoever provided the first argument to provide the hidden premise that does so. If they can’t, then it seems to me that it is procedurally sound to conclude that your parody or counterexample is decisive.

Summary

You can deny an argument by providing a parody or counterexample of that argument. A parody is a new argument that parallels the first that shows that the first has a false premise that is either implicit or explicit. A counterexample is a new argument that parallels the form of the first to show that the first is logically invalid as it stands.

If we deny an argument in either of these two ways, we should look for hidden premises that either make the argument sound (in the case of a parody) or valid (in the case of a counterexample). If, and only if, we have good reasons to think that there aren’t any hidden premises that accomplish this and your opponent hasn’t provided one on request, we can conclude that our parody or counterexample is successful. In a word, don’t be lazy even when you come up with a clever parallel argument.

In the next post I’ll describe how counter-arguments are distinct from counterexamples and conclude that they are one reason that we should suspend judgment on a topic (all things being equal).

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