Leibniz's Law: Legitimate and Illegitimate Uses

Leibniz's Law says that if A and B are one and the same thing, then they have to have all the same properties. If A and B have different properties, then they cannot be one and the same thing. If we find some property that B has but A doesn't, then we can conclude that A and B are not the same thing. Here is an example. We look out the window and see Superman flying by. We wonder who Superman is. Now as Superman flies by outside the window, we see our friend Jimmy Olson standing beside us. We can reason like this:
  1. Superman is flying outside the window.
  2. Jimmy Olson is not flying outside the window, he's standing right beside me.
  3. So Superman has a property that Jimmy Olson doesn't have.
  4. So Superman is not Jimmy Olson.
or like this:
  1. The person who murdered Mr. Body was left-handed.
  2. The butler is not left-handed.
  3. So the butler is not the person who murdered Mr. Body.
Makes sense, right?

Philosophers have tried to use Leibniz's Law to argue for dualism, in a variety of ways. They try to come up with some property that our minds have but our brains and bodies lack, or vice versa. Together with Leibniz's Law, that would enable them to show that our minds can't be the same thing as our brains or bodies.

One such argument uses the property of being divisible. My brain and body can be divided into smaller parts. We then argue that my mind, on the other hand, cannot be divided into smaller parts. If that's right, then my mind can't be the same thing as my brain or my body.

Another argument goes: my brain weighs 5 pounds. My mind doesn't weigh 5 pounds. So my brain and my mind are different things.

What do you think of that argument?

Another argument goes: I am thinking of my Aunt Helen; so I have Aunt Helen in my mind. But Aunt Helen is not in my brain. So my mind and my brain have different properties: one contains Aunt Helen, and the other doesn't. So my mind and my brain are different things.

What do you think of that argument?

Illegitimate Uses of Leibniz's Law

Leibniz's Law is usually a good form of argument---so long as it's the same property that the one object has and the other object lacks.

However, we have to be extremely careful when we're dealing with properties that have to do with what one believes, or has evidence for believing, or doubts, or hopes to be the case. After all, Lois Lane believes that Superman is strong but she does not believe that the mild-mannered reporter Clark Kent is strong. But Superman is one and the same person as Clark Kent. Similarly, Lois can hope that Superman will kiss her, and fail to hope that the mild-mannered reporter Clark Kent will kiss her, even though, again, Superman is the mild-mannered reporter Clark Kent.

Here's another example. Suppose that Lex Luthor thinks he has succeeded in destroying Superman. In fact, Superman is alive and well, and is walking on the street below, dressed as Clark Kent. Lex argues:

  1. I have no doubt about whether that reporter exists. I have very good evidence that he does exist. I can see him right there.
  2. I do have doubts about whether Superman still exists. I think he was destroyed last nights in my clever death-trap. Some people claim to have seen him this morning, but so far these just seem to be unreliable rumors. I do not have very good evidence that Superman still exists.
  3. We've just seen that Superman and that reporter have different properties. They differ with respect to the quality of my evidence concerning them, and what sorts of doubts I have about their existence. So Superman and that reporter must not be one and the same man.

All of the premises of this argument may be true. Yet its conclusion is false. Superman is one and the same man as the reporter Lex sees. So this argument must be invalid. It is invalid because Leibniz's Law cannot be legitimately used when you're dealing with what people believe, or have evidence for believing, or doubt, or hope to be the case.

Here are some other illegitimate uses of Leibniz's Law:

I think that that reporter is wearing a gray suit. I do not think that Superman is wearing a gray suit. So that reporter has some property which Superman lacks: namely, whether he is thought by me to be wearing a gray suit. So by Leibniz's Law, that reporter must not be Superman.

I am afraid of Superman. I am not afraid of that reporter. So Superman has some property that reporter lacks: namely, the property of my being afraid of him. So again, by Leibniz's Law it follows that Superman is not the same person as that reporter.

There are big debates in contemporary philosophy about how we should explain these kinds of properties. We can't get into those debates here. If you take more advanced classes in philosophy of language or philosophy of mind, you will definitely spend time studying these debates. All you need to know for this class is that Leibniz's Law breaks down when you're dealing with these kinds of properties.

Sometimes philosophers have tried to argue for dualism like this:

  1. The existence of my brain and body can rationally be doubted. I'm pretty sure that I have a brain and body, but it's possible that my experiences of them are just an illusion or dream.

    Recall the predicament of people inside The Matrix. Their experiences seem perfectly real to them. They can't tell that it's all just an illusion. At one point in that movie, the character Morpheus says: "Have you ever had a dream, Neo, that you were so sure was real? What if you were unable to wake from that dream? How would you know the difference between the dream world and the real world?"

  2. On the other hand, the existence of my mind cannot rationally be doubted. If I try to doubt whether I exist, or entertain the possibility that I do not exist, I must exist to do so.
  3. So my brain and my body have a property that my mind lacks: they can rationally be doubted.
  4. So, by Leibniz's Law, my mind can't be the same thing as my brain or my body.

However, the property of "being rationally doubtable" is in the same boat as properties about what I believe and doubt and have evidence for believing. (Being rationally doubtable seems just to mean that one lacks unimpeachable evidence for believing it.) So, like the bad arguments we considered a moment ago, this too is an illegitimate use of Leibniz's Law.