Leibniz's Law: Illegitimate Uses

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.

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:

8. I have no doubt about whether that reporter exists. I have very good evidence that he does exist. I can see him right there.
9. 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.
10. 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:

11. 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?"

12. 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.
13. So my brain and my body have a property that my mind lacks: they can rationally be doubted.
14. 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.