Phil 101: Introducing Leibniz’s Law

There’s a principle very often appealed to in philosophy, which says this:

Leibniz’s Law
For any entities x and y, if they’re one and the same thing, then any property that either of them has, the other must have too.

This is named after the philosopher Leibniz pronounced “Libe-nitz”.

Another way to grasp what this principle is saying is to consider its contrapositive formulation:

If there’s some property that x has and y lacks (or y has and x lacks), then x and y can’t be one and the same thing; they must be two different things.

That sounds extremely reasonable. And it doesn’t just sound like something true of physical things, with physical properties. It sounds like it would also be true of non-physical souls (if there are such things), numbers, words, and so on.

An effective way to prove that any two things are really two, and not one, would be to find some property that they differ in.

Here’s an example of using Leibniz’s Law in some natural reasoning:

  1. There goes Superman 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 lacks.
  4. So Superman is not identical to (is not one and the same person as) Jimmy Olson.


  1. The person who murdered Mr Body is left-handed.
  2. The butler is not left-handed.
  3. So the butler is not the person who murdered Mr Body.

These seem to be good pieces of reasoning. In other words, if the premises are true, it seems legitimate to infer that the conclusion will be true too.

Of course, I might be fooled into accepting some of these premises when they’re not true. Maybe the butler is left-handed after all; he’s just managed to fool me into thinking he isn’t. Still, that wouldn’t show that the reasoning is bad. The reasoning wouldn’t have led me astray; it was my being fooled into accepting the premise that did so.

Note that Leibniz’s Law doesn’t say we can infer like this: “x is F, but y is G, so x isn’t identical to y.” Maybe x and y are both F and G. For example, suppose it’s the 1980s and I’ve just met President Reagan. Somebody asks me, Hey do you think he’s that same guy, Ronald Reagan, who acted in the old westerns we’ve been watching? I say “No way! Ronald Reagan is a famous actor. But President Reagan is a politician. So they can’t be identical.” Clearly my reasoning here is mistaken. After all, Ronald Reagan was both a famous actor and a politician. I must have been thinking that President Reagan was a politician instead of being a famous actor; that is, I must have been thinking it’s not possible for someone to be both a politician and a famous actor. But that’s wrong. This is possible. (And Reagan is not the only example.) The lesson here is that when we’re applying Leibniz’s Law, if x does have some property F, we should be checking whether y lacks that property F. The fact that y is G only helps here if its being G is incompatible with its being F.

Some clarifications

Applying Leibniz’s Law in Support of Dualism

Some materialists think that you and/or your mind or self is one individual object m, and your body (or maybe just your brain) is an individual object b, and that m and b are numerically identical. As we discussed before, materialists don’t have to think this. A materialist always has the option of denying that the mind is any kind of substance or object. They could say that minds are more like hikes or dances. What makes them a materialist is that they think all the objects there are, are physical. But they don’t have to say that the minds are some of those physical objects, anymore than we have to say that hikes or dances are.

But suppose you are the kind of materialist who does want to say that m is an object, and that it’s the same object as some physical b, such as your body or brain.

Dualists on the other hand think there’s a non-physical individual object s, namely your immaterial soul, that could in principle exist without your body. Many dualists will say that m = s. (Others may think m = some combination of s and b.) All dualists will say that mb. That is, they’ll think that m and b are not numerically identical, where b is your body or brain or any physical object.

How might the dualist argue that mb? Are there any properties that m and b seem to differ with respect to, so that dualists can appeal to Leibniz’s Law to show this?

They want to argue “minds can’t be physical because physical objects are … but minds aren’t …”

Some ways of filling in the dots that students have suggested in past years include:

A dualist may appeal to properties like these, arguing that physical objects have them but minds lack them, so minds can’t be physical objects.

The materialists who want to identify you and/or your mind or self with some physical object will resist those arguments. I’ll sort the dualist’s arguments into several different groups, differentiated by what kind of complaint these materialist will make about them.

Using Leibniz’s Law, Part 1

Some of the dualist’s arguments may have the same problem as our Ronald Reagan example discussed earlier. Here’s an example:

  1. Physical objects (brains, bodies) all take up space.
  2. But minds are mental.
  3. So minds are not physical objects.

This argument can only succeed if being mental is incompatible with taking up space. And we haven’t yet settled that. That’s just what the dualist and the materialist are arguing about.

So the dualist’s arguments should always have the form of saying the one thing has some property, but the other thing lacks that same property.

Let’s try to fix up the previous argument to say that, explicitly.

  1. Physical objects (brains, bodies) all take up space.
  2. But minds do not take up space.
  3. So minds are not physical objects.

This is an improvement, in that it’s a clear application of Leibniz’s Law. It does seem right that if the premises of this argument are true, the conclusion would have to be true too. The problem though is that the second premise is too controversial to be helpful to use in an argument for dualism. It’s similar to if I were trying to persuade you that God existed, and I argued: “God exists because the Bible says so; and nothing written in the Bible is false.” It may be that everything I’m saying is correct, but you probably wouldn’t be very impressed or moved by my argument. The premise “nothing written in the Bible is false” is just too controversial to appeal to when what we’re trying to establish is whether God exists. Only someone who was already persuaded of the conclusion we’re aiming for is likely to be ready to accept the premise. This is the kind of argument that philosophers call question-begging.

Some arguments are really obviously question-begging, in that you can just see the conclusion among the premises: “God exists, and wrote the Bible. So God exists.” In other arguments, like the one I described a moment ago with the premise “Nothing written in the Bible is false,” the conclusion might not explicitly be among the premises. Still, these arguments are going to be unpersuasive because they start from points which are argumentatively too close to the conclusions they’re trying to establish. A good, persuasive argument should instead have more of a surprising punch to it. It should make its audience see that premises that you’d think are uncontroversial, or at least much less controversial than your conclusion, when taken together in fact do force us to accept that conclusion, after all.

The “Nothing written in the Bible is false” argument wouldn’t do that. And neither would the “Minds do not take up space” argument.

Here are other arguments that have the same feel:

  1. My brain weighs 5 pounds.
  2. My mind doesn’t weigh 5 pounds.
  3. So my mind is not identical to my brain.
  1. Minds are capable of thoughts and sensation.
  2. But physical objects aren’t capable of thoughts and sensation.
  3. So minds are not identical to any physical object.
  1. My brain has physical parts.
  2. My mind doesn’t have any parts. (So we don’t need to argue about whether its parts are physical or not.)
  3. So my mind is not identical to my brain.

In each case, a materialist, or someone who was just trying to make up their mind whether to accept materialsm or dualism, wouldn’t be ready to accept the second premise. If the dualist wants to talk us into their view, they should try harder to find premises that seem plausible even to people who haven’t yet subscribed to dualism.

Using Leibniz’s Law, Part 2

Another group of arguments for dualism at first seem like they might achieve that. Consider:

  1. Aunt Lobelia is in my mind — I’m thinking of her right now.
  2. Aunt Lobelia is not in my brain — how could she fit? Go ahead and cut it open, I guarantee you won’t find her there.
  3. So my mind has a property — having Aunt Lobelia in it — that my brain lacks.
  4. So my mind is not identical to my brain.

Here premise 1 seems to be true, at least as it’s most naturally understood. And premise 2 also seems to be true. And the rest seems to be just an application of Leibniz’s Law. So is this a good argument for dualism?

The problem here is that the sense of “in” where it seems to be clearly true that Aunt Lobelia is in my mind doesn’t seem to be the same sense of “in” where it seems to be clearly true that Aunt Lobelia is not in my brain. For the latter, let’s say “spatially inside.” Sure, Aunt Lobelia is not spatially inside my brain. But is she spatially inside my mind? Don’t think we want to say that. The sense in which she clearly seems to be “in” my mind is that I’m thinking about her, I have thoughts concerning her. It’s not at all clear that that means she is spatially inside my mind. (The dualist wouldn’t want to say she’s spatially inside my mind, anyway! Since they think minds don’t take up space.)

So this argument equivocates. Premise 1 seems plausible if we understand “in” with one meaning (“in my mind” = thinking about). Premise 2 seems plausible if we understand “in” with a second, different meaning (“in” = spatially inside). And if you don’t notice that “in” has changed meanings, it might look like the reasoning structure is correct. But for it really to be correct, we have to understand “in” consistently, with a single meaning.

The dualist may think that the argument is correct with “in” understood consistently in the sense of “thinking about her.” That is, they may accept all the premises and the conclusion of this argument:

  1. My mind is thinking about Aunt Lobelia.
  2. My brain can’t think about Aunt Lobelia, because it’s physical.
  3. So my mind isn’t identical to my brain.

But now we’d once again have an argument whose second premise is too controversial to be persuasive, in an inquiry about whether or not to be dualist.

Using Leibniz’s Law, Part 3

Where have we gotten so far? The materialist complains that if the dualist makes sure to be explicit about what property it is that the mind has but that physical objects lack (or what property it is that physical objects have but the mind lacks), the argument is going to be question-begging. Nobody is going to find the argument persuasive unless they’ve already subscribed to dualism.

The only times so far when it’s seemed otherwise, it was because the dualist was equivocating, and switching the meaning of some word halfway through the argument. One premise sounded plausible with the word understood one way, and the other premise sounded plausible with the word understood the second way.

But it turns out there’s a group of arguments for dualism that seem to be vulnerable to neither of these complaints. These are arguments where the premises do all seem to be plausible, even to people who haven’t yet subscribed to dualism. And it’s not obvious that any words need to switch their meaning halfway through the argument, for the premises to seem obvious in this way.

The general flavor of these arguments can be summarized like this:

  1. I have some “special” or privileged access to my own mind.
  2. I don’t have that kind of access to anyone else’s mind, nor to facts about my brain or body or anything in my physical environment. My access to my own mind is better or different than to those other things.
  3. So my own mind has a property — being accessible to me in this special way — that physical objects lack.
  4. So my mind is not identical to any physical object.

The specifics of these arguments will turn on how they unpack the notion of this “special access.” As we discussed in earlier webnotes, here are some things philosophers say to explain what they mean here.

  1. I and only I can know my own mind’s properties without evidence, observation, or inference.
  2. I can’t be mistaken about my own mind’s properties.
  3. I can’t intelligibly doubt whether my own mind exists: that is, I can’t imagine everything seeming the same right now, but my mind not existing.

The details are controversial, but many philosophers would agree that our access to (at least some parts or aspects of) our own minds is “special” in one or more of these ways. And we don’t seem to have that same special access to anyone else’s mind, nor to facts about our brains or bodies or things in our physical environments.

Perhaps this can be used in an argument by Leibniz’s Law that our minds aren’t identical to anything physical.

For example:

  1. My mind is not “publically accessible”: there are ways to know about it that are in principle only available to me.
  2. Physical objects are all publically accessible: in principle, anybody could get into the best positions to know about them.
  3. So my mind isn’t a physical object.

That argument is reminiscent of the one that Gennaro discusses, which is phrased in terms of what is “knowable through introspection.”

Or here’s the argument that van Inwagen discusses, unpacking “specialness” in our sense (c):

  1. I can’t intelligibly doubt whether my own mind exists right now.
  2. I can intelligibly doubt whether physical objects exist right now. Maybe I’m in some kind of Matrix, and everything seems real, but it’s all an illusion. That at least makes sense.
  3. So physical objects are such that I can intelligibly doubt their existence — I can at least imagine them not existing even though everything seems the same — but my mind lacks that property.
  4. So my mind can’t be identical to any physical object.

These applications of Leibniz’s Law are more challenging than the ones we considered before. It’s less clear how the materialist should respond here. What do you think?