# Phil 101: Introducing Leibniz’s Law

## Logic of Conditionals

If we have a statement of the form If P then Q (which could also be written P → Q or P only if Q), then the whole statement is called a “conditional”, P is called the “antecedent” and Q is called the “consequent”. Some examples:

• If Fred is a dog, then Fred is a mammal. (Or: Fred is a dog only if Fred is a mammal. This conditional is presumably true, regardless of whether Fred is in fact a dog.)

• If Fred is a dog, then Fred is a fish. (This conditional is presumably false.)

The conditional If Q then P (which could also be written P ← Q or Q → P or P if Q) is called the converse of If P then Q. For example:

• The converse of If Fred is a dog, then Fred is a mammal is If Fred is a mammal, then Fred is a dog.

It should be intuitive that this says something different than the original conditional. Presumably the original conditional has to be true, since every dog is a mammal. But If Fred is a mammal, then Fred is a dog might or might not be correct. It’d be correct if, for example, we’re doing detective work on what kind of animal Fred is, and maybe he might be a mammal or a fish, but we’ve settled that if he’s a mammal, then the only species he could be is a dog. But more likely, this conditional would be false. In normal circumstances, the premise that Fred is a mammal leaves it open that he might be any number of species: a dog, a cat, a monkey, and so on…

So in general, a conditional and its converse say different things. They might both be true, or they might both be false, or it might be that one is true and the other is false. There might be special cases where if one of them is true, the other one has to be true too. But we shouldn’t in general expect claims of the form If P then Q and If Q then P to say equivalent things.

Sometimes it is true both that If P then Q and that If Q then P. When both conditionals are true, philosophers express this by saying P if and only if Q, which they often abbreviate as P iff Q. This claim that both conditionals hold is also called a biconditional.

We were discussing the difference between a conditional and its converse. Now, on the other hand, consider these two conditionals:

• If Fred is a dog, then Fred is a mammal.
• If Fred isn’t a mammal, then Fred isn’t a dog.

It should be intuitive that if either of these is true, the other will be true as well. The second conditional is called the contrapositive of the first. That’s a bit of technical logical jargon, but hopefully the basic idea here is clear enough without studying logic. Suppose that Fred is a dog. Then the first claim says that he has to be mammal, and the second implies this too, a bit indirectly, because if he weren’t a mammal, then he couldn’t be a dog. On the other hand, suppose that Fred isn’t a dog. Then the first claim is silent about whether Fred is or isn’t a mammal. And the second claim tells us only that if he isn’t a mammal, then he isn’t a dog, which we were already supposing. So in the case where Fred isn’t a dog, neither conditional adds any new constraints. It’s very natural then to take them to be saying the same thing. They just package it in a different form.

Summarizing:

• the claim If P then Q won’t in general be equivalent to its converse (If Q then P)
• but it will always be equivalent to its contrapositive (If not-Q then not-P)

## Leibniz’s Law

There’s a principle very often appealed to in philosophy. The basic idea is this:

Leibniz’s Law
For any entities x and y, if they’re identical (in the sense of being one and the same thing), then any property one of them has, the other has too.

Sometimes this principle is called the Indiscernability of Identicals (because if the entities are identical, they’re indiscernible — that is, they have the same properties). I’m labeling the principle Leibniz’s Law, and this is a common (and easier to remember) name for it. But awkwardly, this name is sometimes instead used for a different principle, as we’ll see below.

Sometimes this principle is stated in this form:

For any entities x and y, if there’s a property one of them has but the other lacks, then they’re not identical.

That’s roughly the contrapositive of the first claim, and so we can regard these claims as equivalent.

I say “roughly” the contrapositive because these claims aren’t really conditionals of the form If P then Q; instead they have a more complex form For any entities x and y, if … then …

• The kinds of entities we’re talking about aren’t just physical objects, but any kind of entity, including numbers, words, ghosts, and so on (if these things really exist).

• Similarly, the kinds of properties we’re talking about aren’t just physical properties, but any kind of property.

• Also, the kinds of properties we’re talking about aren’t just intrinsic properties (like being 70 inches tall); the principle is supposed to apply to extrinsic properties (like being shorter than Professor Worsnip) too. It’s also supposed to apply to properties like where the object is and has been located in space.

• The kind of “identity” we’re talking about is being one and the same thing, what philosophers usually call numerical identity. Sometimes when we say that objects are “identical” or “the same”, we mean something weaker: just that the objects have many of the same (intrinsic) properties. For example, I might say that you and I bought identical laptops, meaning we have laptops of the same model. Still, we have two laptops. Philosophers usually call that qualitative identity. Leibniz’s Law isn’t talking about that kind of identity. Instead it’s talking about when it’s right to say that my laptop and your laptop is one and the same laptop. (Maybe someone stole it from me and then sold it to you…?)

• What Leibniz’s Law says is that if our laptops are one and the same, then they have to have all the same properties. So if we find a property that one of them has and the other lacks, then we’ve established that the laptops are different.

• Leibniz’s Law is not supposed to rule out the possibility of things changing their properties over time. Perhaps when I owned the laptop, it didn’t have any scratches on it. But when you purchased it, it had a St Pauli logo scratched into it. Leibniz’s Law is compatible with that. It doesn’t say we should argue Prof Pryor’s laptop was unscratched, this laptop is scratched, so therefore they’re two different laptops. Instead, it might be that Prof Pryor’s laptop was unscratched yesterday (but so too was your laptop), and your laptop is scratched today: but if they’re really the same laptop, then Prof Pryor’s laptop is scratched today too.

• Leibniz’s Law is of the form:

• For any x and y, if they’re identical, then they have all the same properties.

Or taking the contrapositive:

• For any x and y, if they don’t have all the same properties, then they’re not identical.

The converse of Leibniz’s Law says instead:

• For any x and y, if they have all the same properties, then they’re identical.

Since “identical” here means “one and the same thing,” another way of saying this is that there can’t be two (or more) distinct objects that have all the same properties. This principle (called the Identity of Indiscernibles) was also favored by the philosopher Leibniz, and by other philosophers, but it is more controversial than the principle we’re calling Leibniz’s Law. It may or may not be true. That depends on answers to other subtle questions and what other metaphysical theories you accept.

What we’re calling Leibniz’s Law on the other hand seems to be a lot more straightforward. (As our discussion proceeds, though, we’ll see that things turn out to be somewhat complicated for Leibniz’s Law, too.)

• Annoyingly, some people use the label “Leibniz’s Law” to refer to this last principle instead of the first. But we’re going to ignore that. Our texts, and we in this course, are going to only use “Leibniz’s Law” to refer to the first principle.

• Experience tells us that, regardless of what they’re called, beginning philosophy students often get confused about the difference between these two principles, so put extra effort into trying to understand and keep track of the difference.

• Also put extra effort into trying to understand and keep track of the difference between sameness in the sense of numerical identity, and sameness in the sense of qualitative identity. Here’s an example. Tamar has two children, Alicia and Abe. For Christmas, she bought them each a new red bike. She made sure to get two bikes of the same design and color, else the kids would get upset. So now: Alicia and Abe have the same bikes, and also the same mother. They “have the same bikes” in the sense of qualitatively identical bikes—there are still two bikes, one for each child. They “have the same mother” in the sense of numerical identity. Unlike the bikes, there is just one mother, that they have to share.

• Whenever we say “one and the same thing,” as in Leibniz’s Law, we’ll be talking about numberical identity.