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:

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:

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:

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.


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 …

Some comments and clarifications about this principle / law: