When we talk about a binary relation some authors say “dyadic instead” we mean a relation that one thing can stand in to a second thing (which might just be the first thing again, but doesn’t have to be). For example, whereas sleeping is a property that a single thing has or lacks, sleeping longer than is a relation that one thing can stand in to a second thing. So it’s a binary relation. That example is not a relation that anything can have to itself, but another binary relation is sleeping in the same bed as, and that’s a relation that things do have to themself (and sometimes to other things).
Some relations involve more than two arguments (for example, sleeping longer than __ but shorter than __), but we’ll just be thinking about binary relations.
We say that a binary relation is reflexive if every thing does stand in that relation to itself. Things are allowed to also stand in the relation to other things, but they don’t have to.
We say that a binary relation is symmetric if, whenever x stands in the relation to y, then y must also stand in the relation to x.
We say that a binary relation is transitive if, whenever x stands in it to y, and y stands in it to z, then x must also stand in it to z.
Some relations have none of these features; others have some of them but not others. An interesting group of relations has all three features. Some examples of these are:
Everybody has the same height as themself; and if I have the same height as you, then you must also have the same height as me; and if I have the same height as you, and you have the same height as Carla, then I must also have the same height as Carla.
When individuals stand in such relations to each other, they count as “equivalent” in some respects (where-they-sleep-wise, or height-wise). That’s compatible with them also being non-equivalent in other respects. You and Carla may be equivalent height-wise, but you have different names, jobs, and so on.
When we are talking about the relation of numerical identity, that is a binary relation that one thing can stand in to a second thing, but only when the second thing is one and the same as the first thing. This relation has the three features listed above. But when things stand in this relation, they aren’t just equivalent in some respects. Since they’ll always be one and the same thing, they must be equivalent in every respect.
Philosophers use some fancy vocabulary to express this idea. They use the word indiscernible to mean “having all the same properties.” And they talk about a principle often called Leibniz’s Law which says that whenever x is numerically identical to y, then x and y must be indiscernible. Or in other words:
The principle is named after the philosopher Leibniz pronounced “Libe-nitz”.
It seems to be true not just for physical things, but for any kind of individual object, including immaterial souls (if there are such things), numbers, words, and so on.
The kinds of properties we’re talking about aren’t just physical properties, but any kind of property the object (or objects) have. And not just intrinsic properties (like being 70 inches tall), but also relations to other objects (like being shorter than Professor Worsnip) and where the object is and has been located in space.
Leibniz’s Law says that if x and y are one and the same individual object, then they’ll have all the same properties. Another way to think of this is: if we find a property one of them has and the other lacks, then we’ve established that x and y aren’t numerically identical. Let’s unpack this.
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.)
If P then Q
may sound like P
has to come first; whereas P only if Q
may sound like it’s Q
that has to come first. But as philosophers use these words, they’re not making any commitment about which comes first.
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:
If Fred is a dog, then Fred is a mammalis
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 must 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 a 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:
If P then Qwon’t in general be equivalent to its converse (
If Q then P) — though there may be special cases where they both must have the same truth-value
If not-Q then not-P)
If x is numerically identical to y, then any property that either of them has, the other must have too.
If there’s some property that x has and y lacks (or y has and x lacks), then x and y aren’t numerically identical (they are two objects).
If x and y have all the same properties, then they are numerically identical.
Another way of saying this is “There can’t be two individual objects that have all the same properties.” This principle (sometimes called the “Identity of Indiscernibles”) was also favored by Leibniz, and by other philosophers, but it is more controversial than the principle we’re calling Leibniz’s Law. It may or not be true. That depends on answers to other subtle questions and what other metaphysical theories you accept.
Annoyingly, some philosophers use the label “Leibniz’s Law” to refer to the more controversial principle. Experience tells us that, regardless of what they’re called, beginning philosophy students often confuse these principles. So put extra effort into trying not to.
Our principle Leibniz’s Law will never deliver the result that two things are numerically identical. It only tells us, if they’re identical, what has to follow. Or it can tell us that two things aren’t numerically identical.
Here’s an example of using Leibniz’s Law in some natural reasoning:
Similarly:
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.
Something we’ve stressed when talking about numerical identity is that a thing’s being numerically identical to itself is not supposed to rule out the possibility of its changing over time. If some properties are essential to x, then x can’t lose those properties and continue to exist. But for properties that aren’t essential, it can. It will depend on which of x’s properties are essential to it. (And this will often be a question that there’s controversy over.)
So Leibniz’s Law should be compatible with a thing’s continuing to be one and the same (numerically identical) over time, even if it changes some of its properties.
But consider this case.
Suppose in 2020 we spend a pleasant afternoon sitting underneath a little tree — call it Junior. In 2040 we return to the same location and find a much taller, fuller tree — call it Senior. People living nearby tell us that no tree was ever cut down or removed from that spot. Does Leibniz’s Law allow us to say that Junior and Senior are one and the same tree? It looks like the trees have different properties. Junior was little, but Senior is very tall and full. Since they don’t have the same properties, how can they be one and the same tree?
Intuitively, we want to say that Junior is the same tree as Senior, even though Senior is much taller and fuller than Junior. That is, Senior is NOW much taller and fuller than Junior WAS. Perhaps this is important. After all, can’t we say:
We have to pay attention to the time at which things have various properties. Senior isn’t just tall, period. He’s tall in 2040. But it also seems true to say that Junior is tall in 2040. There was a time when Junior wasn’t tall. That was back in 2020. But Senior wasn’t tall then, either.
So Junior and Senior can have all the same properties, after all — if we pay attention to the time at which they have them. Neither of them has the property of being tall in 2020. Both of them have the property of being tall in 2040. So Leibniz’s Law will permit us to say that Junior and Senior are one and the same tree, after all.
If x is one and the same thing as y, then: for every time t, if x exists and has some properties at t, then y must also exist and have those properties at t, and vice versa.