A formulation of Leibniz’s Law / the Indiscernibility of Identicals

In discussing the Trinity or Incarnation, I often have an exchange which goes like this:

• someone: Jesus is God.
• me: You mean, Jesus is God himself?
• someone: Yeah.
• me: Don’t you think something is true of Jesus, that isn’t true of God, and vice-versa?
• someone: Yes. e.g. God sent his Son. Jesus didn’t. God is a Trinity. Jesus is not a Trinity.
• me: Right. Then in your view, Jesus is not God.
• someone: But he is.
• me: So, you think he is, and he ain’t?!
• someone: [silent puzzlement]

In this post, I want to explain the part in italics. First: a point of clarification. The second and third lines are important. When many say “Jesus is God” they just mean that in some sense or other Jesus is “divine.” (This could mean a lot of things, depending on one’s assumed metaphysics.) But this sort of person (line 3) understands Jesus to be “divine” in the sense of just being one and the same as God – that Jesus is God himself – one person, so just one (period).

In the italicized line, I’m applying  something called Leibniz’s Law, or the Indiscernibility of Identicals. I sometimes put this roughly as, some x and some y can be numerically identical only if whatever is true of one is true of the other. That’s a sloppy way to put it.

In logic, a more precise way of stating it (used e.g. by Richard Cartwright) is:

(x)(y)(z) ( x= y only if (z is a property of x if and only if z is a property of y))

Literally: for any three things whatever, the first is identical to the second only if the third is a property of the first just in case the third is a property of the second.

The basic intuition is that things are as they are, and not some other way. So if x just is (is numerically the same as) y, then it can’t be that x and y qualitatively differ. This seems undeniable.

There are a few problems, though, with the above formula, which any person trained in philosophy may spot.

First, don’t things change? e.g. Last year you weighed 200, and now you weight 210 lbs. But does this mean that the you of 2010 is not numerically the same as the you of 2011? Ridiculous! Things can qualitatively change while remaining numerically the same. That’s just common sense.

Second, what about this property: being believed by Laverne to be innocent. Suppose that Casey Anthony’s lawyer mounted this defense:

Ladies and Gentlemen of the jury. I submit that we can be sure that Caylee’s killer is not one and the same as my client Ms. Anthony, for the killer has a feature she does not: being believed by Ms. Laverne Shirley of Milwaukee, Wisconsin, to be guilty of murder. I’ve got Ms. Shirley right here, so you can just ask her.

This would move the defendant to tears, as it is such an obviously stupid defense.

Strictly, then, there are two sorts of counterexamples to the principle I stated above.

I haven’t concerned myself much with this so far in print or online. I’m more concerned with the fundamental intuition (that is, the evidence) imprecisely gestured at above, than I am with coming up with a counter-example-proof universal principle. When arguing about God and Jesus, I just stick with differences which are at a time (or eternal) and which are intrinsic, or which are relational, but don’t have to do with intentional attitudes of third-parties, like beliefs. (I take mental states or actions or stances to be intrinsic to the self whose they are.)

Thus, one of my favorite examples has been that one night in the Garden of Gethsemane: at that time, Jesus didn’t want Jesus to be crucified (he was asking that this may be averted, if God would so permit) yet God did want Jesus to be crucified (that was his plan all along). Or again: if you think God is triune – that would be intrinsic to God and eternal – either something which holds at all times, or in timeless eternity. If one thing is always triune and the other ain’t – we’re not talking about one thing here.

But here’s a crack at a principle which is necessary, exceptionless, and self-evident: sadly, it requires 4 variables. It requires in addition just one 3-place predicate W (a, b, c). This is to be read: a is a way b is at c, or a is a mode of b at c, or a is how b intrinsically is at c. Here then, is my preferred version of Leibniz’s Law:

(w)(x)(y)(z) ( x = y -> (W(z, x, w) <-> W(z, y, w)))

Literally: for any four things, the second and third are identical only if the fourth is a way the second is at the first just in case the fourth is a way the third is at the first.

Dang, that’s ugly. Perhaps better to use the variables:

For any w, x , y, and z: x just is y only if: z is a way x is at w if and only if z is a way y is at w.

Call these, from left to right: z, y, x. (click for image credit)

This gets around the two sorts of problems noted. This principle would not let us infer that I’m not numerically the same as my slimmer, younger self. Nor would it license a foolish juror to exonerate Anthony on the grounds that the killer, but not she, has the feature being thought by Laverne to be guilty, for that phrase does not pick out any intrinsic way Casey Anthony ever has been, is, or will be.

The predicate W (_,_,_) can be interpreted in different ways. Not believing in properties, but believing in primitive modes of things, I like to understand it as above. But if you like, interpret W in terms of property-instantiations (if you believe in universal properties) or individual properties (tropes) if you believe in those. It is just meant to refer to things we’re all familiar with, like my being happy now, that ball’s present redness, etc.

It seems to me that this formula captures the intuition gestured at above, and that it is self-evident – something a normal, adult human knows to be true upon coming to understand what it means.

Dear reader: you understand it, with a bit of effort, no? And so, does it not have that obvious shine of truth to it that this has:

For any x, y, and z: if x is bigger than y, and y is bigger than z, then x is bigger than z.

You know this to be true not just of the cartoon here, but of any three things there are or could be, right?

And so, if you’re very sure this version of Leibniz’s Law is true, you should be as sure that Jesus and God are two, as you are that they have ever, do, will, or just could differ (be different ways). You too are now a dastardly “denier of the divinity of Christ,” but only in the sense described at the top of this post. Whether Christ is divine in some other sense, e.g. possessing the divine nature, or being the member of a perfect society, is another question.

Back to the conversation above, last two lines. You think them to have differed. Yet, things which have differed can’t be numerically one. Ergo, in a sense, you already believe them to differ. Or at least, you’re already in a sense committed to their non-identity. If you don’t believe this outright, you should, once it is pointed out that it follows by a self-evident truth from things you do believe. You then have a choice – revise those old beliefs, so they no longer imply this new one, or accept this new belief, and adjust other beliefs according. I suggest that your beliefs about = and Leibniz’s Law are not good candidates for revision, though.

Interestingly, I never have these sorts of conversations with people who understand what identity is and believe there are truths about it – i.e. philosophers, philosophy majors, theologians who have studied a bit of philosophy and logic. They simply accept that Jesus and God aren’t one and the same (aren’t identical), and go on to theorize accordingly.

Dear philosophical readers: can you provide a counterexample to my version of Leibniz’s Law?