Empty Labels

[Eliezer Yudkowsky]

Consider (yet again) the Aristotelian idea of categories. Let’s say that there’s some object with properties A, B, C, D, and E, or at least it looks E-ish.

Fred: “You mean that thing over there is blue, round, fuzzy, and—”
Me: “In Aristotelian logic, it’s not supposed to make a difference what the properties are, or what I call them. That’s why I’m just using the letters.”

Next, I invent the Aristotelian category “zawa”, which describes those objects, all those objects, and only those objects, which have properties A, C, and D.

Me: “Object 1 is zawa, B, and E.”
Fred: “And it’s blue—I mean, A—too, right?”
Me: “That’s implied when I say it’s zawa.”
Fred: “Still, I’d like you to say it explicitly.”
Me: “Okay. Object 1 is A, B, zawa, and E.”

Then I add another word, “yokie”, which describes all and only objects that are B and E; and the word “xippo”, which describes all and only objects which are E but not D.

Me: “Object 1 is zawa and yokie, but not xippo.”
Fred: “Wait, is it luminescent? I mean, is it E?”
Me: “Yes. That is the only possibility on the information given.”
Fred: “I’d rather you spelled it out.”
Me: “Fine: Object 1 is A, zawa, B, yokie, C, D, E, and not xippo.”
Fred: “Amazing! You can tell all that just by looking?”

Impressive, isn’t it? Let’s invent even more new words: “Bolo” is A, C, and yokie; “mun” is A, C, and xippo; and “merlacdonian” is bolo and mun.

Pointlessly confusing? I think so too. Let’s replace the labels with the definitions:

“Zawa, B, and E” becomes [A, C, D], B, E
”Bolo and A” becomes [A, C, [B, E]], A
”Merlacdonian” becomes [A, C, [B, E]], [A, C, [E, ~D]]

And the thing to remember about the Aristotelian idea of categories is that [A, C, D] is the entire information of “zawa”. It’s not just that I can vary the label, but that I can get along just fine without any label at all—the rules for Aristotelian classes work purely on structures like [A, C, D]. To call one of these structures “zawa”, or attach any other label to it, is a human convenience (or inconvenience) which makes not the slightest difference to the Aristotelian rules.

Let’s say that “human” is to be defined as a mortal featherless biped. Then the classic syllogism would have the form:

All [mortal, ~feathers, bipedal] are mortal.
Socrates is a [mortal, ~feathers, bipedal].
Therefore, Socrates is mortal.

The feat of reasoning looks a lot less impressive now, doesn’t it?

Here the illusion of inference comes from the labels, which conceal the premises, and pretend to novelty in the conclusion. Replacing labels with definitions reveals the illusion, making visible the tautology’s empirical unhelpfulness. You can never say that Socrates is a [mortal, ~feathers, biped] until you have observed him to be mortal.

There’s an idea, which you may have noticed I hate, that “you can define a word any way you like”. This idea came from the Aristotelian notion of categories; since, if you follow the Aristotelian rules exactly and without flaw—which humans never do; Aristotle knew perfectly well that Socrates was human, even though that wasn’t justified under his rules—but, if some imaginary nonhuman entity were to follow the rules exactly, they would never arrive at a contradiction. They wouldn’t arrive at much of anything: they couldn’t say that Socrates is a [mortal, ~feathers, biped] until they observed him to be mortal.

But it’s not so much that labels are arbitrary in the Aristotelian system, as that the Aristotelian system works fine without any labels at all—it cranks out exactly the same stream of tautologies, they just look a lot less impressive. The labels are only there to create the illusion of inference.

So if you’re going to have an Aristotelian proverb at all, the proverb should be, not “I can define a word any way I like,” nor even, “Defining a word never has any consequences,” but rather, “Definitions don’t need words.”

[NS]

You seem to be under the impression that Aristotle was a nominalist about definitions. To my knowledge, that is false. Most of the Posterior Analytics is devoted to showing how definitions can be discovered. He does not believe they are simply arbitrary tags. From where in his corpus are you deriving this interpretation?

[Eliezer Yudkowsky]

I am not talking about nominalism at all, actually; nor Aristotle’s notion of horismos which is often translated as “definition” but better translated as “essence”.

Rather, I am speaking about the Aristotle-influenced view (still held by many Traditional Rationalists today) of what we would call “categories” or “definitions”, in terms of individually necessary and together sufficient properties for membership; and of what may be inferred from these by way of what we would call “syllogisms”. (Aristotle’s sullogismos being more properly translated as “deduction”.)

In particular, it is the idea of categorization-based inference as a matter of logically valid deduction, that has given rise to the notion of being able to define a term “any way you like”; this is an Aristotelian notion but not necessarily Aristotle’s notion.

I should note that, being unwilling to put up with Aristotle’s writing style, my understanding of his work is derived from secondary sources such as the Stanford Encyclopedia of Philosophy. Sorry, but seriously, bleah.

Anyone who did not understand the above comment, my advice is not to bother.

[anonymous12]

Here the illusion of inference comes from the labels, which conceal the premises, and pretend to novelty in the conclusion.

Surely you aren’t suggesting that Aristotelian categorization is useless? Assigning arbitrary labels to premises is the only way that humans can make sense of large formal systems—such as software programs or axiomatic deductive systems. OTOH, trying to reason about real-world things and properties in a formally rigorous way will run into trouble whether or not you use Aristotelian labels.

[Jonah]

I have to agree with anonymous. Having read your discussions of “true-by-definition” and arguments about labels for the past couple of weeks, I wonder what ax you are grinding against Aristotle. Who is making the claim that logical inference yields empirically significant inferences? Why do you see the lack of empirically significant inferences as some kind of point against Aristotelian syllogism? Aristotle was one of the first, if not the first, to attempt to formalize reasoning. Sometimes when I read these posts, I feel like your are failing to distinguish between an inference and an induction. As Hume argues (forcefully, in my opinion), induction based on empirical observations can never be certain. I don’t take this as a point against induction, but rather as a caution against those who use it thoughtlessly. Finally, the fact that logical inference can never yield an empirically significant result may not be equivalent to saying that logical inference is pointless. Unlike the classic proof of socrates’ mortality, there are many tautologies that are not obviously tautological. The most famous of these may be “If A is a formal system that allows the development of arithmetic , then there is no set of axioms, B, such that all true statements in A are provable from B.” This is a hasty statement of Goedel’s incompleteness theorem. This statement is tautological, but does that make it an unimpressive inference? This is a tautology that has been extremely empirically helpful, if only insofar as it freed up the time of those struggling to prove the completeness of arithmetic.

[Lewis_Powell]

I don’t think mortal is included in the definition of human.

Shouldn’t the syllogism be rendered:

All [~feathers, bipedal] are mortal. Socrates is a [~feathers, bipedal]. Therefore, Socrates is mortal.

Which is at least a little bit more interesting than you’ve indicated.

Compare also Mill’s discussion of finding out that diamonds are combustible from “A System of Logic”

[Lewis_Powell]

I should have phrased that as saying that I don’t think Aristotle included mortal in the definition of human.

[bigjeff5]

This wasn’t actually about Aristotle’s definition of a human. It was about deducing items already given in the definitions of Aristotlian labels.

I believe Aristotle’s actual definition of a human was [rational, animal]. The point Eliezer is making is that, given this definition, it’s an empty argument to say “Socrates is human, all humans are animals, therefore Socrates is an animal.” This is blindingly obvious and completely unhelpful when you replace “human” with [rational, animal].

In other words, it sounds like a major insight, but that Socrates must be an animal if he is human is in the very definition of human. It did not give you any new insight in any way if you already knew Aristotle’s definition of animal.

There are other things you can deduce logically from these definitions, but it’s dumb to deduce something that is already given in the definition. That’s the point.