# Empty Labels

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 flawwhich 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.”

• 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.

• 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”

• 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.

• 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?

• 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.

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

• 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.