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.”