Variable Question Fallacies

Albert: “Every time I’ve listened to a tree fall, it made a sound, so I’ll guess that other trees fal­ling also make sounds. I don’t be­lieve the world changes around when I’m not look­ing.”
Barry: “Wait a minute. If no one hears it, how can it be a sound?”

While writ­ing the di­alogue of Albert and Barry in their dis­pute over whether a fal­ling tree in a de­serted for­est makes a sound, I some­times found my­self los­ing em­pa­thy with my char­ac­ters. I would start to lose the gut feel of why any­one would ever ar­gue like that, even though I’d seen it hap­pen many times.

On these oc­ca­sions, I would re­peat to my­self, “Either the fal­ling tree makes a sound, or it does not!” to re­store my bor­rowed sense of in­dig­na­tion.

(P or ~P) is not always a re­li­able heuris­tic, if you sub­sti­tute ar­bi­trary English sen­tences for P. “This sen­tence is false” can­not be con­sis­tently viewed as true or false. And then there’s the old clas­sic, “Have you stopped beat­ing your wife?”

Now if you are a math­e­mat­i­cian, and one who be­lieves in clas­si­cal (rather than in­tu­ition­is­tic) logic, there are ways to con­tinue in­sist­ing that (P or ~P) is a the­o­rem: for ex­am­ple, say­ing that “This sen­tence is false” is not a sen­tence.

But such re­s­olu­tions are sub­tle, which suffices to demon­strate a need for sub­tlety. You can­not just bull ahead on ev­ery oc­ca­sion with “Either it does or it doesn’t!”

So does the fal­ling tree make a sound, or not, or...?

Surely, 2 + 2 = X or it does not? Well, maybe, if it’s re­ally the same X, the same 2, and the same + and =. If X eval­u­ates to 5 on some oc­ca­sions and 4 on an­other, your in­dig­na­tion may be mis­placed.

To even be­gin claiming that (P or ~P) ought to be a nec­es­sary truth, the sym­bol P must stand for ex­actly the same thing in both halves of the dilemma. “Either the fall makes a sound, or not!”—but if Albert::sound is not the same as Barry::sound, there is noth­ing para­dox­i­cal about the tree mak­ing an Albert::sound but not a Barry::sound.

(The :: idiom is some­thing I picked up in my C++ days for avoid­ing names­pace col­li­sions. If you’ve got two differ­ent pack­ages that define a class Sound, you can write Pack­age1::Sound to spec­ify which Sound you mean. The idiom is not widely known, I think; which is a pity, be­cause I of­ten wish I could use it in writ­ing.)

The vari­abil­ity may be sub­tle: Albert and Barry may care­fully ver­ify that it is the same tree, in the same for­est, and the same oc­ca­sion of fal­ling, just to en­sure that they re­ally do have a sub­stan­tive dis­agree­ment about ex­actly the same event. And then for­get to check that they are match­ing this event against ex­actly the same con­cept.

Think about the gro­cery store that you visit most of­ten: Is it on the left side of the street, or the right? But of course there is no “the left side” of the street, only your left side, as you travel along it from some par­tic­u­lar di­rec­tion. Many of the words we use are re­ally func­tions of im­plicit vari­ables sup­plied by con­text.

It’s ac­tu­ally one heck of a pain, re­quiring one heck of a lot of work, to han­dle this kind of prob­lem in an Ar­tifi­cial In­tel­li­gence pro­gram in­tended to parse lan­guage—the phe­nomenon go­ing by the name of “speaker deixis”.

“Martin told Bob the build­ing was on his left.” But “left” is a func­tion-word that eval­u­ates with a speaker-de­pen­dent vari­able in­visi­bly grabbed from the sur­round­ing con­text. Whose “left” is meant, Bob’s or Martin’s?

The vari­ables in a vari­able ques­tion fal­lacy of­ten aren’t neatly la­beled—it’s not as sim­ple as “Say, do you think Z + 2 equals 6?”

If a names­pace col­li­sion in­tro­duces two differ­ent con­cepts that look like “the same con­cept” be­cause they have the same name—or a map com­pres­sion in­tro­duces two differ­ent events that look like the same event be­cause they don’t have sep­a­rate men­tal files—or the same func­tion eval­u­ates in differ­ent con­texts—then re­al­ity it­self be­comes pro­tean, change­able. At least that’s what the al­gorithm feels like from in­side. Your mind’s eye sees the map, not the ter­ri­tory di­rectly.

If you have a ques­tion with a hid­den vari­able, that eval­u­ates to differ­ent ex­pres­sions in differ­ent con­texts, it feels like re­al­ity it­self is un­sta­ble—what your mind’s eye sees, shifts around de­pend­ing on where it looks.

This of­ten con­fuses un­der­grad­u­ates (and post­mod­ernist pro­fes­sors) who dis­cover a sen­tence with more than one in­ter­pre­ta­tion; they think they have dis­cov­ered an un­sta­ble por­tion of re­al­ity.

“Oh my gosh! ‘The Sun goes around the Earth’ is true for Hunga Hun­ter­gath­erer, but for Amara Astronomer, ‘The Sun goes around the Earth’ is false! There is no fixed truth!” The de­con­struc­tion of this sopho­moric nitwit­tery is left as an ex­er­cise to the reader.

And yet, even I ini­tially found my­self writ­ing “If X is 5 on some oc­ca­sions and 4 on an­other, the sen­tence ‘2 + 2 = X’ may have no fixed truth-value.” There is not one sen­tence with a vari­able truth-value. “2 + 2 = X” has no truth-value. It is not a propo­si­tion, not yet, not as math­e­mat­i­ci­ans define propo­si­tion-ness, any more than “2 + 2 =” is a propo­si­tion, or “Fred jumped over the” is a gram­mat­i­cal sen­tence.

But this fal­lacy tends to sneak in, even when you allegedly know bet­ter, be­cause, well, that’s how the al­gorithm feels from in­side.