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Variable Question Fallacies

21 Post author: Eliezer_Yudkowsky 05 March 2008 06:22AM

Followup toWords as Mental Paintbrush Handles

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.

 

Part of the sequence A Human's Guide to Words

Next post: "37 Ways That Words Can Be Wrong" (sequence guide)

Previous post: "Words as Mental Paintbrush Handles"

Comments (32)

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Comment author: Yelsgib 05 March 2008 06:52:02AM -2 points [-]

"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."

I don't really know how to read this sentence. Are you claiming that there is a fixed, stable reality? Are you claiming that the postmodernist professor is implicitly claiming the existence of a fixed reality?

I think the more articulate postmodernist professor would claim "we cannot make reference to a fixed interpretation of phenomena outside of an assumed cultural reference." -You're- the one talking about "reality."

You are using terms like "proposition," "question," etc. very loosely. Could you please clarify what the pertinent "question" that the huntergatherer and the astronomer are trying to "answer" is? What "propositions" do they assert?

I would make two claims. First, I claim that everyday people going about their everyday business are not trying to answer claims/make propositions. Second, I think that "truth" as a linguistic concept exists only in very specific contexts.

Comment author: Caledonian2 05 March 2008 01:49:03PM 0 points [-]

"Have you stopped beating your wife?" has well-defined true-or-false answers. It's just that people are generally too stupid to understand what the no-answer actually indicates.

"Is this sentence false?" is problematic only if we presume that it's meaningful. All things are dividable into the categories of sensible and nonsensical. The sensible portion is then further dividable into the categories of true and false. Nonsense is outside the bounds of the true-false distinction.

Comment author: Patrick_(orthonormal) 05 March 2008 05:51:31PM 3 points [-]

Actually, you can't quite escape the problem of the excluded middle by asserting that "This sentence is false" is not well-formed, or meaningful; because Gรถdel's sentence G is a perfectly well-formed (albeit horrifically complicated) statement about the properties of natural numbers which is undecidable in exactly the same way as Epimenides' paradox.

Mathematicians who prefer to use the law of excluded middle (i.e. most of us, including me) have to affirm that (G or ~G) is indeed a theorem, although neither G nor ~G are theorems! (This doesn't lead to a contradiction within the system, fortunately, because it's also impossible to formally prove that neither G nor ~G are theorems.)

Comment author: Luke_A_Somers 19 February 2012 07:27:32PM 0 points [-]

No, it's not the same. Gödel sentences can be resolved by adding axioms. You can't add axioms to resolve 'This sentence is false'.

Comment author: Patrick_(orthonormal) 05 March 2008 06:33:14PM 0 points [-]

More to the point: (P or ~P) isn't a theorem, it's an axiom. It is (so far as we can tell) consistent with our other axioms and absolutely necessary for many important theorems (any proof by contradiction— and there are some theorems like Brouwer's Fixed Point Theorem which, IIRC, don't seem to be provable any other way), so we accept a few counterintuitive but consistent consequences like (G or ~G) as the price of doing business. (The Axiom of Choice with the Banach-Tarski Paradox is the same way.)

OK, I've said enough on that tangent.

Comment author: Tom_Breton 05 March 2008 11:29:25PM 0 points [-]

"Have you stopped beating your wife?" has well-defined true-or-false answers. It's just that people are generally too stupid to understand what the no-answer actually indicates.

It's usually given as "Have you stopped beating your wife yet?" (Emph mine). The problem is the presupposition that you have been beating your wife. Either answer accepts (or appears to accept) that presupposition.

It's a different sort of bad question than the underconstrained questions. The Liar Paradox OTOH is a case of underconstrained question because it contains non-well-founded recursion.

Comment author: Frank_Hirsch 06 March 2008 09:02:13AM 5 points [-]

I think the trouble about "Have you stopped beating your wife?" is that it is not about a state but about a state transition. It asks "10?", and the answer "no" really leaves three possibilities open (including that the questionee has recently started beating his wife). The sentence structure implies a false choice between answers 10 and 11, because we are used to asking (and answering) yes/no questions about 1-bit issues while here we deal with a 2-bit issue. But you probably knew all that... =)

Comment author: Frank_Hirsch 06 March 2008 09:09:32AM 1 point [-]

Oh, and the Liar Paradox makes much more sense once we overcome our obsession about recursion: If we take the equally valid stance of viewing it as an iteration, it is easy to see that the whole problem is that the proposition does not converge; that's all there is to it.

Comment author: TheOtherDave 03 November 2010 06:23:53PM 12 points [-]

it's not as simple as "Say, do you think Z + 2 equals 6?"

Except, of course, when it is. And sometimes that isn't simple either.

A friend tells the story of being asked (by a stranger at a bus stop) "Is ten percent about two dollars?" "It depends," says she. "Ten percent of what?" "A mop," the stranger explains helpfully. "Um," says she, re-evaluating her understanding of the conversation. "How much does the mop cost?" "Twenty dollars." "Well, then, yes. Ten percent of twenty dollars is two dollars."

"Well," comes the huffy reply, "why didn't you say so in the first place, then?"

This is admittedly largely irrelevant to the point of your post, but I often remember that story when conversations seem to break down. Sometimes the word we have different interpretations of didn't even get spoken in the first place.

Comment author: MoreOn 14 December 2010 02:26:03AM *  1 point [-]

"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.

Am I correct that this sophomoric nitwittery can be solved by taking Earth as a fixed point? Then sun really will go around it. So will the moon. All other planets will go around the sun.

If not, well... you can imagine why I didn't get an A in that philosophy where a teacher meant it literally (as in relativism)

Comment author: bigjeff5 11 February 2011 05:20:49AM 0 points [-]

Basically, yes. From the earth's perspective, the earth spinning on its axis while the sun is (relatively) stationary looks exactly the same as the sun going around the earth.

So they look identical, but Amara the Astronomer knows that this illusion is simply a misunderstanding of how the heavenly bodies interact.

Hunga Huntergatherer doesn't know the earth spins. If he did he'd probably be able to figure out that the sun isn't circling the earth if he thought about it for a while.

Comment author: MoreOn 12 February 2011 12:38:34AM *  1 point [-]

Well in that case Earth doesn't really go around the sun, it just goes around the center of this galaxy on this weird wiggly orbit and the sun happens to always be in a certain position with respect to...... ouch! See what I did? I babbled myself into ineptness by trying to be "absolutely technically correct." I just can't. Even if I finished that "absolutely technically correct" sentence, I'd probably be wrong in some other way I haven't even imagined yet.

So let's accept the fact that not everything that is said which is true is "absolutely technically correct." (True with respect to The Simple Truth, ugh, this semantics is tiring so I'll quit).

The not-technically-correct truth for Hunga Huntergatherer and the not-technically-correct truth for Amara Astronomer seem to verbally contradict each other in the same way that Albert::sound verbally contradicts Barry::sound. Is the solution to it that one is false and other is true? You take the side of Amara Astronomer (and so do I) because the maps in our heads resemble this view better than the other.

The fact that these two notions seem contradictory is not because they are contradictory, but because our minds are trying to map them both into the same spot.

Your solution brings us back to analyzing maps. Its analogue is defining Albert::sound to be correct. I don't believe that the point of the article was to define truth. It's practically impossible to do so (see my fumble above). I think the point of the article was that contradictions in our ill-defined language (and concepts and maps that come with it) do not imply contradictions in reality.

Comment author: bigjeff5 12 February 2011 04:37:05AM 0 points [-]

I believe you missed my point entirely.

I was simply discribing why Hunga Huntergatherer might not have realized that it is the earth that goes round the sun.

Hunga's map is still extremely useful, particularly for getting your bearings. The old saying "the sun rises in the east and sets in the west" is still useful even though it is the earth spinning to create the effect rather than the sun actually moving around the earth (which is implied in the saying).

It's worth noting that Hunga's map is included in Amara's map, not eliminated by it. Albert's map also includes Barry's map, just like Einstein's map of gravity includes Newton's map.

They're all still just maps though, and should be treated as such.

Comment author: Perplexed 12 February 2011 04:45:47AM 1 point [-]

It's maps all the way down.

Comment author: bigjeff5 14 February 2011 12:08:29AM 0 points [-]

The map is not the territory.

Comment author: Perplexed 14 February 2011 02:07:30AM 3 points [-]

True, but what you imagine to be territory may just be another layer of maps.

If you need to think that there is a territory down there somewhere in order to keep from drowning in relativism, then go ahead and think that. But be careful not to imagine that you have actually seen the territory. You haven't. All you have access to (by way of science) are some mighty fine maps.

Comment author: nshepperd 14 February 2011 03:02:07AM *  1 point [-]

I pause. “Well…” I say slowly. “Frankly, I’m not entirely sure myself where this ‘reality’ business comes from. I can’t create my own reality in the lab, so I must not understand it yet. But occasionally I believe strongly that something is going to happen, and then something else happens instead. I need a name for whatever-it-is that determines my experimental results, so I call it ‘reality’. This ‘reality’ is somehow separate from even my very best hypotheses. Even when I have a simple hypothesis, strongly supported by all the evidence I know, sometimes I’m still surprised. So I need different names for the thingies that determine my predictions and the thingy that determines my experimental results. I call the former thingies ‘belief’, and the latter thingy ‘reality’.”

The map is not the territory, and the territory is not the map. My hypotheses about it might be wrong, but the territory is still the territory. How would a map determine my experimental observations?

Comment author: Perplexed 14 February 2011 04:32:19AM 1 point [-]

That is a great quote from The Simple Truth. And what is more, it is perfectly responsive to what I was trying to say. Thank you.

As you may already know, Eliezer quoted that passage in Quantum Non-realism because QM makes it necessary to modify that argument slightly. The trouble is that in QM, your experimental results are no longer "determined" or at least not in the same sense. Oh, I agree with the basic message of that Quantum Non-realism posting that QM creates no problems for realism that MWI and a little fine print can't fix. But I think that the fact that QM forced a change to the argument does suggest that there may be even more changes needed down the road.

I need a name for whatever-it-is that determines my experimental results, so I call it ‘reality’. This ‘reality’ is somehow separate from even my very best hypotheses.

If you want to call the whatever-it-is 'reality', that is fine with me. The whatever-it-is is definitely different from the best map that you know of. But it is possible, is it not, that the whatever-it-is is the whole tower of maps - including the maps you know of and the maps you don't even imagine yet.

How would a map determine my experimental observations?

A map doesn't determine observations. A whole tower of maps determines observations (modulo the necessary QM/MWI fine print). In much the same way that map-towers determine theoretical predictions. Maps, predictions, and observations are all made out of the same kind of 'stuff'. There is nothing mysterious about it. You only get into trouble if you somehow begin to imagine that experimental observations are somehow built out of some kind of 'reality stuff' which is ontologically different from map-tower stuff. They are not. Observations are very theory-laden.

Logical positivism had all this stuff covered fairly satisfactorily by 1970 or so (IMHO) but then somehow there was a change in the Zeitgeist and everyone agreed that positivism is dead. I am a contrarian who thinks something like it can be revived - along with a number of more academically serious anti-realist philosophers working in philosophy of science.

Comment author: bigjeff5 15 February 2011 06:06:52AM 1 point [-]

Is there any real evidence of this? I hear interesting conjecture but not one bit of evidence.

You know the saying, big claims require big evidence. These are very big claims.

Comment author: nshepperd 15 February 2011 06:55:13AM 2 points [-]

I'm not sure what you mean by this.

How does one make maps into a tower? What would such a tower of maps look like? How is this different from a "territory" containing a tower of maps?

Comment author: lessdazed 07 March 2011 10:41:59AM 0 points [-]

I think the main problem is in "goes around", although HH::the Sun != AA::The Sun and HH::the Earth probably != AA::the Earth, the latter two shouldn't matter as much.

HH believes "HH:(text X)" is true and AA believes "AA::(text Y)" is true, which isn't interesting. What's interesting are three things: that AA::(text X) has meaning, AA::(text X) != AA::(text Y), and if AA::(text X) then NOT AA::(text Y).

Comment author: raptortech97 12 April 2012 11:12:05PM *  -1 points [-]

Have you stopped beating your wife?

I'd just like to point out that there is a definite answer to this. If a person has never started beating his or her wife, then they cannot stop and the answer must be no. Is there a flaw in this reasoning? Or am I not using the common definitions?

Martin told Bob the building was on his left.

Here, too, I see a definite answer. The word "left" is possessed by the word "his." In the English language, the pronoun "his" (and similarly "him," "her," "it," etc.) always refers to the nearest possible preceding sensible noun. In this case, "building" is not a sensible word for "his" to refer to. The next nearest noun is "Bob," which does make sense for "'his" to refer to. Therefore, "his left" must refer to Bob's left. Of course, given context, the interpretation of "sensible" could change. If, say, Bob was giving Martin directions, and Bob just asked Martin to tell Bob what Martin saw (note that pronouns would more typically be used; I used names to allow a more certain answer upon careful reading), then "his left" would refer to Martin's left. Of course, this example is widely more open to interpretation, and I myself am not convinced.

Comment author: thomblake 12 April 2012 11:24:34PM *  0 points [-]

In the English language, the pronoun "his" (and similarly "him," "her," "it," etc.) always refers to the nearest possible preceding sensible noun.

Maybe where your from. In English where I'm at, Jim will use 'him' to refer to Bob if he wants.

Or am I not using the common definitions?

An answer of 'no' to that question would normally be interpreted "I am still beating my wife".

Descriptivism FTW.

Comment author: Desrtopa 13 April 2012 12:21:25AM 3 points [-]

I'd just like to point out that there is a definite answer to this. If a person has never started beating his or her wife, then they cannot stop and the answer must be no. Is there a flaw in this reasoning? Or am I not using the common definitions?

It's technically accurate, but it fails to provide useful information. The question isn't impossible to answer on its own terms, it just turns a simple negative into non-Gricean communication.

Comment author: Taurus_Londono 12 April 2013 05:15:00PM 0 points [-]

"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.

An apt way to put it. That this worthless dimestore philosophy so often underlies contemporary contemplative discourse by relatively intelligent people never ceases to bewilder and sadden me. (see example below)

Comment author: Document 12 April 2013 06:02:25PM 0 points [-]

Below for who?