Variable Question Fallacies
Followup to: Words 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.
Comments (7)
"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.
"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.
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.)
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.
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.
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... =)
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.