An unmanageable inferential mess? Evidence gone wild? Not necessarily.
Maybe U is “Speaks a language,” V is “Two arms and ten digits,” W is “Wears clothes,” X is “Poisonable by hemlock,” and Y is “Red blood.” Now if you encounter a thing-in-the-world, that might be an apple and might be a rock, and you learn that this thing speaks Chinese, you are liable to assess a much higher probability that it wears clothes; and if you learn that the thing is not poisonable by hemlock, you will assess a somewhat lower probability that it has red blood.
Now some of these rules are stronger than others. There is the case of Fred, who is missing a finger due to a volcano accident, and the case of Barney the Baby who doesn’t speak yet, and the case of Irving the IRCBot who emits sentences but has no blood. So if we learn that a certain thing is not wearing clothes, that doesn’t screen off everything that its speech capability can tell us about its blood color. If the thing doesn’t wear clothes but does talk, maybe it’s Nude Nellie.
This makes the case more interesting than, say, five integer variables that are all odd or all even, but otherwise uncorrelated. In that case, knowing any one of the variables would screen off everything that knowing a second variable could tell us about a third variable.
But here, we have dependencies that don’t go away as soon as we learn just one variable, as the case of Nude Nellie shows. So is it an unmanageable inferential inconvenience?
Fear not! For there may be some sixth variable Z, which, if we knew it, really would screen off every pair of variables from each other. There may be some variable Z—even if we have to construct Z rather than observing it directly—such that:
P(U|V,W,X,Y,Z) = P(U|Z)
P(V|U,W,X,Y,Z) = P(V|Z)
P(W|U,V,X,Y,Z) = P(W|Z)
…
Perhaps, given that a thing is “human,” then the probabilities of it speaking, wearing clothes, and having the standard number of fingers, are all independent. Fred may be missing a finger—but he is no more likely to be a nudist than the next person; Nude Nellie never wears clothes, but knowing this doesn’t make it any less likely that she speaks; and Baby Barney doesn’t talk yet, but is not missing any limbs.
This is called the “Naive Bayes” method, because it usually isn’t quite true, but pretending that it’s true can simplify the living daylights out of your calculations. We don’t keep separate track of the influence of clothed-ness on speech capability given finger number. We just use all the information we’ve observed to keep track of the probability that this thingy is a human (or alternatively, something else, like a chimpanzee or robot) and then use our beliefs about the central class to predict anything we haven’t seen yet, like vulnerability to hemlock.
Any observations of U,V, W, X, and Y just act as evidence for the central class variable Z, and then we use the posterior distribution on Z to make any predictions that need making about unobserved variables in U,V, W, X, and Y.
Sound familiar? It should; see Figure 177.1.
Figure 177.1: Network 2
As a matter of fact, if you use the right kind of neural network units, this “neural network” ends up exactly, mathematically equivalent to Naive Bayes. The central unit just needs a logistic threshold—an S-curve response—and the weights of the inputs just need to match the logarithms of the likelihood ratios, et cetera. In fact, it’s a good guess that this is one of the reasons why logistic response often works so well in neural networks—it lets the algorithm sneak in a little Bayesian reasoning while the designers aren’t looking.
Just because someone is presenting you with an algorithm that they call a “neural network” with buzzwords like “scruffy” and “emergent” plastered all over it, disclaiming proudly that they have no idea how the learned network works—well, don’t assume that their little AI algorithm really is Beyond the Realms of Logic. For this paradigm of adhockery, if it works, will turn out to have Bayesian structure; it may even be exactly equivalent to an algorithm of the sort called “Bayesian.”
Even if it doesn’t look Bayesian, on the surface.
And then you just know that the Bayesians are going to start explaining exactly how the algorithm works, what underlying assumptions it reflects, which environmental regularities it exploits, where it works and where it fails, and even attaching understandable meanings to the learned network weights.
Disappointing, isn’t it?
*
178
Words as Mental Paintbrush Handles
Suppose I tell you: “It’s the strangest thing: The lamps in this hotel have triangular lightbulbs.”
You may or may not have visualized it—if you haven’t done it yet, do so now—what, in your mind’s eye, does a “triangular lightbulb” look like?
In your mind’s eye, did the glass have sharp edges, or smooth?
When the phrase “triangular lightbulb” first crossed my mind—no, the hotel doesn’t have them—then as best as my introspection could determine, I first saw a pyramidal lightbulb with sharp edges, then (almost immediately) the edges were smoothed, and then my mind generated a loop of flourescent bulb in the shape of a smooth triangle as an alternative.
As far as I can tell, no deliberative/verbal thoughts were involved—just wordless reflex flinch away from the imaginary mental vision of sharp glass, which design problem was solved before I could even think in words.
Believe it or not, for some decades, there was a serious debate about whether people really had mental images in their mind—an actual picture of a chair somewhere—or if people just naively thought they had mental images (having been misled by “introspection,” a very bad forbidden activity), while actually just having a little “chair” label, like a LISP token, active in their brain.
I am trying hard not to say anything like “How spectacularly silly,” because there is always the hindsight effect to consider, but: how spectacularly silly.
This academic paradigm, I think, was mostly a deranged legacy of behaviorism, which denied the existence of thoughts in humans, and sought to explain all human phenomena as “reflex,” including speech. Behaviorism probably deserves its own write at some point, as it was a perversion of rationalism; but this is not that write.
“You call it ‘silly,’” you inquire, “but how do you know that your brain represents visual images? Is it merely that you can close your eyes and see them?”
This question used to be harder to answer, back in the day of the controversy. If you wanted to prove the existence of mental imagery “scientifically,” rather than just by introspection, you had to infer the existence of mental imagery from experiments like this: Show subjects two objects and ask them if one can be rotated into correspondence with the other. The response time is linearly proportional to the angle of rotation required. This is easy to explain if you are actually visualizing the image and continuously rotating it at a constant speed, but hard to explain if you are just checking propositional features of the image.
Today we can actually neuroimage the little pictures in the visual cortex. So, yes, your brain really does represent a detailed image of what it sees or imagines. See Stephen Kosslyn’s Image and Brain: The Resolution of the Imagery Debate.1
Part of the reason people get in trouble with words, is that they do not realize how much complexity lurks behind words.
Can you visualize a “green dog”? Can you visualize a “cheese apple”?
“Apple” isn’t just a sequence of two syllables or five letters. That’s a shadow. That’s the tip of the tiger’s tail.
Words, or rather the concepts behind them, are paintbrushes—you can use them to draw images in your own mind. Literally draw, if you employ concepts to make a picture in your visual cortex. And by the use of shared labels, you can reach into someone else’s mind, and grasp their paintbrushes to draw pictures in their minds—sketch a little green dog in their visual cortex.
But don’t think that, because you send syllables through the air, or letters through t
he Internet, it is the syllables or the letters that draw pictures in the visual cortex. That takes some complex instructions that wouldn’t fit in the sequence of letters. “Apple” is 5 bytes, and drawing a picture of an apple from scratch would take more data than that.
“Apple” is merely the tag attached to the true and wordless apple concept, which can paint a picture in your visual cortex, or collide with “cheese,” or recognize an apple when you see one, or taste its archetype in apple pie, maybe even send out the motor behavior for eating an apple . . .
And it’s not as simple as just calling up a picture from memory. Or how would you be able to visualize combinations like a “triangular lightbulb”—imposing triangleness on lightbulbs, keeping the essence of both, even if you’ve never seen such a thing in your life?
Don’t make the mistake the behaviorists made. There’s far more to speech than sound in air. The labels are just pointers—“look in memory area 1387540.” Sooner or later, when you’re handed a pointer, it comes time to dereference it, and actually look in memory area 1387540.
What does a word point to?
*
1. Stephen M. Kosslyn, Image and Brain: The Resolution of the Imagery Debate (Cambridge, MA: MIT Press, 1994).
179
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.
*
180
37 Ways That Words Can Be Wrong
Some reader is bound to declare that a better title for this essay would be “37 Ways That You Can Use Words Unwisely,” or “37 Ways That Suboptimal Use Of Categories Can Have Negative Side Effects On Your Cognition.”
But one of the primary lessons of this gigantic list is that saying “There’s no way my choice of X can be ‘wrong’” is nearly always an error in practice, whatever the theory. You can always be wrong. Even when it’s theoretically impossible to be wrong, you can still be wrong. There is never a Get Out of Jail Free card for anything you do. That’s life.
Besides, I can define the word “wrong” to mean anything I like—it’s not like a word can be wrong.
Personally, I think it quite justified to use the word “wrong” when:
A word fails to connect to reality in the first place. Is Socrates a framster? Yes or no? (The Parable of the Dagger)
Your argument, if it worked, could coerce reality to go a different way by choosing a different word definition. Socrates is a human, and humans, by definition, are mortal. So if you defined humans to not be mortal, would Socrates live forever? (The Parable of Hemlock)
You try to establish any sort of empirical proposition as being true “by definition.” Socrates is a human, and humans, by definition, are mortal. So is it a logical truth if we empirically predict that Socrates should keel over if he drinks hemlock? It seems like there are logically possible, non-self-contradictory worlds where Socrates doesn’t keel over—where he’s immu
ne to hemlock by a quirk of biochemistry, say. Logical truths are true in all possible worlds, and so never tell you which possible world you live in—and anything you can establish “by definition” is a logical truth. (The Parable of Hemlock)
You unconsciously slap the conventional label on something, without actually using the verbal definition you just gave. You know perfectly well that Bob is “human,” even though, by your definition, you can never call Bob “human” without first observing him to be mortal. (The Parable of Hemlock)
The act of labeling something with a word disguises a challengable inductive inference you are making. If the last 11 egg-shaped objects drawn have been blue, and the last 8 cubes drawn have been red, it is a matter of induction to say this rule will hold in the future. But if you call the blue eggs “bleggs” and the red cubes “rubes,” you may reach into the barrel, feel an egg shape, and think “Oh, a blegg.” (Words as Hidden Inferences)
You try to define a word using words, in turn defined with ever-more-abstract words, without being able to point to an example. “What is red?” “Red is a color.” “What’s a color?” “It’s a property of a thing.” “What’s a thing? What’s a property?” It never occurs to you to point to a stop sign and an apple. (Extensions and Intensions)
The extension doesn’t match the intension. We aren’t consciously aware of our identification of a red light in the sky as “Mars,” which will probably happen regardless of your attempt to define “Mars” as “The God of War.” (Extensions and Intensions)
Your verbal definition doesn’t capture more than a tiny fraction of the category’s shared characteristics, but you try to reason as if it does. When the philosophers of Plato’s Academy claimed that the best definition of a human was a “featherless biped,” Diogenes the Cynic is said to have exhibited a plucked chicken and declared “Here is Plato’s Man.” The Platonists promptly changed their definition to “a featherless biped with broad nails.” (Similarity Clusters)
Rationality- From AI to Zombies Page 70
