Dr. Paranoia's Problem

Dr. Paranoia had a problem. His clients arrived in his consulting room with torn and stained clothing, long bloody scratches and fang marks, and wide, staring eyes. This made it hard to examine them. Dr. Paranoia decided to start by checking his waiting room. It contained a couch, a coffee table with some 23rd century magazines, a live tiger, and a large potted plant. Nothing unusual there. After giving the potted plant a searching look, the doctor was stumped. The doctor decided to call in the Mad Wizard, who will use causal reasoning to investigate the problem. Causal reasoning starts with some particular event (called the "effect") and attempts to establish which other event or condition, (called the "cause") made the effect happen. Causes are important because, by changing them, we can change future events. Such as skittish patients who bleed on your examining room carpet.
"Well," said the Mad Wizard, "the problem is obviously caused by your waiting room. Two thousand dollars please."
"Wait. How'd you figure that?"
"I used John Stuart Mill's method of agreement, which means looking for a factor that's present in all the cases where the effect is present. (Such a factor is called a correlate) I asked all your patients what else they did before coming to you. Alaric Aggis went to the Hypermall, Cafe Spittoon, and the Annoying Gift Shoppe. Bilbo Baggins went to the Hypermall, Cafe Spittoon and Unpleasant Cards. Campy Cyses went to the Hypermall, Noodles Up Your Nose and the Annoying Gift Shoppe. Finally, Dez Distillate went to Noodles Up Your Nose, Unpleasant Cards and the Annoying Gift Shoppe. At first I thought it must be the hypermall, but at the last minute I remembered that they all sat in your waiting room before being examined. Now look at this chart I have tattooed on my left kneecap."

 Hypermall Cafe Spittoon Annoying Gifts Unpleas. Cards Noodles / Nose waiting room Alaric Aggis * * * * Bilbo Baggins * * * * Campy Cyses * * * * Dez Distillate * * * *
"As you can see," said the wizard importantly, "the waiting room is the only factor that correlates with the problem."
"Well, that looks pretty convincing," Dr. Paranoia nodded, "but how do you know it wasn't a combination of things? Maybe it was a combination of things, or maybe there's two things that cause it?"
"How'd you figure?" Asked the Wizard.
"Well, maybe both Cafe Spittoon and Noodles Up Your Nose will cause the effect." Dr. Paranoia snapped his fingers. "Or some other combination. Say my waiting room had nothing to do with it, but both Unpleasant Cards and the Annoying Gift Shoppe caused torn clothing and serious skin punctures, then we'd get the same picture, wouldn't we?"
"I suppose." The wizard thought for a while. Green smoke drifted from his ears. "Okay, let's use Mill's method of difference. Call your next patient and have him... "
"Her."
"... have her come in through the window instead of your waiting room."
And that's what they did. Of Dr. Paranoia's next two patients, Ezy Edna visited the Hypermall, Cafe Spittoon, Unpleasant Cards, Noodles Up Your Nose, the Annoying Gift Shoppe and climbed the seven floors to come in through the doctor's window. ("You could have opened it!") Fdor Fiff also visited the Hypermall, Cafe Spittoon, Unpleasant Cards, Noodles Up Your Nose, the Annoying Gift Shoppe but came in through the doctor's waiting room. Ezy Edna was fine, but Fdor Fiff was all torn up. Here's the Wizard's right kneecap.

 Hypermall Cafe Spittoon Annoyin' Gifts Unpls. Cards Noodles/Nose waiting room effect Ezy Edna * * * * * Fdor Fiff * * * * * * *

"According to Mill's method of difference, it has to be the waiting room, because that factor is present when the effect is present and absent when the effect is absent. Thus the waiting room correlates with the effects, and the other factors don't correlate. Therefore that factor is the cause. Two thousand dollars please."
"But what if it's an intermittent effect?" asked the doctor. "Maybe it's the hypermall and the noodle shop, but each only does it sometimes. Then the correlation between my waiting room and the effect would just be a coincidence."
"That's what makes this an inductive argument. You can never be absolutely sure."
"Still, I'd be happier if we could do more to rule out posssibilities like that."
"Okay, you could try the joint method of agreement and difference."
"How do you do that?"
"It's easy. Just get a lot of data and apply both methods at once." The wizard giggled. "Here, have a look at this problem I'm also working on."

"Okay, I'll make a little chart," said the doctor. "Darn, how do you manage to tattoo on your knees so well?"
"Practice."
"Well, I'm having trouble."
"Okay then, hold your knee up to the screen here and have the reader tattoo in the information."
Here's the knee. Get tattooing.

 Stinkball Snotweed Slimeflower Toadrecliner Bugwort Frogcelery Porkbush effect Groo _________ _________ _________ _________ _________ _________ _________ _________ Hubpf _________ _________ _________ _________ _________ _________ _________ _________ Idjit _________ _________ _________ _________ _________ _________ _________ _________ Jerp _________ _________ _________ _________ _________ _________ _________ _________ Klom _________ _________ _________ _________ _________ _________ _________ _________ Lugg _________ _________ _________ _________ _________ _________ _________ _________ Mrnnk _________ _________ _________ _________ _________ _________ _________ _________ Nah-Nah _________ _________ _________ _________ _________ _________ _________ _________
A factor is positively correllated  with the effect if it is always present when the effect is present.  A factor is negatively correllated with the effect if it is always absent when the effect is absent. A factor that is neither positively nor negatively correlated with the effect is independent. A factor that is both positively and negatively correlated with the effect is fully correlated. Here is a correlation chart that the Mad Wizard filled in on his own earlobe.
 positively correllated  with the effect negatively correllated with the effect independent of the effect fully correllated with the effect Stinkball no no yes no Snotweed no yes no no Slimeflower no no yes no Toadrecliner yes yes no yes Bugwort yes no no no Frogcelery no yes no no Porkbush yes no no no
"Okay," said the doctor, "let me take it from here. Stinkball and Slimeflower are independent of the effect, so we can eliminate them."
"Well, we can say they're very unlikely to be the cause," put in the wizard.
"But each of them is absent at some time when the effect is present, and each is present at some time when the effect is absent, so it looks like they have nothing to do with it."
"Probably."
"Yeah, yeah. Anyway Bugwort and Porkbush are only positively correlated, which means each of them is present whenever the effect is present, but each is still present at some time when the effect is absent, so we can eliminate them too."
"We can say they're unlikely to be the cause."
"Sure, and the same thing for Snotweed and Frogcelery because they're only negatively correlated with the effect, which means that they're always absent when the effect is absent, but absent at other times too."
"Fine, Snotweed and Frogcelery are unlikely too," added the wizard.
"Which leaves Toadrecliner, which is the only factor fully correlated with the effect. So it's the cause."
"So it's likely to be the cause." replied the wizard.
"What else could it be?" the doctor asked.
"Well, it could be something else they had in common, or a combination of things, or something else we don't know about. The thing to do is make up a dose of herbal viagra that only contains Toadrecliner, and sell it to the next CEO that comes in. If he turns into a badger, we'd be pretty sure it was the Toadrecliner." The wizard glowed thoughtfully. "These methods would only give us certainty if we were certain that these were the only factors involved, certain that it wasn't an intermittent effect, certain that there's only one cause, and so on. We can't know things like that for sure, so we can never be absolutely certain about a cause."
"What if we eliminated every other factor?"
"Even so!" Purple smoke puffed from the wizard's nostrils. "Look at the herbal viagra chart. For instance, there's such a thing as multiple causes. What if Snotweed and Frogcelery both caused badgerism? That would account for the chart, wouldn't it?"
"Erm, but what about Toadrecliner being fully correlated?"
"Coincidence. That's why we have to test Toadrecliner without Snotweed and Frogcelery."
"Could be, I suppose..."
"What if Stinkball and Slimeflower were such that each will cause badgerism except when the other is present." The wizard's teeth shone with an uncanny blue fire as he interrupted." Stinkball alone - badger! Slimeflower alone - badger! Stinkball and Slimeflower together - no badger! If that was true, we'd get the same chart, wouldn't we?"
"Errr..."
"We would. Trust me!"
"I'll look at the chart later."
"Or there could be an intermittent cause." The wizard's ears expanded in excitement. "Think about Bugwort and Porkbush. If we assume that one of these sometimes causes badgerism, then it could be any one of them!
"AAAAAAAAAGH!"
"Oh sorry. I don't know how that dragon got out of my pocket."
After they crammed the dragon back into the wizard's pocket, the two friends went to a local bar to get drunk. After several shots of Sheep Dip, Dr. Paranoia leaned over to the Mad Wizard and said, "Can we talk about my problem?"
"Well, the causal factors in your case are the air in the waiting room, the waiting room floor, the walls, your patients, your practice and your patients's lack of suits of armor. Remove one of these and your problem's solved."
"I don't want to remove any of those. Isn't there another way?"
"Well, maybe it's not just the waiting room that's causing the problem."
"What do you mean?"
"What if it was something in the waiting room that was affecting your patients?"
"You mean, like the potted palm?"
"Or the couch."
"But how will Mill's methods help us here? Nothing in the room has a known rate for causing this problem, and everyone who comes in the room is exposed to all the objects."
"Can't you remove the different objects at different times, so that different patients are exposed to different combinations?"
"No." Dr. Paranoia shook his head. "The Feng Shui in that room is linked to the fashion industry. If I change anything major, pedal-pushers will come back into style."
"Ouch!"
"So what do we do?"
"We could partition the room so that your patients are exposed to each object seperately."
"How would that help?"
"If each patient was given a different exposure to each factor, then we could use Mill's method of concomitant variation."
"How's that work?"
"The factor that most closely varies with the effect is most likely to be the cause. If the effect is greater when more of one factor is present, and less when less of that factor is present, then that factor is our best bet."
"Let's try it."
That weekend, they partitioned the waiting room into cubicles with time locks so that each patient was exposed to a different object in the waiting room for a different time period than the other patients. The next week, Pot Pzebie was exposed to the couch for 6 minutes, the coffee table for 9 minutes, the tiger for 2 minutes and the potted plant for 5 minutes and turned up with 3 scratches. Quo Quardis was exposed to the couch for 2 minutes, the coffee table for 3 minutes, the tiger for 7 minutes and the potted plant for 3 minutes and had 1 scratch. Rwert Rod was exposed to the couch for 8 minutes, the coffee table for 6 minutes, the tiger for 1 minute and the potted plant for 2 minutes and had 2 scratches. Finally, Sto Sart was exposed to the couch for 4 minutes, the coffee table for 12 minutes, the tiger for 3 minutes and the potted plant for 4 minutes and had 4 scratches.Here's a chart.
 couch coffee table tiger potted plant scratches Pot Pzebie 6 minutes 9 minutes 2 minutes 5 minutes 3 scratches Quo Quardis 2 minutes 3 minutes 7 minutes 3 minutes 1 scratches Rwert Rod 8 minutes 6 minutes 1 minute 2 minutes 2 scratches Sto Sart 4 minutes 12 minutes 3 minutes 4 minutes 4 scratches
Now, here's a bar graph. Couch is red, coffee table blue, tiger orange and potted plant green.
Pot Pzebie   Quo Quardis   Rwert Rod   Sto Sart
 . CT CT CT C CT C CT P C CT P C CT P S C CT T P S C CT T P S

 . T T T T CT T P C CT T P C CT T P S

 .. C C C CT C CT C CT C CT C CT P S C CT T P S

 CT CT CT CT CT CT CT CT C CT P S C CT T P S C CT T P S C CT T P S
"Look," cried Dr. Paranoia. "It's got to be the potted plant. Rwert Rod spent two minutes with the plant and got two scratches and Sto Sart spent four minutes with the plant and got four scratches!"
"But Pot Pzebie spent five minutes with the plant and got three scratches, and Quo Quardis spent three minutes with the plant and only got one scratch. So the injuries didn't vary with exposure to the plant." The wizard's head rotated thoughtfully. "No, what we need is a factor that varies in constant ratio to the scratches!"
"Well, let's look at the ratios then. Let's make a graph that says exactly how many minutes of exposure to each object correspond to just one scratch for that particular person. That will put everything on a common scale and let us do one-to-one comparisons."
"Okay, here that graph."
Pot Pzebie Quo Quardis Rwert Rod Sto Sart
 couch coffee table tiger potted plant 6/3 = 2/1 9/3 = 3/1 2/3 = 0.6/1 5/3 = 1.6/1
 couch coffee table tiger potted plant 2/1 = 2/1 3/1 = 3/1 7/1 = 7/1 3/1 = 3/1
 couch coffee table tiger potted plant 8/2 = 4/1 6/2 = 3/1 1/2 = 0.5/1 2/2 = 1/1
 couch coffee table tiger potted plant 4/4 = 1/1 12/4 = 3/1 3/4 =.75/1 4/4 = 1/1
"Now," the wizard said with blazing elbows, " which factor varies with the effect?"
"Huh?"
"Which factor exists in the same ratio to the effect in every case?"
"THE COFFEE TABLE!" Dr. Paranoia leaped to his feet. "We must kill it immediately! Here's your two thousand dollars."
Carrying two Bergmann submachineguns which the wizard happened to have under his cloak, they rushed into the waiting room and fired 46 nine-millimetre rounds into the coffee table, killing it dead and riddling the 23rd century magazines."
"Do you mind?" said the tiger. "I was reading those magazines."
"Well, the coffee table was attacking people," Dr. Paranoia apologized. "I just paid the Mad Wizard two thousand dollars to find that out."
"I could have told you that," the tiger said in an aggrieved voice.
"I'm sorry, but 'ask the tiger' isn't one of Mill's methods." replied the wizard.
After solving the mystery of the mauled patients, the two friends went to a local bar to get drunk. After several shots of Sheep Dip, Dr. Paranoia leaned over to the Mad Wizard and said, "what if you've got both intermittent and multiple causes?"
"Well... "
"I mean, think about all the things one does while driving. Like drinking beverages, fiddling with the radio and elective surgery..."
"And magic tricks," put in the wizard helpfully
"... and magic tricks... "
"And knitting,"
"... and knitting." The doctor waved his drink to emphasize the points. "Each of those activities seems perfectly safe, doesn't it?"
"Uh-huh."
"And yet, each of them is responsible for a certain number of accidents every year!"
"Really?"
"Really!"
"Well, I'd better be more careful with my beverages!"
"So how do Mill's methods deal with things like that?"
"Well, there is the method of residuals. Say we wanted to determine the number of car accidents caused last year by UFOs trying to abduct people while they were driving."
"While the people were driving."
"Exactly. Say that last year there were 275 accidents per million miles driven by American drivers. And say that we know that drinking beverages accounts for 47 accidents per million miles, fiddling with the radio causes 61 accidents per million miles, elective surgery gives us 8 accidents per million miles, magic tricks 19 accidents per million miles, and knitting 12 accidents per million miles driven by American drivers. If these were the only other possible causes of accidents, then we'd be able to calculate the number of accidents caused by ill-timed attempts at alien abduction by subtracting the number of accidents from those causes from the total number of accidents. Like this."
 drinking beverages 47 fiddling with the radio 61 elective surgery 8 magic tricks 19 knitting 12 Total non-abduction accidents 147 Total accidents 275 Minus non-abduction accidents 147 Total abduction related accidents 128
So if the causes listed here are the only ones operating, and the figures for non-abduction accidents are right, then it follows that alien attempts to abduct the drivers of moving vehicles account for 128 accidents per million miles driven by American drivers."
"Wow." said the doctor. "Does Pep Boys sell alien repellant?" "Now," said the wizard, "about my two thousand dollars."
"Wait, wait," replied Paranoia. "You've just identified the cause - my waiting room - but you haven't told me how to fix the problem."
"Actually, there's several factors involved. Removing any one of them will solve the problem."
"Explain."
"Well, there's the air in the waiting room. Probably the thing that's attacking your patients couldn't live without air, so if you remove the air, that will solve the problem. There's also the waiting room floor, which holds up the attacker. Remove that, and there will be no attacks. And the walls prevent your patients from escaping, so removing them would also solve the problem. And there's your practice. If you didn't see patients, there would be no problem...."
"Yes, yes, but..."
"...and your patients have such fragile skin, so if they would just wear suits of armor, that would take care of it too."
"but those things aren't the cause of the problem."
"Why not?"
"But they don't make the effect happen!"
"Sure they do. If any one of them was absent, the effect wouldn't happen!"
"But, but... "
"Think about this. Say in room number one you have a pile of really dry kindling, or really dry leaves, or crumpled paper, that's resting on a really hot surface, say 500 degrees Fahrenheit in an airtight room that has no atmosphere whatsoever. What's going to happen?"
"A fire?"
"Without oxygen?"
"Yeah. Well, I guess the kindling's just going to get really hot."
"But no fire?"
"No fire."
"Room number two, just the same, full of air, but no kindling?"
"Again, no fire."
"Room three, full of air, with kindling, but cold floor?"
"No fire."
"Room four, full of air, kindling, hot floor, but the kindling is soaking wet?"
"Umm, no fire?"
"Now say we take room one, with hot floor, dry kindling, but no atmosphere, and add air. What do we get?"
"Fire."
"Room two, hot floor, full of air. Now add dry kindling?"
"Fire."
"Room three, dry kindling, full of air. Now make the floor hot?"
"Fire."
"Room four, wet kindling, full of air, hot floor. Now dry out the kindling?"
"Fire."
"So, what caused the fire in room one?"
"What caused the fire in room two?"
"What caused the fire in room two?"
"Making the floor hot."
"What caused the fire in room two?"
"Drying the kindling."
"But the rooms were identical when the fires started!" The wizard levitated in his excitement. "How can four identical fires in four identical rooms have four different causes?"
"Um, I dunno."
"It's because "cause" is a social concept as well as a physical concept. Phyisically the situations are identical. Physically, each of the fires was caused by the same combination of material conditions. Socially, the situations were very different. Each one started with a fairly safe condition which was made dangerous by adding a previously missing causal factor. From a social point of view, each fire was caused by the action of 'adding the missing causal factor.' Since the factor added was different in each case, we say the cause was different in each case."
"Ooooooo," replied Dr. Paranoia.