I use the conjunction fallacy for my first illustration.
Seems to me that all that would do is reinforce someone's opinion that probability theory is irrelevant to the real world.
I personally would start with confirmation bias, partly because there are lots of clear examples in pop culture. Like: last night I was watching a rerun of "Glee." Will Schuester, a teacher and the glee-club advisor, is trying to quash a student's crush. He sings her (Rachel) a medley of songs in which the singer is trying to deflect a much younger woman's advances. (Actually, both songs -- "Don't Stand So Close to Me" and "Young Girl" -- are actually about the singer unsuccessfully trying to resist the temptation of the younger woman, but in the episode the lyrics are changed and edited so that they ostensibly work.) So he sings, and the whole time Rachel is clearly hearing the opposite of the intended message. After the song, Will asks Rachel what his message was, and she says, almost giddily, that his message was clear: "I'm very young and it's hard for you to stand close to me."
I thank the Ravenclaw Harry Potter for bringing me here. I've been lurking for a couple of weeks. My first clue that I'd feel at home here was learning that Eliezer taught himself physics by reading the Feynman lectures.
I'm an evolutionary ecologist by training, and a self-taught Python programmer and GIS analyst. I currently work at a community college, where I do a lot of one-on-one biology-teaching. I spend a lot of time thinking about where students go wrong when they're thinking about science, and how to help them think more about their own thinking. (In my department we call it metacognition.) I'm also the father of a four-year-old, and so I also spend a good part of my home-life confronting and responding to some pretty fascinating cognitive and philosophical puzzles. (Her latest interest: the origins and arbitrariness of names.)
I've been developing as a rationalist (without the label) since who-knows-when during childhood, but I trace my more careful, articulated thinking about my own thinking to my early grad-school days, when I spent a lot of time fretting over how scientists should think about nature and problem-solving.
I'm looking forward to learning some new cognitive habits (my current thing is to think of -- and teach -- many cognitive skills as habits) and reinforcing some that I already have.
I believe that in the first example, "A" is supposed to be right. In the second example, is "A" or "B" supposed to be right? B is doing the math, but assumes that fuel required is proportional to mass, which is wrong, due at least to engine size and air resistance. (Consider the (mass x miles)/gallon of a 2005 RST1000 Futura motorcycle (565 x 42 = 23730), a Smart car (1808lb x 40mpg = 72320), a 2010 Honda Civic DX-VP (2709 x 36 = 97524), and a 2010 Toyota Camry SE (3329 x 33 = 109857). All MPG are EPA highway estimates.)
By default, I expect your examples to take the same form (e.g., the counterargument is right in both cases, or wrong in both cases). Deviations from that pattern should be pointed out. Cases where doing math does not qualify as "doing the math" due to incompleteness should be pointed out.
(BTW, reminds me that Brad Templeton, founder of rec.humor.funny and the Oracle, gave a talk at the 2009 Singularity Summit in which he showed data claiming that mass transit typically has the same fuel efficiency as a car with 1.5-3 people in it. Because a mass transit trip (at least the ones I take) usually require you to travel a longer distance than you would by car, mass transit loses to one person in a fuel-efficient car for fuel efficiency. And the cost of mass transit is much higher per person-mile; and the time taken is about double (in the DC metro area). These facts combined suggest that mass transit is neutral or bad for the environment, bad for the passenger, and bad for the economy.)
I'd meant A to be right in both cases. And of course -- against my own remonstration -- I did none of the math myself. I was unfamiliar with the Templeton data. I looked it up, and it's interesting. I'd note that while Templeton agrees that transit (by the system, not by the fully utilized vehicle) is less efficient than fuel-efficient personal transportation, he still thinks people should make use of existing transit systems.
I ride a bike.
Here's something that comes up in many, many discussions of climate change and anything else where a lot of arguments come from models or simulations: sometimes you have to do the math to make a valid (counter-)argument.
Example:
A: ...And so you, see, as CO2 increases, the mean global temperature will also increase.
B: That's bullshit, and here's why: as CO2 increases, there will be more photosynthesis -- and the increased plant growth will consume all that extra CO2.
Another example (the one that motivated this comment):
A: And so, as long as the bus is carrying six or more passengers, it'll be more efficient than the passenger-equivalent number of cars.
B: That's bullshit! Buses are ten times heavier than cars, so it's got to be ten or more bus passengers.
People often think that in discussions of quantitative phenomena, it's enough to make arguments based purely on directional drivers/phenomena, when really the magnitudes of those drivers are hugely important. Of course there are negative feedbacks, countervailing forces, etc., but (a) usually they're already dealt with in the original model and so B isn't telling anyone anything new, and (b) magnitude matters.
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Another frequent sin - not necessarily an inherent one - of the enterprise is to explain phenomena by reference to a virtus dormitiva - to say "here is an observed fact x about humans/undergraduate psychology majors, here is a plausible story for why evolution would select for x," and thus conclude that x occurs because of an x instinct or x module. This is equivalent to saying that we observe x, here is a plausible story for why capitalism/patriarchy/society at large stands to benefit from x, so capitalism &c. creates a "social process" to realize x. That's not an explanation; it's not even a hypothesis. You have to talk about mechanisms.
Again, this isn't inherent in evolutionary psychology, but I think until our understanding of behavioral genetics becomes much better it will be a more common vice than its equivalent in social explanations. If we found ways to genetically reconstruct our ancestors at various points in our specific development that would go a long way as well.
Mechanisms -- good point. I have another way of putting it.
In TV police shows, detectives try to establish a suspect's means, motive, and opportunity for committing a crime.Sloppy detectives (and jurors!) tend to focus just on motive and its lurid details. In evolutionary bio/psych, we try to establish genetic variation, selective advantage, and heritability. Sloppy evolutionary hypotheses tend to focus just on the selective advantage and ignore the other two.
People are drawn to stories. We give evidence extra weight if it comes in the form of a good narrative. (Is there a name for that bias?)