Fallacies

I love fallacies and recently they seem to be cropping up a lot around me. Fallacies come in many types, and each can be fun in its own way. I think the pleasure comes from creating a conclusion that is so intensely convincing and yet, at the same time, is completely illogical. Your brain tosses back and forth between agreeing and disagreeing until some meta-process comes in and stops that loop from continuing.

(as an aside, there is a great story called “The Riddle Of The Universe, And It’s Solution” in Douglas Hofstadter and Daniel Dennett’s anthology ‘The Mind’s I” about what might happen if you could come up with a fallacy so dense that you could not eject)

The simple logical loop is created by a statement such as “This sentance has threee errors”. There are only two obvious errors (the two typos) which means that the sentence is wrong, but the fact that the sentence has miscounted means that there are actually three errors and thus the sentence is correct, however that means that the third error no longer exists and once again the sentence is incorrect. And so on, and so on. Another good example is the statement “I cannot assert this sentence.” If you agree that you cannot assert the sentence then really the sentence is wrong since you can assert it, but that means the sentence is right that you cannot assert it, and so on, and so on.

More fun, is when the concrete field of math comes into play. I remember when Mort Anderson, my AP calculus teacher in high school was frustrated with our class one day. We were distracted and not paying attention. He put the following on the board and then just sat at his desk while we tried to figure out what he had just done. Follow this factorization if you will (you’ll need to recall some algebra for this one):

X = X (take two numbers)

X2 = X2 (square them both)

X2 – X2 = X2 – X2 (subtract the square from both sides)

(X + X) * (X / X) = X * (X / X) (factor each side)

(X + X) = X (divide both sides by (X – X))

now, replace X with 1 and you just proved that 1 + 1 = 1.

Every step in that equation appears to be legal algebraic maneuvers but the result is impossible. The reason is simple. When you divide both sides by (X – X) you’re forgetting that X – X is actually 0 and you cannot divide by 0. Now you know why you can’t divide by 0 – addition ceases to function and the universe will begin to cave in on itself.

Probability is a great area for logical traps. I love the old statistics joke “How do you ensure that a terrorist doesn’t bring a bomb onto your plane? Bring your own bomb! The probability that your plane has two bombs on it is so remote it could never happen!”

That joke relies on an inversion of the base rate fallacy. It assumes that the probabilities are linked when, in fact, they’re not. For example: if you flip a coin, the likelihood of getting heads is 1/2. If you flip a coin twice, the likelihood of getting heads on both flips is 1/2 * 1/2 or 1/4. You can figure out the probability of getting X number of heads in a row by calculating 1/2^X (e.g. X = 4, 2^4 = 2 * 2 * 2 * 2 = 16, therefore getting heads 4 times in a row has a 1/16 likelihood). So getting heads 100 times in a row is astronomically unlikely. However the likelihood that you get heads on the hundredth flip is still 1/2 because that flip doesn’t know that there have been 99 before it. So too does the terrorist not know that there is already a bomb on that plane so the likelihood of his act taking place is not affected by yours.

Recently, my friend Dewitt Clinton posted the following on Google+:

According to this, our galaxy has 100,000,000,000 planets. And there are at least 100,000,000,000 galaxies in the observable universe.

Which averages out to over 10,000,000,000,000,000,000,000 planets in total (all things being equal).

Which means that even if there’s just a .00000000000000000000001% chance of life on any given planet (about 3 quadrillion times less likely than you being eaten by a shark this year), then there’s life on at least one more out there. = )

I realized there was an implication with that probability that I wasn’t entirely comfortable with:

Since I’m 99.9% certain that there is other life in the universe, and it’s 3 quadrillion times more likely that I’ll be eaten by a shark this year then, by the transitive property, I am absolutely certain to be eaten by a shark this year.

Sonofabitch.

The likelihood is so high, in fact, that if I sat in a hut in the middle of the sahara, a shark would spontaneously appear from thin air and eat me. An event that would be deeply unfortunate for me and incomprehensibly surreal for any by-standers.

This is an example of an appeal to probability. In fact, just because something could happen, does not mean it will. If I were living in a hut in the sahara, the likelihood of being eat by a shark would approach 0 (there is still some chance, just not much…).

What are your favorite fallacies?

I’ll leave you with a recent XKCD cartoon that shows the unsustainability of using the word sustainable at current rates. Enjoy: