Why Fallacies are False 003 -- Meet Linda, the Conjunction Fallacy

I love Linda. You will, too, if you pay attention, because Linda is why conspiracy theories are fallacies.

You may have heard of the Linda problem.

You create a dataset describing Linda. You can say anything you want about her. It doesn’t even have to be true.

Then you try to decide which of two statements is more likely to be true:

1.     Linda is X.

2.     Linda is X and Y.

There is no relationship between X and your dataset about Linda. There is also no relationship between Y and your dataset, or between Y and X. Based on the dataset, you have no idea about how likely it is that X is true about Linda, or Y is true about Linda.

Linda’s formal name is the Conjunction Fallacy. “X and Y” is a conjunction. That will ring a bell if you know anything about Boolean algebra, which you should if you have a degree in computers. The rest of us not so much. I met Boole decades ago when I got into a computer internship program where I worked, because my old boss declared my skill area non-essential so I couldn’t get advancement or CEUs any more. (Eleven people jumped his ship and he had a shreck and tried to stop them, and his bosses said they’re non-essential now, so you can’t do that. It’s called hoist with your own petard.)

Anyway, for one thing, Boolean algebra tells you the results of combining two bits of data, where each bit can have one of two values. I’m going to call the possible values zero and one to make the relationship to math clear.

Boolean algebra has operations, the two most important of which right now are conjunction and disjunction. Conjunction means that our two bits, A and B, have to both be 1 for the answer to be one. Disjunction means that A and B have to both be zero for the answer to be zero.

In words, a conjunction is AND while a disjunction is OR. Yes, I know it’s more complicated than that, but people can go get into the other details if they want. For now, look at statement 2 above. It’s a conjunction, right? So what’s the answer?

We can’t tell.

Since neither X nor Y has any relationship to Linda’s dataset, we don’t know their values.

What we do know is that, since there is no proof for or against their truth, the probability that they are true can’t be either 100% or 0%. It’s between 100% and 0%. And if you know percentages, you know that 100% is 1.0. If the answer has to be between 1 and 0, it’s a fraction less than 1. You can express it as 0.50, 0.167, whatever. Its value is not the issue; the issue is the fact that it’s a fraction less than one.

What’s the algebra for a conjunction?

A	B	A OR B		A	B	A AND B
1	1	1		1	1	1
1	0	1		1	0	0
0	1	1		0	1	0
0	0	0		0	0	0

 

If you changed the words to arithmetic operators, you would have to use a plus sign for the OR and a multiplication sign for the AND. (You can’t go above a probability of 1 and that’s why adding 1 and 1 is still 1.)

Do you remember your grade school arithmetic? If X is 0.50 and Y is also 0.50, what’s the answer for a conjunction?

So for statement one, the answer is X or 0.50. But for statement 2, the answer is the product of X and Y, which is 0.25. It’s smaller.

Whichever of X or Y is smaller, the product will be smaller than that. 0.25 AND 0.50 is 0.125.

So statement 2 cannot be more probable.

When you meet up with a conspiracy theory, break it down.

·       Identify the dataset. If it contains false data, you’re done. The theory is false.

·       If the dataset is true, examine the statements about it. Do they have a natural organic relationship to the dataset such that it supports their truth or undercuts it? If the latter, the probability for statement 1 is higher, but it still may not reach 100%.

·       If any of the statements are conjunctions, then if the dataset does not support them or there’s no relationship to the dataset, those statements are LESS LIKELY TO BE TRUE.

You’re not trying to prove that they’re not true, which would be a value of zero. You’re proving that they are not likely to be true, with a value below 100%, below 50%, below 25%, or worse. The least likely term of the conjunction, is the top limit.

The problem with the Conjunction Fallacy is that most people will pick the conjunction as more likely. You basically have to give them a quick course in probability math to show them why that’s not true. And then, when they meet another conjunction fallacy, they are STILL likely to pick the wrong answer. There could be a number of reasons: normal forgetfulness; not realizing that a new problem could be a conjunction fallacy; or it says something they want to believe and they throw logic out the window. They STILL pick the wrong statement.