Joel seemed happily married. He killed his wife.

01 Feb 2012

Which of these two statements is more likely?

Joel seemed happily married. He killed his wife.

Joel seemed happily married. He killed his wife to get her inheritance.

Most people would choose the second but in his book The Black Swan, Nassim Taleb claims that this is a “pure mistake of logic”. His argument is simple enough:

Statement one is a superset of statement two as it includes any reason for Joel killing his wife and must therefore be at least as likely as statement two.

I disagree. When we say ‘how likely is this thing to happen?’, we can mean either: - how frequently does this type of thing happen, or - given some evidence, how does this affect my knowledge of this type of thing happening.

Unless the thing is some indeterminate process found in quantum physics, our reference to ‘likely’ refers to types of events, not particulars: after the fact, the chance of this particular thing happening is 100% because given that it did happen, it did happen; likewise for future events, if some particular in a determinate world will happen, it will happen - its our knowledge that is lacking.

Back to our problem: how often does this particular Joel kill this particular wife? Obviously just once - our ex post knowledge of his motives didn’t change its happening. Which of the two statements were more likely given both happened? They are the same - both are certain.

What Taleb seems to be thinking is this: “What is more likely to be true about the next person you meet……” and from this perspective he is right. But equally, the question could be interpreted as “The next person you meet is called Joel and seems happy: does wanting the inheritance so much he is prepared to kill for it make him more likely to kill his wife?”. If that’s the case, then Taleb is clearly wrong. The difference is:

    p( J & H & K ) > p( J & H & K & I )
    p( K | J & H ) < p( K | J & H & I )


J = There exists a person called Joel
H = There exists a person who is happily married
K = There exists a person who killed his wife
I = There exists a person who wanted his wife’s inheritance enough to kill for it.

To contrast this even more, lets make a few assumptions to simplify things: a persons name has nothing to do with their desire for inheritance or propensity to kill; and killers are just as likely to fake being happy as happy people are to be happy. Then:

    P( K ) > P( K & I ), and
    p( K ) < p( K | I )

The problem for Talib’s account is that killing is highly dependent on the desire to kill. My knowledge of Joel’s desire does increase my ability to predict his killing (or assess the truthfulness of the claim), and does so far more than my knowledge of his perceived happiness or name. Humans do think conditionally, so much so it seems to be a primitive concept (although its pretty clear our conditional thinking is not always coherent).

So this is not an example of irrational human thought processes (unlike the -n-, ing experiment), rather its an example of the vagueness of language (that evidently Taleb conflates).