Let us start from where we ended yesterday, the P -> Q problem. Remember the truth table?
| P | Q | P -> Q |
| true | true | true |
| true | false | false |
| false | true | true |
| false | false | true |
The way to read the truth table is:
If P is true, Q has to be true. It is called direct reasoning. Q false with P true is a violation of the rule, and therefore if Q is false, P has to be false (indirect reasoning). Finally, if P is not true, Q can be true or false.
Re-look at the earlier rain problem – if it rains, I will take an umbrella. The only thing that is not possible is rain and no umbrella. The statement, it doesn’t rain now, and therefore, I should not have an umbrella [1] is not true; I can have an umbrella, whether it rains or not. Also, the statement, I am carrying an umbrella, and therefore it is raining [2] is also wrong.
The above statements numbered 1 and 2 mark two widespread logical errors. [1] is called the fallacy of the inverse. An example is: if it’s a dog, it has a tail. The fallacy is if it is not a dog, it can not have a tail. But what about a cat?
[2] is called the fallacy of the converse. An example is catholic priests are men. He is a man and, therefore, must be a catholic priest.
From the examples so far, it seems the inverse and converse errors are easy to spot and escape; until you reach more complex situations such as the equation of life! Yes, the Bayesian way of interpreting evidence. Take our famous example, suppose the probability of having a test is positive given the person has the disease (sensitivity of the equipment) is 95% or P(+|D). We know from our earlier discussions that this is just one variable in the equation. We need more data to estimate our ultimate quest, i.e. P(D|+) or the probability of having the disease given the test is positive. Yet, most people jump to conclude that the probability of getting the disease is 95%. Let’s re-phrase the equation to the PQ format. If the person has the disease, there is a 95% chance that the instrument will test +ve (P ->Q). But, the public presumes the converse, i.e. the device has tested +ve, and therefore the person has a 95% chance of having the disease, which is utter nonsense for a rare disease (low prior).
