So I already have two posts that go over the notion that logical fallacies aren’t necessarily fallacies of probability. The issue with logical fallacies is that in deduction, the conclusion has to follow necessarily from the premises. But we don’t live in a world of deductive certainty; we live in a world of uncertainty: the world of probability.
The thing about post hoc ergo propter hoc is that it is an inductive inference. That being the case, post hoc fallacies should be easily explained using probability theory, thus Bayes’. Thinking about this fallacy intuitively (that is, using quick Bayesian format), it seems that this fallacy is an instance of the Base Rate fallacy. Of course, given that some cause is the reason for some effect, the cause has to come before the effect (unless you live in the world of quantum physics, which none of us do).
This means that the conditional probability, or success rate, of a post hoc argument would necessarily be 1.00, or P(B Happened After X | B Caused By X) = 1.00. But the argument itself is trying to prove P(B Caused By X | B Happened After X); the cause is the hypothesis and what happens is the evidence. Sure, given that god answers prayer there’s a 100% chance you would get a job after praying for it. But that’s a Base Rate fallacy; we are not trying to establish P(Get A Job After Praying To God | God Answers Prayers) but P(God Answers Prayers).
One hundred percent of all effects (in the macro world) are preceded by their causes. Concluding that because the conditional probability is 100% that it means that it is actually the reason is, like I said, a Base Rate fallacy, because we aren’t taking into account the prior probability.
But there’s a second factor that has to be taken into account: The alternative hypothesis. Or, what about an effect that just happens after the “cause” by chance or some other cause? In other words the false positive rate? This, surely, must also be a high number but it doesn’t necessitate 100% certainty like the success rate that denominates post hoc logic. Given this, it seems that the Likelihood Ratio, or dividing the success rate by the false positive rate, returns a very very low quotient. If the success rate is 100%, and the false positive rate is 98%, then this is only a Bayes’ Factor of 1.02 decibles. This means that if we had a 50/50 spread between the hypothesis and the alternative, the post hoc ergo propter hoc logic in this example would only increase our probability to 50.5%.
If we go back to my original example P(Get A Job After Praying To God | God Answers Prayers), we would have to include the alternative hypothesis. There are various alternatives, but let’s just go with P(Get A Job After Praying | Economy Improves). Of course, there’s not a 100% chance that you would get a job when the economy improves, but an improving economy in and of itself has a much higher prior probability than the existence of god. Therefore, in this case, the prior probability of P(God Answers Prayers) doesn’t get much of a boost due to the small difference between P(Get A Job After Praying To God | God Answers Prayers) and P(Get A Job After Praying | Economy Improves).
So post hoc ergo propter hoc is weak (possibly very weak) probabilistic evidence. It’s not strong enough evidence to rest an entire argument on; you would need much more evidence. Or you would need an argument or situation where there is a huge disparity between the success rate and false positive rate, which most post hoc ergo propter hoc arguments never attempt to ascertain.
The god hypothesis, of course, also suffers due to its lack of falsifiability.