> It emphasises how logic alone might not lead us to the right conclusions, because there are more things at play in reality than in our mental models.
> It suggests that we choose actions by carefully studying outcomes rather than based on what ought to yield the best outcome.
> It tells us that differences in outcomes may not be a signal of differences in controllable antecedents: it is often just the natural variation of the process.
As a statistician, perhaps the most useful feature of statistical thinking is an explicit rejection of point estimate. Always, always, always, give an interval estimate, or a five point summary if you have that. This was the one lesson that was drilled into me repeatedly.
Ofcourse, you head out into the real world & it is full of point estimates! The article gives a very good example of real-world statistical illiteracy - "If the gas mileage of a car is 40 miles per gallon, and I drive 20 miles, I will have used half a gallon of gas". While I'm sure that the HN audience prides itself on statistical literacy, they will certainly have trouble coming up with a good interval estimate for that scenario.
I think the comment is alluding to the scientific method, more or less. Some kinds of knowledge are simply not accessible through logic alone. Facts about the world have to be obtained through observation. We then create mental models based on logic that can often explain and predict more facts about the world, but these mental models themselves often turn out to be incomplete or based on faulty assumptions. If we make observations contrary to the prediction of the model, we then have to revise the model.
I'll give a rather silly example: I recently saw a couple of people debating whether betting odds should give more accurate predictions of the outcomes of sporting events than other predictions. One person claimed that people are more risk-averse when their money is on the line and therefore should be expected to consider the probabilities more rigorously, while the other person claimed that it shouldn't matter whether one is putting up money. In other words, they were both making a priori arguments for why betting odds would or would not be most predictive. But in fact it is empirically verifiable that betting odds are more predictive than other models; no amount of a priori reasoning will lead you to the correct answer, only observation will.
The downside to logical reasoning is that it only deals with absolutes. If you mistake an uncertain trend in the world for an absolute fact then logical reasoning also suffers from garbage in garbage out.
All models have prerequisites, and logic requires propositions, which are either true or false. If you are trying to talk about models themselves (which is what most of math is), then logic is useful because it's easy to come up with propositions about models, and then you can explore what else must be true, or what else can't be true.
Probability and statistics are much more useful for explaining the world than logic. But we use logic to prove things about the models that we use in probability and statistics.
Logic is correct, you aren't going to find something logically true and then see it violated empirically. You are going to instead discover that the real world thing can't be neatly modeled as propositions. The error will be in producing the inputs to the model, not the model itself.
"They have a really good startup idea so they will be successful."
"We need to fire a salesperson so we'll pick the one with the lowest total contract value."
"The bus trip takes 43 minutes so I will be there at 12.17."
You'll note that these are not failures of logic (which is somewhat self-consistent and cannot fail) but rather of small-ish world views that don't allow for uncontrolled factors, which I think is a distinguishing factor of statistical thinking.
As you intimated, none of these are logically provable. They seem like sound logical conclusions on their face but, if someone has studied logic, they know that further evidence is required to make these deductions.
You're going off a very strict "logically provable" definition which is generally incompatible. We use "logically provable" like this in cases for math equations and algorithms (which has a lot of value!) but not for every day things.
"My cat is sad" is not logically provable, but we feel comfortable that this idea is basically real enough. "vorpalhex has an internal monologue" is not logically provable, but we feel comfortable that it's probably true if I say it's true.
To go back to your original ask, for something that can be shown via Bayesian thinking but not logical deductive reasoning: "Intelligent alien life has never visited Earth". This can't be logically proven because of the observer paradox - the strongest statement we could defend logically would be "There is no known evidence that intelligent alien life has ever visited Earth" (which is a weaker claim). If we allow Bayesian reasoning, then we can defend that the likelihood that intelligent alien life has visited Earth and never left behind evidence is much, much smaller than the likelihood that intelligent alien life has simply never visited us.
> It emphasises how logic alone might not lead us to the right conclusions, because there are more things at play in reality than in our mental models.
> It suggests that we choose actions by carefully studying outcomes rather than based on what ought to yield the best outcome.
> It tells us that differences in outcomes may not be a signal of differences in controllable antecedents: it is often just the natural variation of the process.