> 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.
> Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language.
Logical thinking is a good foundation, but one of its main problems is that it is rapidly overwhelmed in the real world by massive amounts of data and the need to make decisions even if you can't logic through the problem for a variety of reasons. So: probabilistic thinking. Logical thinking is a foundation here, probabilistic thinking done correctly still involves the basics of logic, but you may have to take what is in logic a huge net of propositions and data and collapse it down to a single number.
Statistical thinking, which I will draw a for-this-message-only distinction (that is, don't reply with some angry denunciation of this difference, I'm not globalizing it) as when you have high-quality data and statistics can draw out some non-obvious conclusion. The previous paragraph is about very real-world situations where you don't even have high quality data and has its own flavor to it. This can generally be more specialized to people going into a science or engineering track; for most people what I am thinking of in the previous paragraph is already more than they are currently operating with.
Financial thinking. Time value of money, investments, compound interest and the much less common understanding of why you can't just model it as "8% a year" in the real world because the black swans happen on large enough time frames, investment, information about basic business. This dips into politics as well with the ever-important question qui bono? There's also some generalized "planning thinking" that goes into this, like, if you want to be in a particular place in 10 years, how do you get there? e.g., in a particular job, in a particular relationship state, in a particular financial position, etc. I witness so many people who clearly have goals or desires for these sorts of things yet simply have no idea how to take such things down to "what do I do this year, this month, this week, today to advance that goal" with the result that progress is never made except by accident and the goals are so frequently missed as they just live day-by-day without advancing their goals. (Note I am not critiquing day-by-day living itself. I am not Type-A and don't plan obsessively. But I do try to make sure that if I have long-term goals I make progress towards them at a pace that means I'll get there.)
Mathematical thinking; another specialized one that not everyone needs, but the basics of "starting from these axioms where can we go" and "given this real world thing how can I reformulate it into this mathematical system"? This is, by and large, not what is already being taught in school. Which is not necessarily bad on its own terms, but grinding through quadratic equations isn't really what I mean here.
Not intended as an exclusive list. The main point is, to quote a great philosopher, "Logic is the beginning of wisdom, not the end." - https://www.youtube.com/watch?v=A4XPTmmvVow
What would be on that list?