Does Correlation "Sometimes" Imply Causality? I am an MBA student taking courses in statistics.
I am sure that at some point, we have all heard the famous expression - "Correlation Does Not Imply Causality".
When we are being introduced to this concept, we are often shown an absurd example of two variables that are highly unlikely to have a causal relationship, yet seemingly appear to be correlated with one another. For example - it is highly unlikely that there might be a causal relationship between the "letters in the words of spelling bees" and the "number of people killed every year by spiders":

However, there are other examples in which it might be reasonable to assume that correlation does in fact imply causality. For example, it is quite likely that there might be some relationship between "life expectancy" and "GDP" - countries with higher GDP likely have stronger economies, stronger economies likely have stronger health sectors, countries with stronger health sectors likely have more hospitals/doctors ... using this logic, although there might be outliers and exceptions, it does not seem completely unreasonable to expect that there might be some causal correlation between income and life expectancy:

This leads me to my question - when we are told that "Correlation Does Not Imply Causality", does this statement indirectly mean that there might be some instances in which "Correlation DOES Imply Causality"?
For instance, suppose I have some data (for different countries) which shows a clear linear relationship between  "the number of doctors per 1000 people" and the "life expectancy" of those countries (i.e. people who live in countries with more doctors tend to live longer on average) - although there some philosophical arguments that suggest we might never be able to "truly" attribute causality between two concepts (e.g. perhaps tomorrow we find out that countries with more doctors have some newly discovered vitamins in their water and this is why they live longer, not actually due to more available doctors, yet this is highly unlikely) - would be fundamentally wrong and morally unreasonable to prematurely suggest that "there theoretically might exist some level of causal relationship between the number of doctors and life expectancy"?

The problem here is logical. Nobody can deny that causality can lead to correlation, and so there are instances in which you have causality and also correlation. The statement "correlation implies causality" however is a general logical statement; it means that you have causality whenever there is correlation, and the "whenever" doesn't make sense when discussing a single instance. So no, as long as counterexamples exist, correlation does not imply causality, and this statement is not contradicted by the existence of instances where you have both (or even where correlation can be legitimately taken to indicate the possibility of causality).
I add that one might think that even if there is no logical implication, there could be a strong probabilistic indication, i.e., when we see correlation, very often there is causation, and counterexamples are few and far between. This, however, can hardly be made precise (even before investigating this empirically, decisions would need to be made what to count, and it isn't at all obvious how to do that), and my impression is that there are for sure enough counterexamples and enough ways to explain how correlation arises without causation, that one should neither think that this "implication" holds "with high probability".
