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"?

I think there are possibly two related issues causing confusion.
First, "imply" means to "suggest."  And correlation is a clue that 2 things might be causally related.  It doesn't prove it, but it hints that a cause might be there. So the statement "Correlation Does Not Imply Causality" is frankly wrong.  Correlation does imply ("suggest") possible causation.  The statement should be "Correlation Does Not Prove Causality."
Comments from Michael MacAskill and V C pointed out that "imply" in this case refers to logical implication: if p then q.  In that context, the statement is definitely not "frankly wrong."  However, the meaning is still "Correlation Does Not Prove Causality."
Which brings up the 2nd issue.  When learning the difference between correlation and causation, students may get the impression that they are opposites or mutually exclusive.  They are not.  Correlation certainly does not rule out causation. If confounding variables are eliminated, then all causally related data will be correlated (not necessarily in a linear fashion), but not all correlated data will be causally related:

This is just another example that correlation does suggest (imply) but not prove causation. Or to put it another way, lack of correlation does prove lack of causation.  Correlation is necessary but not sufficient to prove causation.  (As long as confounding variables are controlled for. I'm being idealistic here for teaching purposes.  It may not always be practical to control for confounds.  Even if A and B are causally related, a scatterplot of A and B may still appear random if other factors also affect A and/or B.)
