Yeah there's the logical imply, but that's not exactly what we're after here (I don't think) since the truth table can be tricky and doesn't have to correspond to anything in reality. The truth table for "If P, then Q" (P->Q) evaluates to "True" in all cases of P and Q except when P is True and Q is False. (That is, we don't want true premises to lead to false conclusions, so p->q is false when p is true but q is false.)
I'm actually more interested in the mathematics of causality which is formally contained in Probability Theory. (See Judea Pearl's "Causality" for a full treatment...) One expression can be in the form of Prob(Y = y | do(X = x)) (essentially invoking a "do" operator in your given information), another form of representation is with graphs and arrows; electrical engineers have no problems with causality (at least for normal things). Here's my speculation, but I'd say that in the framework of probability theory, a correlative implication can be expressed as background information. And so you ask if prob(short life given unhealthy and some common set of bg info) > prob(short life given just the bg info) to see if there's an "implication" there you may want to investigate and perhaps infer a causation by performing a do() operation (e.g. controlled experiment).
Also I find it interesting that physics doesn't have any notion of "causality" in the equations, they just state a relationship. "F = m*a" works in time-reverse as well as time-forward (though this isn't always the case for all the laws, some change under mirror reflection or charge differences).
Anyway, I'm not a formal mathematician, someone more knowledgeable can correct me on anything. Read Pearl, Jaynes, etc. if you want more probability theory!
I'm actually more interested in the mathematics of causality which is formally contained in Probability Theory. (See Judea Pearl's "Causality" for a full treatment...) One expression can be in the form of Prob(Y = y | do(X = x)) (essentially invoking a "do" operator in your given information), another form of representation is with graphs and arrows; electrical engineers have no problems with causality (at least for normal things). Here's my speculation, but I'd say that in the framework of probability theory, a correlative implication can be expressed as background information. And so you ask if prob(short life given unhealthy and some common set of bg info) > prob(short life given just the bg info) to see if there's an "implication" there you may want to investigate and perhaps infer a causation by performing a do() operation (e.g. controlled experiment).
Also I find it interesting that physics doesn't have any notion of "causality" in the equations, they just state a relationship. "F = m*a" works in time-reverse as well as time-forward (though this isn't always the case for all the laws, some change under mirror reflection or charge differences).
Anyway, I'm not a formal mathematician, someone more knowledgeable can correct me on anything. Read Pearl, Jaynes, etc. if you want more probability theory!