I'm sympathetic to your point of view; in some cases it seems nearly impossible to rigorously exclude reverse-associations when there are so many complexities.

The authors respond to this objection as follows:

The finding that infections are associated with decreased general cognitive ability after a latency period might be an epiphenomenon and not a biologically-mediated relationship. For instance, a reverse association could have existed so that lower cognitive ability may be a risk factor for acquiring infections. Furthermore, studies have indicated that immune related genes might be implicated in cognition [34], and individuals with genetic liability towards a lower general cognitive ability might also be more genetically vulnerable towards infections. However, although a genetic or reverse association could have partly explained some of the observed associations, this probably does not account for the observed temporal associations or the associations with severity of the infection.

The association between infections and cognitive ability remained significant after adjusting for other factors known to influence a decrease of general cognitive ability, such as gestational age, birth weight [35], and having more older siblings (i.e. being later in a sibship), which often correlate with socio-economic gradients [36]. ..Nevertheless, the findings remained significant also after adjustment for an individual or family history of psychiatric disorders and substance abuse which could be associated with several of these unmeasured risk factors as well as socio-economic factors. Furthermore, the possible effect of maternal smoking during pregnancy should to some degree be captured by the included birth weight and gestational age in the adjustments. Moreover, there may be diseases that are associated with both increased susceptibility to infections and somewhat lower cognition (either the disease itself or its treatment).

How do the authors manage to just wave away all the genetic effects with just "this probably does not account for the observed temporal association"? Either the authors have some evidence or else it is not a hypothesis that they can rule out. All their correction factors are non-genetic factors.

"temporal association" is the point that shortly after infections, there is a stronger effect on intelligence. So you're saying "maybe people who get hit in the head with 2x4s are just dumber" and they're saying "that doesn't explain why they do so bad on cognitive tests right after they got hit". It's much stronger evidence that way.

>this reverse association could not account for the observed temporal correlations or the correlations with the severity of infection. Moreover, the correlation between infections and cognitive ability retained significance even after adjustment for other parameters known to influence general cognitive ability...which are also associated with socio-economic gradients.

Could possibly be both.