Why do grandparent's love their grandchildren so much, when they can usually take or leave other people's children? Is it natures way of making sure that should something happen to the natural parents, the offspring will be raised by someone who cares?

The evolutionary explanation in terms of genetic fitness (kin selection) goes roughly like this: Grandchildren share 1/4 of each of their grandparent's genes (parents and siblings share 1/2), so genes that help to "code" for traits that lead you to give up X/4 amount of your fitness (your chance of reproductive success) to increase your grandchildren's fitness X amount would spread through the population more than genes that "code" for more selfish behavior. So, all else being equal, we should expect to see selection for genes that lead grandparents to be nice to their grandkids.

Of course, things are messier than this. Such traits won't be selected for in organisms that disperse such that grandparents aren't near their grandkids. Conversely, since grandparents (e.g., in humans) are typically past reproductive age, genes that code for even more generosity might be selected for--that might explain why grandparents spoil their grandkids rotten!

Of course, it would be hard to code for something as specific as "calculate if your action has a benefit to your kin that is greater than the cost to you multiplied by your relatedness to that kin." So, we see heuristics at work: strong emotional bonds with kin, emotions that are triggered less by actual relatedness than by a good proxy for that--e.g., whom you are around a lot. Hence, grandparents (parents) typically love their adopted grandkids (kids) as much as their biological ones.

Now, this somewhat cold genetic explanation may lead us to worry that we love our kids and grandkids for (ultimately) selfish reasons or even that we don't really love them. But that's bad reasoning. I really do love my children and would (I hope) die for them, and it's not for the sake of myself or my genes (whatever that would mean). Rather, the evolutionary explanation (combined with lots of cultural explanation too) accounts for why I am the sort of creature that really loves my children and really sacrifices a lot for them--yes, it really is a sacrifice of other interests and obligations I have. (Compare: suppose the reason we love our kids is because God created us to have the relevant emotions. Would that historical explanation for why we have those emotions mean that we don't really love our children?)

Finally, your first question could be read with a normative, rather than a descriptive "why?" Granted, grandparents do love their grandchildren more than other people's kids (I've just explained why that might be), but should they? I'm inclined to say yes, but it'd take a while to try to justify that answer, especially to someone who's a devout utilitarian and thinks we should just maximize overall happiness: If there's a situation where Grandpa can either save his granddaughter or save five other kids, whom should he save?

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