To what extent is common sense innate versus influenced by interactions with others? If someone has grown up with minimal interaction with other people, would their common sense be different?
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It is difficult to say that "common sense" is rigidly defined enough to be studied in the way stated above. There are interesting topics concerning common sense in Psychology, but most don't come from the angle you are suggesting. For instance, here is a really decent article discussing cognitive bias and common sense:
You also have to consider that common sense doesn't tend to employ rigid enough requirements for theories; we also allow for conflicting theories to serve as 'valid' explanations. Take for instance:
Birds of a feather flock together vs. opposites attract.