What, according to the functionalist, is the difference between things which merely have functional roles and things whose functional roles amount to their having a mind?
Different people may interpret identical events or stimuli differently due to their having a mind that differs from your own.
Which part of English grammar stipulates using their in these sentences? For me it should be either "to them having a mind" or "to their having of a mind."
Both constructions are acceptable.
Indeed for most of the 20th century only the construction with their would have been acceptable in formal discourse: having was by many ‘authorities’ regarded as a nominal form (a gerund) which could not take a subject but only a "possessor".
Today we by and large have a better understanding of how the language works. However, there are still old-school folks around who may be tolerant of explicitly objective forms in this position, but are still uncomfortable with it. I recommend using the genitive with pronouns, as in your examples, NOT because it’s in any sense more correct but because it avoids jarring any of your readers. Consider that they’re mostly old farts like me and may be in Very Senior Positions, and wait another twenty years: they’ll all be dead by then, or retired, and then you can do it your way.
