First off, let me say that I think there is about zero chance that the six change.
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From the basic list of Strength, Constitution, Dexterity, Intelligence, Wisdom, and Charisma, are there any that you would like out in the next edition? Are there any others you would like in?
I don't have a problem with them as a starting point for the kind of simulation D&D kinda grew into, but I'm not sure they work as the basis for all skill checks. There's overlap/confuddlement with:
Strength as the base for both raw power and Athletic tests.
Dexterity counting for Agility and Fine Motor Skills (some can be shifted to Int, I suppose.)
Wisdom also counting for Perception
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Old School Hack uses a pretty good set for this new purpose:
Brawn - sheer size, encumbrance, physical intimidation
Cunning - sneaky stuff, trickery
Daring - buckling swashes, courage
Charm - ability to inspire or sway others to see your way, social grace
Commitment - devotion and intensity, shaking off influence
Awareness - alertness and perception, reflexes, reading between the lines
Again, I have little hope of seeing this in the core.
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What would each of the abilities mean? What would an increase of them give to an unclassed character, for example? How would certain classes particularly benefit from certain ability scores?
Hard to say without the rest of the mechanics. I would imagine the relationships wouldn't change beyond the obvious. I think the point of ability scores is to give a quick snapshot of the character's capabilities.
I know it may sound crazy, but I'm in favor of divorcing ability scores and combat stats. Give all that to the classes, and then you don't have to worry if you've maxed out your primes. Eliminate that little balancing act.
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How much would you want the ability scores to define your character? How much should this change over time? How much should ability scores change, if at all?
On a scale of 1 to 10, I think Ability scores should come in about a 2. Class, race, and background should do most of it, but ability scores can differentiate between two similar characters a bit.