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It's an or statement. I can either not hold a shield OR not wear armor. Right?
EDIT: Here's the full text:
You have a +1 bonus to Armour Class and Saving Throws as long as you are not wearing armour or holding a shield.
Monk's unarmored defense has the same wording.
The "or" implies wearing armor and holding a shield are two separate fail states.
Then it should be a AND statement.
OR implies one or the other.
That's the point. AND would imply that you must not be wearing armor AND holding a shield to benefit, which is clearly not the intent.
You would be correct if this were boolean logic, but it isn't.
The problem seems to be that you think the description is a logical OR, when it's actually logical NOR. The statement is structured "You will not get the benefit if A or B."
With a NOR statement: