| | Re: as well
Look at what I said again - as well = too = so They're connectives of addition , meaning they can only be used to join two ideas which are the same - as in Swan's example:
You can have a cake and you can (have a cake) too/as well.
You can have a cake and so can you.
The meaning is yes + yes. For no + no you have the negative variants either/neither.
You can't have a cake and you can't (have a cake) either
But your first example is not yes+yes - it's no + yes. The ideas are different, not the same. So you cannot use connectives of addition - you need a connective of contrast - eg but
You can have a cake but you can't (have a cake) (No/Yes)
John didn't stop but Mary did. (John no, Mary yes)
John stopped, but Mary didn't. (John yes, Mary no)