Positive does not mean permissive. We just have different ideas about the necessary nature of the negative consequence. When one is needed, positive trainers are most likely to use negative punishment (taking away a good thing), rather than positive punishment(the application of a bad thing). As an adjunct to that, we counsel the generous use of management to prevent the dog from practicing (and getting rewarded for) undesirable behaviors.
The result? Since all living things repeat behaviors that are rewarding, and those behaviors that aren’t rewarded extinguish (go away), the combination of negative punishment and management creates a well-trained dog at least as easily as harsh or painful corrections and without the very real potential for relationship damage that is created by the use of physical punishment.