How can I show contacts that DO NOT have an activity of type X in last Y months?
The process is slightly roundabout:


Create a smart group that captures all contacts called "All Contacts".
Create a second smart group consisting of people who have the activity type "Confirm Contact Details".
Use the "Include/Exclude Search" (Search menu > Custom Searches) to include the first smart group and exclude the second smart group.


Once you've done this, you can save it as a smart group for one-step searches moving forward.