Couple of things...
1. You can't delete projects from within Logic. The only way you can trash them is to go into the Finder, find the project file (or folder) and drag it to the trash. Emptying the trash will then permanently delete the file.
BTW, another way to delete files when in the Finder (desktop) is to select the file and hit CMD (Apple Key)+DELETE.
2. The only thing you can delete from within a Logic project (while it's up and running) are audio files. To do that, you'd go into the Bin (audio bin), select the file, and then go up to the Bin's local Audio File menu and use the "Delete Audio File(s)" function.
If you're not able to delete files from the Finder then the only thing I can think of is that you don't have "permission" to do that, meaning, that on Macs, if you're not an 'administrator' then you may not be able to delete files.
So when you try to delete files from the Finder (desktop), do you get an error message or something?