There's something wrong with your files. Seem to still be in version 1.4.2 (at least some of them). For example, the fatal error about method ProjectMessages::getWorkspaceString being private, is because the method was private on version 1.4.2 but it is no longer private in 1.5. So probably file 'application/models/project_messages/ProjectMessages.class.php' wasn't updated correctly and is still on the previous version. The other error seems to be the same problem but with file 'application/models/application_logs/ApplicationLogs.class.php'.
So try to replace those two files with the ones in the 1.5 zip and you should not get those error messages anymore (maybe others due to other files being corrupt).
Also, just in case you moved your installation folder, try deleting file 'cache/autoloader.php'.