Hide blocks in node page using context
This should work... Context allows contexts to be fired based on the presence of other contexts. So in your "global" context, you can add the condition 'Context(all)' and then add ~[taxonomy_context_id]. Where taxonomy_context_id is the id of the taxonomy context. Make sure that "Require all conditions" is checked. And save. 

Now the "global" context should only show when the the other isn't available. 

Recommendation: If you are looking to switch logos based on taxonomy term, you can use template_preprocess_page() to switch out the path of the logo based on the current menu_object that is loading. This allows for you to have a global context that persists and doesn't duplicate the work/blocks and provides an override to the page.tpl.php variables instead of multiple possibly conflicting context.