Inverting the filter

I need to invert the functionality of this plugin, so instead of only showing posts that *do* have a certain taxonomy, I need it to filter them out and show only those posts which *don’t* have it. Is this possible?

Secondly, if so, is there a way to make that view the default when a user visits the ‘all posts’ page?

https://www.ads-software.com/plugins/custom-taxonomy-filter/