Widget logic has been causing 500 errors to appear for bots (Google Webmaster Tools specifically) for several weeks.
From server error log:
[Thu Jan 23 20:53:17 2014] [error] [client 18.104.22.168] PHP Parse error: syntax error, unexpected T_CONSTANT_ENCAPSED_STRING in /nas/wp/www/cluster-2010/chlaw/wp-content/plugins/widget-logic/widget_logic.php(286) : eval()'d code on line 4
When I disabled the plugin that all went away and everything worked perfectly, so I know it's the plugin. But I'm not sure what exactly within the plugin is causing this.
It's been happening since shortly after upgrading to 3.8 (as far as I can tell in error logs and in GWT).