Stephen Toulmin wrote some excellent books that discuss using non-symbolic logic. He argues that people, in general, don't communicate using syllogisms or symbolic logic. He's not totally dismissing symbolic logic, just that it's not practical or realistic in the vast majority of human communication. For instance, in the Court Room, lawyers, judges, etc, usually don't use symbolic logic. I recommend supplementing your readings of Stephen Toulmin with books on Venn Diagrams, symbolic logic, and syllogisms. No one approach is a panacea.
I tend to believe that the following quote best describes the individual pursuing correct thinking:
"A syncretistic individual implementing synergistic integration in order to posit the impossibility of the implausibility."