Put epistemology to the side. Logical consequence applies to all of your beliefs, no matter how you came to arrive to them. For example, you wouldn't believe (p

Put epistemology to the side. Logical consequence applies to all of your beliefs, no matter how you came to arrive to them. For example, you wouldn't believe (p & ~p) or (p -> ~p), independent of what your beliefs are. The fact that people can compartmentalize different inconsistent mental models really has no bearing on whether any given proposition is true or not. Humans are complex and prone to believing in superstition.