It's really hard (for me, and I imagine, for everyone else) to not put myself into my views and opinions. Like, when someone shows me that I'm wrong, it's natural for me to feel attacked, instead of just taking it as a learning moment. Noticing when this happens and working with it has been my main struggle in learning how to be more rational. Those views and opinions really don't need to be a part of what I consider…
> Allowing multiple beliefs to overlap This doesn't seem very rational. If your beliefs are in conflict and you're content to not resolve that, then pretty much by definition you're accepting a logical inconsistency. If resolving the intersection doesn't lead to a new stable belief system, then aren't you basically going with "whatever I'm feeling that day"?
However, the drive for total and pure consistency is also misguided in my judgement. One reason why we usually feel so motivated and conflicted (to the point where it can lead to depression) with inconsistency is the psychological effect of cognitive dissonance. It's not clear to me that the only way to quieten cognitive dissonance is to resolve the dissenting thoughts.
Another way is to accept that not everything needs to be resolved. This can be great for mental health - again, just in my experience. Don't let the (sometimes irrational) effects of cognitive dissonance override your decision making. Resolution can work, but so can acceptance.