Reflective equilibrium is the simple but compelling concept that a person reflects on conflicts between different beliefs that they hold, and revises their beliefs to reduce the conflicts, or to justify/unify some beliefs in terms of others.
Has anyone made a serious attempt at devising a logic that models this process? It seems that such a logic could be a closer match to human reasoning than any deductive logic. Conflict and unity could be measured as real numbers to optimize, so that we are not always dealing with black-and-white contradiction. We might also want some notion where we optimize only a small set of propositions at a time, rather than all of them at once, because that's what humans do.
In order for the process to converge towards a stable set of beliefs, there could be a need for real-valued propositional attitudes, where we do not simply hold P, we hold P with some level of resistance to changing it. Some beliefs that are foundational or encode sensory information might be immutable.