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A useful 3-valued provability predicate Demos is defined, that can provably simulate non-classical logics and connectives (like autoepistemic logic and negation-as-failure) and that, despite its 3-valued semantics, is able to capture thoroughly classical notions of provability and unprovability. Demos is largely tolerant to self-reference, and can be axiomatised consistently. Its semantics provides a guide to the development of effective axiomatisations and meta-interpreters that do not fall under the scope of the well known negative results on incompleteness and inconsistency.