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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
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Locally consistent theories
Is there a notion in the literature of a theory which is locally consistent, in some sense wherein choosing a "vantage" yields a theory which is consistent, but the entire theory may not be consistent …