Non-Classical Negations and the Classical Recapture: A Tour of the Land of Formal Inconsistency and its Neighborhood
Upon introducing his theory of inconsistent formal systems, in the 1960s, Newton da Costa designed axiomatic calculi whose object-language negation ceased to be a contradictory-forming operator. In particular, the validity of the inference according to which from a given sentence A and its negation not-A any other sentence would logically follow was restricted: that specific kind of irrelevant reasoning, ratified by intuitionistic logic and its extensions, was only to be allowed in case the sentence A could be assumed to behave consistently. In the decades to come, many aspects of that theory were further detailed and developed: its conceptual basis, its presentations, its semantics, the notion of undeterminedness as the dual to inconsistency, its algebraic aspects, and its many applications.
The present tutorial will be an introduction to the Logics of Formal Inconsistency and Formal Undeterminedness (LFIs and LFUs), and it is divided into five parts:
Session 1. On the abstract meaning of paraconsistency and paracompleteness, as the natural deviations from the classical approach to negation.
Session 2. On the logics capturing the dual properties of negation-consistency and negation-determinedness.
Session 3. Multi-valued semantics and modal semantics for formal inconsistency and formal undeterminedness.
Session 4. Interpretations by way of swap structures and Fidel structures.
Session 5. Consistency and Inconsistency in the foundations of mathematics: on paraconsistent set theory.