The paraconsistent system CPQ-ZFC/F is defined. It is shown using strong non-finitary methods that the theorems of CPQ-ZFC/F are exactly the theorems of classical ZFC minus foundation. The proof presented in the paper uses the assumption that a strongly inaccessible cardinal exists. It is then shown using strictly finitary methods that CPQ-ZFC/F is non-trivial. CPQ-ZFC/F thus provides a formulation of set theory that has the same deductive power as the corresponding classical system but is more reliable in that non-triviality is (...) provable by strictly finitary methods. This result does not contradict Gödel's incompleteness theorem because the proof of the deductive equivalence of the paraconsistent and classical systemss use non-finitary methods. (shrink)

A new technique for proving realisability results is presented, and is illustrated in detail for the simple case of arithmetic minus induction. CL is a Gentzen formulation of classical logic. CPQ is CL minus the Cut Rule. The basic proof theory and model theory of CPQ and CL is developed. For the semantics presented CPQ is a paraconsistent logic, i.e. there are non-trivial CPQ models in which some sentences are both true and false. Two systems of arithmetic minus induction are (...) introduced, CL-A and CPQ-A based on CL and CPQ, respectively. The realisability theorem for CPQ-A is proved: It is shown constructively that to each theorem A of CPQ-A there is a formula A *, a so-called “realised disjunctive form of A ”, such that variables bound by essentially existential quantifiers in A * can be written as recursive functions of free variables and variables bound by essentially universal quantifiers. Realisability is then applied to prove the consistency of CL-A, making use of certain finite non-trivial inconsistent models of CPQ-A. (shrink)

It is shown that logical contradictions are derivable from natural translations into first order logic of the description and background assumptions of the Soros Game, and of other games and social contexts that exhibit conflict and reflexivity. The logical structure of these contexts is analysed using proof-theoretic and model-theoretic techniques of first order paraconsistent logic. It is shown that all the contradictions that arise contain the knowledge operator K. Thus, the contradictions do not refer purely to material objects, and do (...) not imply the existence of inconsistent, concrete, physical objects, or the inconsistency of direct sensory experience. However, the decision-making of rational self-interested agents is stymied by the appearance of such intensional contradictions. Replacing the rational self-interest axioms with axioms for an appropriate moral framework removes the inconsistencies. Rational moral choice in conflict-reflexive social contexts then becomes possible. (shrink)