Complete Sets of Connectives for Generalized Łukasiewicz Logics
نویسنده
چکیده
While ∧,∨,¬ form a complete set of connectives for classical propositional logic, this does not hold for Łukasiewicz’s three-valued propositional logic, nor its generalization to n-valued logic. We define a unary connective r so that ∧,∨,¬, r form a complete set of connectives for n-valued Łukasiewicz logic. We discuss generalizations of this to infinitary logics. If we allow infinite conjunctions and disjunctions of arbitrary size, this provides a complete set of connectives for real-valued Łukasiewicz logic. Restricting to countable conjunctions and disjunctions, the truth functions expressible with these connectives are precisely the Borel functions.
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