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Relation Between the Almost Truth of the Principle of Charity and Uniform Continuity for Monoidal T-norm Based Logics
Noah Walker
It is known that the Sorites paradox can be resolved by working in fuzzy logic and making one of the premises of the argument close to true rather than fully true [2]. We call this premise the Principle of Charity and introduce an additional condition on it, desiring for it to be “almost true” and not just close to true. We proceed to analyze the resolution of the Sorites paradox under various monoidal t-norm based fuzzy logics and determine which of these logics establishes the Principle of Charity as “almost true”. In particular, we desire the Principle of Charity to be “almost true” iff the function mapping objects to their truth values is uniformly continuous. We prove that this relationship holds exactly when the underlying t-norm is nilpotent. This result provides a characterization of both nilpotent t-norms and uniformly continuous functions onto [0, 1].
Keywords: Fuzzy logic, left-continuous t-norms, monoidal t-norm based logic, nilpotent t-norms, sorites paradox, principle of charity, almost true, uniform continuity