Combining Supervaluation and Degree Based Reasoning Under Vagueness [chapter]

Christian G. Fermüller, Robert Kosik
2006 Lecture Notes in Computer Science  
Two popular approaches to formalize adequate reasoning with vague propositions are usually deemed incompatible: On the one hand, there is supervaluation with respect to precisification spaces, which consist in collections of classical interpretations that represent admissible ways of making vague atomic statements precise. On the other hand, t-norm based fuzzy logics model truth functional reasoning, where reals in the unit interval [0, 1] are interpreted as degrees of truth. We show that both
more » ... ypes of reasoning can be combined within a single logic SŁ, that extends both: Łukasiewicz logic Ł and (classical) S5, where the modality corresponds to '. . . is true in all complete precisifications'. Our main result consists in a game theoretic interpretation of SŁ, building on ideas already introduced by Robin Giles in the 1970s to obtain a characterization of Ł in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion. In our case the experiments are replaced by random evaluations with respect to a given probability distribution over permissible precisifications. 1 One should mention that the extent to which supervaluation leads to classical validity and consequence relations is hotly debated. See, in particular [22] for proofs that compactness,
doi:10.1007/11916277_15 fatcat:qprykkz2unfs5adgwr6edfps5a