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Peterson, Clayton; Kulicki, Piotr (2016): Conditional Normative Reasoning with Substructural Logics. In: Roy, Olivier; Tamminga, Allard; Willer, Malte (eds.) : Deontic Logic and Normative Systems: 13th International Conference, DEON 2016, Bayreuth, Germany, July 18-21, 2016. College Publications. pp. 220-236
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This paper extends the results presented in [22,20] and explores how new paradoxes arise in various substructural logics used to model conditional obligations. Our investigation starts from the comparison that can be made between monoidal logics and Lambek's [17] analysis of substructural logics, who distinguished between four different ways to introduce a (multiplicative) disjunction. While Lambek's analysis resulted in four variants of substructural logics, namely BL1, BL1(a), BL1(b) and BL2, we show that these systems are insufficient to model conditional obligations insofar as either they lack relevant desirable properties, such as some of De Morgan's dualities or the law of excluded middle, or they satisfy logical principles that yield new paradoxes. To answer these concerns, we propose an intermediate system that is stronger than BL1 but weaker than BL1(a), BL1(b) and BL2.