Description:5.1 Kripkean incompatibility semantics -- 5.1.1 Possible worlds in IS -- 5.1.2 How to Kripke IS -- 5.1.3 What it means to Kripke IS -- 5.2 Non-monotonic incompatibility semantics -- 5.2.1 Ranges of counterfactual robustness -- 5.2.2 Non-monotonicity -- 5.2.3 Relevant reasoning -- 5.2.4 Defeasible reasoning -- 5.2.5 Preferential incompatibility semantics -- 5.3 The expressive role of incompatibility -- 6. From inferentialism to idealism, and back -- 6.1 Inferentialism -- 6.2 Pragmatism -- 6.3 Idealism -- 6.4 The revision of material incompatibilities -- 7. Conclusions -- Index of names -- Index of subjects