normality of their witness worlds, and the potential cause (or causes) with the most normal witness is selected. Since the world in which the gardener does not water her plants is more normal than, say, the worlds in which soil does not provide nutrients, or the sun does not shine, the gardener watering her plants is the cause of their survival. Notice that we are now in a different regime from the plain modified account: in the plain modified account, and in the first version of normality, each potential cause is assessed by itself: with graded causation, potential causes are in competition. Returning to Billy and Sally, graded approach also fails. Suzy throws first and breaks Halpern’s the bottle, but the graded approach to normality fails to deem her a cause for two reasons: the most normal witness to Suzy not being the cause of the bottle breaking is the circumstance in which neither of them throw; and as before, by AC2(am) there is no allowable witness to Suzy's throw as an actual cause of the bottle breaking. We have the same problem as with the first normality proposal: Billy's Sally's throw is the conjunctive cause. Halpern goes on to and introduce a third alternative approach to incorporating normality. 6.3 Graded Causation with Probabilities of Events This the approach differs from the first primarily in that he puts the normality ordering on of sets contexts instead of worlds. A context be identified with a that is, an “can complete assignment,” assignment of values to all of the variables (80). Here is the formalization, fixing the causal model, and using a pre-ordering of normality of contexts for and interventions on variables M M in : M There is a partition of (the set of endogenous variables) into two disjoint subsets V and with and a setting and of the variables in and respectively, Z W X Z x’ w X W, such that: [[X [[u]] and (M, |= [X (97) = x', W = w, u]] u) W w] x’, φ
normality of their witness worlds, and the potential cause (or causes) with the most normal witness is selected. Since the world in which the gardener does not water her plants is more normal than, say, the worlds in which soil does not provide nutrients, or the sun does not shine, the gardener watering her plants is the cause of their survival. Notice that we are now in a different regime from the plain modified account: in the plain modified account, and in the first version of normality, each potential cause is assessed by itself: with graded causation, potential causes are in competition. Returning to Billy and Sally, Halpern’s graded approach also fails. Suzy throws first and breaks the bottle, but the graded approach to normality fails to deem her a cause for two reasons: the most normal witness to Suzy not being the cause of the bottle breaking is the circumstance in which neither of them throw; and as before, by AC2(am) there is no allowable witness to Suzy's throw as an actual cause of the bottle breaking. We have the same problem as with the first normality proposal: Billy's and Sally's throw is the conjunctive cause. Halpern goes on to introduce a third alternative approach to incorporating normality. 6.3 Graded Causation with Probabilities of Events This the approach differs from the first primarily in that he puts the normality ordering on sets of contexts instead of worlds. A context “can be identified with a complete assignment,” that is, an assignment of values to all of the variables (80). Here is the formalization, fixing the causal model, M and using a pre-ordering of normality of contexts for M and interventions on variables in M : There is a partition of V (the set of endogenous variables) into two disjoint subsets Z and W with X Z and a setting x’ and w of the variables in X and W, respectively, such that: [[X = x', W = w, u]] [[u]] and (M, u) |= [X x’, W w] φ (97)