A Model-Invariant Tieory of Singular Causation J. Dmitri Gallow - PDF document

A Model-Invariant Tieory of Singular Causation J. Dmitri Gallow Counterfactual Causal Models 2. diagram is correct. are causally determined by each other. I’ll assume throughout that the canonical model of a neuron default, fjring deviant), and a true system of equations describing how the values of those variables (a) Given a neuron diagram, let the canonical model be the one that has, for each neuron, a binary 1 Causal Models 1 Causal Models 1. We will represent causal determination structure with a causal model , or a structural equations model , Causal & Explanatory Reasoning · Venice International University · November 13, 2017 A causal model � = ( � , ⃗ u , � , � , � ) is a 5-tuple of (a) A vector, � = ( U 1 , U 2 ,..., U M ) , of exogenous variables; (b) An assignment of values, ⃗ u = ( u 1 , u 2 ,..., u M ) , to � ; (c) A vector � = ( V 1 , V 2 ,..., V N ) , of endogenous variables; and (d) A vector � = ( ϕ V 1 , ϕ V 2 ,..., ϕ V N ) of structural equations , one for each V i ∈ � . (e) A specifjcation, � , of which variable values are default and which are deviant . � : ( A , C ) B   E : = B ∨ D ⃗ u : ( 1 , 1 ) D : = C � :   B : = A ∧ ¬ C � : ( B , D , E ) Figure 1: Preemptive Overdetermination . (For all variables, the value 0 is default, and the value 1 is deviant.) variable taking the value 1 if the neuron fjres and the value 0 if it doesn’t fjre (where not fjring is Given a causal model � , and an assignment v of values to the variables in V , we can defjne a counterfactual model � [ V → v ] . Given a causal model � = ( � , ⃗ u , � , � ) , including the variables V , and given the assignment of values v to V , the counterfactual model � [ V → v ] = ( � [ V → v ] , ⃗ u [ V → v ] , � [ V → v ] , � [ V → v ] , � [ V → v ]) is the model such that: (a) � [ V → v ] = � ∪ V (b) ⃗ u [ V → v ] = ⃗ u ∪ v (c) � [ V → v ] = � − V (d) � [ V → v ] = � − ( ϕ V i | V i ∈ V ) (e) � [ V → v ] = �

Figure 2: Symmetric Overdetermination factual dependence within some appropriate counterfactual model. 1 variable values, see Halpern (2008). For the problem, see Hiddleston (2005) and Hall (2007), and for a more careful statement of the solution in terms of default 2 (forthcoming) all take this general form. Tie difgerences between them have to do with what makes the variable assignment Tie accounts of Hitchcock (2001), Halpern & Pearl (2001, 2005), Woodward (2003), Halpern (2008, 2016), and Weslake 1 In the case of Symmetric Overdetermination from fjgure 2, the counterfactual counterfactual account says 5. From this, we may recover a familiar 2-place causal relation: (a) Tiis theory gives us a 4-place causal relation: The Counterfactual Counterfactual Theory and Tie majority of theories of causation formulated in terms of causal models say that causation is counter- The Counterfactual Counterfactual Theory Figure 3: Bogus Prevention 3. Using counterfactual models, we may provide a semantics for causal counterfactuals: Causal Counterfactuals 4. 2 2 a e In a causal model � , containing the variables in V , the causal counterfactual V = v � → ϕ is true ifg ϕ is true in the counterfactual model � [ V → v ] , � | = V = v � → ϕ ⇐⇒ � [ V → v ] | = ϕ In a causal model � , C ’s taking on the value c , rather than c ∗ , causes E to take on the value e , rather than e ∗ , ifg there is some suitable vector of variables O with suitable values o such that, if O is held fjxed at o , then both (I) C would have taken on the value c and E would have taken on the value e , � [ O → o ] | = C = c ∧ E = e (II) Had C taken on the value c ∗ , E would have taken on the value e ∗ , � [ O → o ] | = C = c ∗ � → E = e ∗ Cause ( C = c , C = c ∗ , E = e , E = e ∗ ) Cause ( C = c , E = e ) ⇐⇒ ∃ c ∗ ∃ e ∗ Cause ( C = c , C = c ∗ , E = e , E = e ∗ ) that C ’s fjring caused E to fjre. However, the same system of equations, and the same values, may be used to model to the case of Bogus Prevention from fjgure 3. (a) Tie solution is to say (roughly) that, in order to be suitable , the variable values O = o must be at least as default as the values O take on in the actual model. 2 O = o suitable .

3 7. tual account. (b) So we cannot accept Exogenous Reduction , Model Invariance , and the counterfactual counterfac- the following principle: 9. Figure 4: Symmetric Overdetermination 8. get by: 6. Ideally, a theory of singular causation would satisfy the following principle: Figure 5: Short Circuit 2.1 Model Variance B E B Model Invariance Given any two causal models, � and � † , which both contain the variables C and E , if both � and � † are correct, then C = c caused E = e in � ifg C = c caused E = e in � † . (a) In the canonical model of the neuron diagram from fjgure 4, � 4 , the counterfactual counterfactual account says that C ’s fjring caused E to fjre, since � 4 [ A → 0 ] | = C = 1 ∧ E = 1 , � 4 [ A → 0 ] | = C = 0 � → E = 0 , and A = 0 is more default than the actual value A = 1 . (b) However, if we remove the exogenous variable A from the model, getting the model � − A 4 , which contains the equation E : = ¬ B ∨ C , then the counterfactual counterfactual account will say that C ’s fjring didn’t cause E to fjre. u , � , � , � ) is a causal model with U ∈ � , then let � − U be the model that you In general, if � = ( � , ⃗ (a) Removing U from � (b) Removing U ’s value from ⃗ u (c) Exogenizing any variables in � whose only parent was U (d) Replacing U for its value in every structural equation in � (e) Removing default information about U from � . If every equation in � − U is surjective, then say that U is an inessential variable. Tien, we should endorse Exogenous Reduction If a causal model � = ( � , ⃗ u , � , � , � ) is correct, and U ∈ � is inessential, then � − U is also correct. (a) In the model � 4 , A is an inessential exogenous variable, and the counterfactual counterfactual ac- count tells us that C = 1 causes E = 1 in � 4 , but not in � − A 4 . 10. (a) In the canonical model of the neuron diagram from fjgure 5, � 5 , the counterfactual counterfactual account says that C ’s fjring caused E to not fjre. For � 5 [ B → 1 ] | = C = 1 ∧ E = 0 , � 5 [ B → 1 ] | = C = 0 � → E = 1 , and B = 1 is as default as the actual value (because it is the actual value).

4 by: 3 Preemptive Overdetermination 3.1 (a) I’ll build up the theory by progressing through some familiar cases from the literature. Endogenous Reduction , Exogenous Reduction , and Model Invariance . 13. I will present a theory of causation in terms of structural equations models which is consistent with A Model Invariant Theory of Causation Figure 6: Preemptive Overdetermination 3 terfactual account. ii. So we cannot accept Endogenous Reduction , Model Invariance , and the counterfactual coun- i. (c) Tien, we should accept the following principle: � : ( A , C ) B � E : = B ∨ C � ⃗ u : ( 1 , 1 ) � : B : = A ∧ ¬ C � : ( B , E ) (b) However, if we remove the endogenous variable B from the model—getting the model � − B 5 , which contains the equations E : = C ∧ ¬ D and D : = C —then the counterfactual counterfactual account will say that C ’s fjring didn’t cause E to not fjre. u , � , � , � ) is a causal model with V ∈ � , then let � − V be the model that you get 11. In general, if � = ( � , ⃗ (a) Leaving � alone (b) Leaving ⃗ u alone (c) Removing V from � (d) Removing ϕ V from � , and replacing V with ϕ V ( PA ( V )) wherever V appears on the right-hand-side of an equation in � 3 (e) Removing default information about V from � 12. (a) If V has a single parent and a single child, then say that V is an interpolated variable. ... P → V → C ... (b) If all the equations in � − V are surjective, then say that V is inessential . Endogenous Reduction If a causal model � = ( � , ⃗ u , � , � , � ) is correct, and V ∈ � is an inessen- tial, interpolated variable, then � − V is also correct. In the model � 5 , B is an inessential, interpolated variable, and the counterfactual counterfactual account tells us that C = 1 caused E = 0 in � 5 , but not in � − B 5 . 14. (a) In the canonical model, � 6 , of Preemptive Overdetermination shown in fjgure 6, E = 1 does not counterfactually depend upon C = 1 . PA ( E ) are E ’s causal parents in the model—those variables which appear on the right-hand-side of E ’s structural equation ϕ E .