Appendix A Lewis s Counterfactuals I will briefly describe David Lewis s possible world semantics of counterfactual conditionals (1973, p. 13). Let us understand a possible world simply as a way things could have been. First, consider the truth conditions of a counterfactual conditional: Counterfactual conditional. A counterfactual conditional A B is true in a possible world w if and only if either (i) proposition A is false at every world, or (ii) in all the worlds that are closest to w at which A holds, B also holds. The first condition is uninteresting for our purposes since it is fulfilled only when the antecedent of the conditional is logically inconsistent. The second condition is more important. It involves the concept of closeness, which can be understood using a similarity metric. Thus, when a world w 1 is more similar to w than is w 2,we also say that w 1 is closer to w than is w 2. In order to understand the concept of similarity, suppose a system of spheres S w i.e., A set of possible worlds with its centre on w. If spheres S 1 and S 2 are subsets of S w, then either S 1 is a subset of S 2 or S 2 is a subset of S 1. In other words, the system is concentric. We can now consider the limit assumption on the basis of which one of the truth conditions of the counterfactual conditional is given (Lewis 1973, p. 19): Limit assumption. For every world w and every antecedent A, there is a smallest sphere S where A is true. The smallest sphere around w is the sphere of all worlds that are most similar to w. We can also define the concept of similarity as follows: Comparative similarity. For every pair of worlds w 1 and w 1 of a system S w with its centre at w the following holds: If w 1 is in a smaller sphere than w 2, then w 1 is more similar (closer) to w than is w 2. Lewis (1986, p. 47) classifies the factors that should be taken into account to evaluate similarity and difference between worlds as follows: The Author(s) 2016 E. Céspedes, Causal Overdetermination and Contextualism, SpringerBriefs in Philosophy, DOI 10.1007/978-3-319-33801-9 75
76 Appendix A: Lewis s Counterfactuals Priorities of similarity. (i) First importance: Avoid big, widespread, diverse violations of law. (ii) Second importance: Maximise the spatio-temporal region in which a perfect match of a particular fact prevails. (iii) Third importance: Avoid small, localised, simple violations of law. (iv) Fourth importance: Secure approximate similarity of a particular fact. We may say, for instance, according to this table of priorities, that if w 1 differs from w only regarding some particular fact and if there is widespread difference between w 2 and w regarding laws, then w 1 is more similar to w than is w 2. References Lewis, D. (1973). Counterfactuals. Blackwell Publishers. Lewis, D. (1986). Philosophical papers: Volume II. Oxford University Press.
Appendix B The Regularity Theory of Causation B.1 INUS Conditions The regularity theory of causation is based on causes being regularly followed by their effects. In a sense, such a principle is equivalent to Hume s first definition of the concept of causation, as given in the first chapter: An object causes another when objects similar to the first are followed by objects similar to the second. A further characterisation of this aspect of causation was developed by John Stuart Mill, who stated the following: The Law of Causation, the recognition of which is the main pillar of inductive science, is but the familiar truth, that invariability of succession is found by observation to obtain between every fact in nature and some other fact which has preceded it. (Mill 1843, Book III, Chap. V, 2) One of the clearest approaches to this idea was developed by John Mackie. According to Mackie s theory (Mackie 1974, p. 62; Maslen 2012), the notion of cause can be defined as follows. B.1.1 INUS condition. Event c is a cause of e if and only if c is an insufficient but necessary, non-redundant part of an unnecessary but sufficient, fully actualised condition for e. This concept of a cause involves the notion of a sufficient condition. Simply put, an event e is a sufficient condition of an event f if and only if the proposition E describing e, in conjunction with some laws, logically entails the proposition F, which describes f. This entailment is a key in this definition of the concept of cause. The regularity is captured in the laws that permit the entailment for example, universal propositions like All events similar to a together with events similar to b are followed by events similar to c. Descriptions of necessary conditions also express regularities. Consider Mackie s characterisation of sufficient and necessary conditions (1974, p. 62): The Author(s) 2016 E. Céspedes, Causal Overdetermination and Contextualism, SpringerBriefs in Philosophy, DOI 10.1007/978-3-319-33801-9 77
78 Appendix B: The Regularity Theory of Causation [W]e are now using letters to stand for types of event or situation, and X is a necessary condition for Y will mean that whenever an event of type Y occurs, an event of type X also occurs, and X is a sufficient condition for Y will mean that whenever an event of type X occurs, so does an event of type Y. Consider, for instance, the simple scenario in which Suzy throws a rock at a bottle and breaks it. Let ST be the proposition describing Suzy s throw and BS the proposition describing the bottle being shattered. Surely, the mere event of Suzy s throw is not a sufficient condition for the bottle s shattering. Further conditions should be considered for instance, atmospheric conditions, the bottle s constitution, the absence of obstacles, and the rock s mass and velocity. Let C be the proposition describing all these conditions, and assume that proposition BS follows from C & ST. Then, Suzy s throw and the conditions described by C form a sufficient condition for the shattering of the bottle. Notice that (C & ST) does not represent a necessary condition: The bottle might have been broken by other factors, like falling brushwood or an earthquake. However, Suzy s throw is a necessary part of that particular sufficient condition and is, thus, a cause of the bottle s shattering. B.2 Preemption and Regularity Cei Maslen (2012) examines the problem of preemption and its impact on Mackie s regularity account. She argues that the problem remains unsolved for this theory, despite the various ways of describing it. Consider the following description of a case of late preemption, involving two persons, A and B, throwing a ball (Strevens 2007; Maslen 2012). Late preemption. A and B both want to break a jar. Each throws a ball at the jar, and A s ball arrives first. By the time B s ball arrives, the jar is already broken, but if A s ball had missed, then B s ball would have hit and broken the jar instead (Maslen 2012, p. 423). Disregarding irrelevant details, this is basically the same scenario as the late preemption case of Suzy and Billy throwing rocks. I will use this story in particular to follow Maslen s examination. According to the regularity theory, A s throw is correctly considered as a cause of the jar breaking. Consider the set W 1 as a sufficient but unnecessary condition for the jar breaking: W 1 =[(A threw the ball towards position x with a momentum greater than p) & (Nothing interfered with the trajectory of A s ball) & (Facts about the structure of the jar)] In what follows, the laws will be assumed implicitly. Thus, the description of the jar s breaking follows from W 1, together with the laws. Since A s throw is a necessary part of W 1, it also counts as a cause of the breaking, according to Mackie s regularity account. Remember that the problem originated by late preemption for
Appendix B: The Regularity Theory of Causation 79 the counterfactual account of causation was that A, in this case, does not count as a cause. In contrast, the problem for the regularity account of causation that originates from this case is that B also counts as a cause, which seems clearly unintuitive. Consider now the set W 2, involving B s throw: W 2 =[(B threw the ball towards position x with a momentum greater than p) & (Nothing interfered with the trajectory of A s ball) & (Facts about the structure of the jar)] Since B s throw is also a necessary part of a set of conditions that is sufficient for the breaking of the jar, it is also regarded as a cause of the breaking, which is incorrect. In order to correct this in favour of the regularity theory of causation, consider the following set of conditions describing the same story and involving A s throw (Maslen 2012, p. 424): U =[(A threw the ball towards position x at time t with a momentum greater than p) & (The jar was at position x at time (t + 1)) & (Nothing interfered with the trajectory of A s ball) & (Facts about the structure of the jar)] Again, A s throw counts as a necessary but insufficient part of this sufficient but unnecessary set of conditions: A s throw is a cause of the jar breaking. Now consider the following condition involving B s throw (Strevens 2007; Maslen 2012, p. 424): V 1 =[(B threw towards position x at time (t + 1) with a momentum greater than p) & (The jar was at position x at time (t + 2)) & (Nothing interfered with the trajectory of B s ball) & (Facts about the structure of the jar)] Notice that V 1 is not a fully actualised sufficient condition because the jar was not actually there when B s ball passed through position x: It was already broken. Thus, B does not count as a cause of the breaking. Notice that this step towards helping the regularity account in a case of preemption somehow involves the strategy that I am defending in this work: fine-graining of the set of variables considered. This strategy is crucial to the discussion about whether Mackie s regularity theory can account for preemption cases correctly. In contrast to conditions W 1 and W 2, temporal parameters are included in conditions U and V 1 in order capture the asymmetry of the preemption scenario i.e., that person B is not a cause of the jar breaking. In order to argue against the idea that the regularity account of causation can truly account for the asymmetry of preemption, Maslen (2012, p. 426) constructs a fully actualised sufficient condition for the jar s breaking that includes B s throw as a necessary, non-redundant part. This is condition V 4, below. But let us first look at the steps leading to it. Maslen considers a necessary condition for the breaking of the jar expressed with a counterfactual: Had A s throw been absent, the jar would have been intact at time (t + 2) i.e., the time at which B s ball is supposed to arrive at that position. Since this counterfactual is true, the condition represented by it is also actualised. The following is a condition involving this new part (Maslen 2012, p. 427):
80 Appendix B: The Regularity Theory of Causation V 2 =[(B threw towards position x at time (t + 1) with a momentum greater than p) & (If A s throw had been absent then the jar would have been at position x at time (t + 2)) & (Nothing interfered with the trajectory of the ball) & (Facts about the structure of the jar)] Condition V 2 is a fully actualised condition, with B s throw as a necessary part. But V 2 is not sufficient for the breaking of the jar. The condition that the jar is actually at position x at time (t + 2) is needed. If we added the condition that A s throw was absent, together with the counterfactual, If A s throw had been absent, then the jar would have been at position x at time (t +2), then we would have the condition that the jar is at position x at time (t + 2). The problem is the following: Adding the false condition that A s throw was absent would make the whole set a condition that is not fully actualised. Thus, a further fact must be added: that nothing interfered with the trajectory of A s ball. By adding this fact, Maslen (2012, p. 428) ensures that A s throw is not regarded as absent and that the breaking of the jar follows from the following fully actualised set of conditions: V 3 =[(B threw towards position x at time (t + 1) with a momentum greater than p) & (The jar was at position x at time (t + 1)) & (Facts that make it true that if A s throw had been absent then the jar would have been at position x at time (t + 2)) & (Nothing interfered with the trajectory of balls A or B) & (Facts about the structure of the jar)] Since facts about A s throw are also considered, condition V 3 is sufficient, together with the laws, for the occurrence of the jar s breaking. In order to avoid the inclusion of a counterfactual conditional in the conditions, Maslen now introduces the facts that make the counterfactual true. Hence, the last step is to describe such facts. These are facts about the forces affecting the jar and the absence of any other projectiles (together with the laws of mechanics). If forces on the jar could move it from its position at time (t + 2) or if another projectile was about to hit the jar at time (t + 2), the jar would not be at position x at time (t + 2), even if A s throw was absent. Here is the new set of conditions including these facts (Maslen 2012, p. 428): V 4 =[(B threw towards position x at time (t + 1) with a momentum greater than p) & (The jar was at position x at time (t + 1)) & (Nothing interfered with the trajectory of balls A or B) & (There were no further incoming projectiles) & (Facts about the initial forces on the jar) & (Facts about the structure of the jar)] With V 4, we arrive at a sufficient and fully actualised condition for the jar s breaking, of which B s throw is a necessary, non-redundant part. Thus, B s throw is incorrectly considered as a cause of the breaking in the light of Mackie s regularity theory of causation. I will not discuss Maslen s criticism in greater depth and will regard it as plausible. What really interests me is that, with regard to the strategy of choosing the appropriate fine-graining of variables and events, the regularity account is not an exception. The epistemic context i.e., the considered descriptions of conditions determines the evaluation of our causal claims. As I have shown, the set of
Appendix B: The Regularity Theory of Causation 81 conditions can be changed and fine-grained to describe one and the same situation. On this basis, our answers as to whether some event involved in those conditions counts as a cause of another event will depend on such fine-graining. References Mackie, J. (1974/1980). The cement of the universe: A study of causation. Oxford: Clarendon. Maslen, C. (2012). Regularity accounts of causation and the problem of pre-emption: Dark prospects indeed. Erkenntnis77(3):419 434. Mill, J. S. (1843/1911). A system of logic: Ratiocinative and inductive. London: Longmans, Green (1911). Strevens, M. (2007). Mackie remixed. In Campbell, J., O Rourke, M. & Silverstein, H. Causation and explanation. Cambridge: MIT Press.
Appendix C Fundamental Events Douglas Kutach (2013) develops the notion of a maximally fine-grained event. He begins with the notion of material content (also called matter or stuff ), which is whatever a physical theory about reality postulates such as, for instance, particles, fields or forces (2013, p. 57). An arena is the space that contains the material content and is divided into regions. Kutach s theory distinguishes fundamental events from derivative events. The notion of a fundamental event is characterised by Kutach as follows. C.1 Fundamental event. Stated simply, a fundamental event is an arrangement of fundamental quantities instantiated in some region. [ ] Another way to put this is that fundamental events are maximally fine-grained wherever they define their region and material content. (Kutach 2013, p. 60) This notion of a fundamental event not only fits Dowe s account of physical causation, but also with our requirement about the fine-graining of events being context sensitive. Thus, the fine-graining of the fundamental events will depend on the material contents postulated by our best scientific theories (or by our best epistemic systems). We can define the concept of a derivative event as any event that is not fundamental. Regarding causation, the only kind of derivative event that is of interest is the coarse-grained event (Kutach 2013, p. 60). The notion of a coarse-grained event can be defined as follows. C.2 Coarse-grained event. A coarse-grained event is a set of possible fundamental events. Coarse-grained events are described as the different ways a fundamental event may occur. Kutach exemplifies this as follows: For illustration, let e be some actual fundamental event that instantiates one moment of the first moon landing. A hypothetical alteration of e that slightly shifts just one of its molecules will result in a possible fundamental event that is numerically distinct from e. Yet, for practical purposes, we can often conceive of this moment of the first moon landing as being The Author(s) 2016 E. Céspedes, Causal Overdetermination and Contextualism, SpringerBriefs in Philosophy, DOI 10.1007/978-3-319-33801-9 83
84 Appendix C: Fundamental Events insensitive to the precise position of a single molecule. When we think of a set of multiple possible instances of whatever counts as (near enough) this moment of the first moon landing, we are thinking of it as a coarse-grained event. (Kutach 2013, p. 60) In the intuitive descriptions of the examples discussed so far, we have considered only coarse-grained events. Thus, the fine-graining strategy in preemption cases, for example, consists in considering smaller coarse-grained events for the description of the cause and the effect i.e., considering a set involving fewer possible fundamental events than originally considered. Take, for instance, late preemption. The initial problem is that the bottle s breaking does not causally depend on Suzy s throw: If she had not thrown, the bottle would still have been shattered by Billy s rock. The solution is to acknowledge first that the event described as The bottle shattered is too coarse-grained. We must consider a smaller set of possible fundamental events, such as the event described as The bottle shattered at time (t +1). Now consider the fine-graining of Suzy s and Billy s throw in such a way that Suzy threw her rock at time t and Billy his rock at time (t + 1). Thus, the breaking causally depends on Suzy s throw: If Suzy had not thrown at t, the bottle would not have shattered at (t + 1). Furthermore, if we use only fundamental events to describe this case of late preemption, there will also be a causal dependence. Let c be the fundamental event describing Suzy s actual throw and e be the fundamental event describing the actual shattering of the bottle. Then, if c had not occurred, e would not have occurred, even if Billy s late throw is considered. Reference Kutach, D. (2013). Causation and its basis in fundamental physics. Oxford University Press.
Index A Analysis (causal a.), 2 4, 10, 24, 25, 27, 35, 36, 53, 54, 57, 65, 67, 69, 71 73 C Causal chain, 3, 4, 7 11, 21, 60, 70 Causal claim, 1, 13, 14, 23, 36, 39, 44, 51, 52, 56, 57, 66, 69 71 Causal contextualism, 1, 13, 14, 16 18, 44, 49, 51, 52, 56, 59, 66, 69, 72, 73 Causal dependence, 2 4, 6 9, 11, 22, 24, 30 32, 35, 36, 57, 69, 70 Causal exclusion, 59 61, 65 Causal graph, 19, 21, 35, 42 Causal influence, 11, 12, 19, 22, 23, 70, 72 Causal model, 14, 18, 19, 21 24, 27, 34 36, 40 Causal perspectivalism, 15, 16 Causal process, 14, 49 57, 71, 72 Causal relevance, 35 Causation actual c., 18 20, 35 c. as Connection, 52 counterfactual account/analysis of c., 4 6, 8, 11, 14, 27, 30, 36, 40, 57, 69 direct c., 30, 33, 35 joint c., 5, 6 mental c., 59, 60, 63 probabilistic account/analysis of c., 14, 39, 44, 46, 47, 72 ranking-theoretic account/analysis of c., 14, 30, 31, 35, 40, 46, 71, 72 regularity theory/analysis of c., 2 4, 24, 25, 35, 53, 57, 65, 69, 72, 73 Causation as difference-making, 72, 73 Circularity, 24, 25 Comparative similarity, 75 Consciousness, 66, 67 Conserved Quantity, 49, 51 56, 71, 72 Context, 17, 18 background c., 42 44, 46 c. dependence, 32, 62, 65 c. of explanation, 61, 65 epistemic c., 13, 23, 36, 71, 72, 80 metaphysical c., 69 Correlation, 42 Counterfactual, 2, 3, 6 11, 14, 20, 24, 30, 36, 40, 54, 55, 57, 69, 70, 72, 73 Counterfactual dependence, 3, 10, 14, 20, 22, 24, 32 34, 55 D Degree of belief, 28, 32, 40, 72 Disbelief, 27, 28 E Events, 17, 18 derivative e., 83 fundamental e., 83, 84 Explanation, 15, 60 66, 69, 73 Explanatory accessibility, 63 Explanatory exclusion, 59 61, 62 65 F Fine-graining, 5, 13, 17, 20, 22, 23, 30, 31, 39, 48, 49, 51, 53, 56, 57, 73 Frequency, 39 I Interaction, 50 53, 55, 56, 71, 72 Interpretations of probability, 39 Intervention, 24 Intrinsicness, 8 11, 70 The Author(s) 2016 E. Céspedes, Causal Overdetermination and Contextualism, SpringerBriefs in Philosophy, DOI 10.1007/978-3-319-33801-9 85
86 Index L Limit assumption, 75 M Mark transmission, 49, 50 Merged contexts, 64 O Omission, 54 56, 73 Overdetermination asymmetric o., 6, 11 symmetric o., 5, 6, 10, 13, 14, 17, 31, 34, 47, 48, 65 P Physical explanans, 61 65 Population, 35, 44 Possible world, 25, 39 Preemption early p., 6, 18 20, 30 late p., 7, 12, 21, 31, 47 Prevention preemptive p., 8 10, 54 56, 73 Prima facie causation, 39 Probability, 17, 18 conditional p., 40 Propensity, 39 Q Quasi-dependence, 8, 22 R Ranking function, 17, 18 negative r. f., 28 positive r. f., 28 Reason, 17, 18 necessary r., 29 sufficient r., 29, 30 supererogatory r., 29 weak r., 29 Reduction/reductive, 25, 35, 36, 73 S Signal, 50 Spurious cause, 5, 41 Supererogatory cause, 30, 31, 65, 71 T Theory of relativity, 50 Transitivity of the causal relation/transitive causal relation, 3 Trumping, 11, 12, 21, 23, 32, 45 48, 53, 70 V Variable, 17, 18 endogenous v., 19 exogenous v., 18, 19 W Would-cause-counterfactual, 73