Chapter 2: The HP Definition of Causality
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Chapter 2 The HP Definition of Causality The fluttering of a butterfly’s wing in Rio de Janeiro, amplified by atmospheric currents, could cause a tornado in Texas two weeks later. Edward Lorenz There is only one constant, one universal. It is the only real truth. Causality. Action, reaction. Cause and effect. Merovingian, The Matrix Reloaded In this chapter, I go through the HP definition in detail. The HP definition is a formal, mathematical definition. Although this does add some initial overhead, it has an important advantage : it prevents ambiguity about whether A counts as a cause of B. There is no need, as in many other definitions, to try to understand how to interpret the words. For example, recall the INUS condition from the notes in Chapter 1. For A to be a cause of B under this definition, A has to be a necessary part of a condition that is itself unnecessary but insufficient for B. But what is a “condition”? The formalization of INUS suggests that it is a formula or set of formulas. Is there any constraint on this set? What language is it expressed in? This lack of ambiguity is obviously critical if we want to apply causal reasoning in the law. But as we shall see in Chapter 8, it is equally important in other applications of causality, such as program verification, auditing, and database queries. However, even if there is no ambiguity about the definition, it does not follow that there can be no disagreement about whether A is a cause of B. To understand how this can be the case, it is best to outline the general approach. The first step in the HP definition involves building a formal model in which causality can be determined unambiguously. Among other things, the model determines the language that is used to describe the world. We then define only what it means for A to be a cause of B in model M. It is possible to construct two closely related models M1 and M2 such that A is a cause of B in M1 but not in M2. I do not believe that there is, in general, a “right” model; in any case, the definition is silent on what makes one model better than another. (This is an important issue, however. I do think that there are criteria that can help judge whether one 9 10 Chapter 2. The HP Definition of Causality model is better than another; see below and Chapter 4 for more on this point.) Here we already see one instance where, even if there is agreement regarding the definition of causality, we can get disagreements regarding causality: there may be disagreement about which model better describes the real world. This is arguably a feature of the definition. It moves the question of actual causality to the right arena—debating which of two (or more) models of the world is a better representation of those aspects of the world that one wishes to capture and reason about. This, indeed, is the type of debate that goes on in informal (and legal) arguments all the time. 2.1 Causal Models The model assumes that the world is described in terms of variables; these variables can take on various values. For example, if we are trying to determine whether a forest fire was caused by lightning or an arsonist, we can take the world to be described by three variables: FF for forest fire, where FF = 1 if there is a forest fire and FF = 0 otherwise; L for lightning, where L = 1 if lightning occurred and L = 0 otherwise; MD for match dropped (by arsonist), where MD = 1 if the arsonist dropped a lit match and MD = 0 otherwise. If we are considering a voting scenario where there are eleven voters voting for either Billy or Suzy, we can describe the world using twelve variables, V1, . . . , V11, W, where Vi = 0 if voter i voted for Billy and V1 = 1 if voter i voted for Suzy, for i = 1, . . . , 11, W = 0 if Billy wins, and W = 1 if Suzy wins. In these two examples, all the variables are binary, that is, they take on only two values. There is no problem allowing a variable to have more than two possible values. For example, the variable Vi could be either 0, 1, or 2, where...