Some Thoughts on Causation

Causation is an ambiguous concept. The particular meaning depends on its context of use. I will explore some of these concepts in detail, in particular the counterfactual and configurational views:

If you are trained in Econometrics, Statistics, Epidemiology, or biomedical sciences, causation is normally thought of as a counterfactual conditional within the framework of potential outcomes. This was introduced by Donald Rubin and others when they proposed something known as the Rubin Causal Model. Causation within this framework is based on the idea that there is "no causation without manipulation"; you cannot infer causation without intervening on a model. These methods seek to address the fundamental problem of causal inference: when intervening on a set of units who will receive a treatment, we fundamentally cannot observe the counterfactual state of the world where they did not receive the treatment. This is the basic problem experimental design addresses. Mathematically we can describe it as:

That is, the expected value of the treatment effect as the expectation of the difference between two alternative states of nature. Since an Expectation is, by definition, an average, this estimate is known as the Average Treatment Effect (ATE) across a representative sample. This is what researchers seek to discover when conducting empirical studies whether it be in medicine, social sciences, or public policy. Rubin defines a causal effect as:

Intuitively, the causal effect of one treatment, E, over another, C, for a particular unit and an interval of time from $t_{1}$ to $t_{2}$ is the difference between what would have happened at time $t_{2}$ if the unit had been exposed to E initiated at $t_{1}$ and what would have happened at $t_{2}$ if the unit had been exposed to C initiated at $t_{1}$ : 'If an hour ago I had taken two aspirins instead of just a glass of water, my headache would now be gone,' or 'because an hour ago I took two aspirins instead of just a glass of water, my headache is now gone.' Our definition of the causal effect of the E versus C treatment will reflect this intuitive meaning."

You may also come across something known as do-calculus, formalized by Judea Pearl, which generalizes this notion of intervention over a causal model. The basic idea behind this is that causality is fundamentally unidirectional, but the current formalization based on statistical models rely on equality operators, we can never truly rule out whether X -> Y or Y -> X; Equality allows for both interpretations. Furthermore, it is difficult to account for confounders, but in Pearl's framework you can explicitly model the pathways between hypothetical factors within your model. You can also do this in Structural Equation Modeling (SEM) but Pearl generalizes beyond the assumptions of that set of models. Pearl's framework is rich in that it can encode causal assumptions (making them testable), explicitly controls confounding variables, gives us a way of measuring direct/indirect effects through the use of mediation analysis, allows us to test for external validity and identify sample selection bias, and algorithmically discovers causal effects (assuming the model is properly specified). We can summarize some of these ideas from Rubin and Pearl with Pearl's causal hierarchy:

These frameworks are incredibly powerful and are part of my basic frame of reference when engaging in empirical analyses. However, I am not going to cover this framework in depth right now, even though I hold it dear to my heart, given my training in econometrics and statistics. For an in depth technical presentation of the ideas refer to Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction, this is the most comprehensive overview of the empirical methods based on the Rubin Causal Framework.

(Extra source to clarify terminology: Structural Causal Models (SCMs))

My point of introducing with this causal framework is to provide a benchmark to contrast two alternative conceptions discussed in some of the philosophical literature. I came across these papers as a response to the fact that statistical methods can only provide a probabilistic inference of the effect of some intervention on a population, given the estimated effect in the sample. Remember what the ATE is; it's the expected effect on the outcome if everyone in the population were treated. This explicitly means that the causal effect will not be the same for everyone; the effect of some intervention is statistical implying a distribution around the average. In other words, there are heterogenous effects; individuals are different, so you could imagine that some intervention affects each person in a systematically different way. Mathematically:

The subscripts i,j simply represent the i'th and j'th individual within your population of interest. Furthermore, there are random effects that are path dependent. You could imagine a scenario where someone receiving some dose of medicine to treat hypertension will react differently than someone without hypertension, based on their medical history. I am not saying this to be unnecessarily skeptical. These are well known problems when analyzing interventions and there are loads of methods used to reduce the effect of these sources of bias.

What I became particularly interested in was the A in ATE. Averages are used to reduce a set of information into a single informative measure that represents some property of the sample, in order to generalize over that sample. Generalizations necessarily abstract away specific details that may be relevant to understand in particular situations. So for example, if I wanted to discover the cause of a house fire (denote the event as Y), by referring to a statistical analysis of common factors typically associated with house fires, I can infer probabilistically (with some level of confidence) that the cause of individual fire Y was due to the presence of some set of factors Z=(X,W,....). This is just what an expectation is; in the absence of case-specific information, the expected value would be your best bet (assuming tight variance, no skewness, and other facts about the distribution). But we actually still don't know what caused the event in that particular instance. Probabilistic methods necessarily cannot provide that information. This is the difference between singular and general causation which began bothering me. However, we use the word "cause" when referring to both inferential scenarios.

Consider a hypothetical scenario of singular causation. Would we want to use a statistical framework for discovering what "the cause" was? Some person unfortunately is murdered. We have some facts discovered at the scene of the crime and a set of possible suspects. We run a statistical analysis and find that, with high confidence, suspect A is most likely the person to commit the crime. We therefore find them guilty. Rather than identifying case specific information implicating them of their guilt, we use our best guess given relative frequencies and proportions. There is obviously something odd about this. When I was in graduate school learning about the Rubin Causal Model, this dichotomy became immediately apparent to me, so I began reading literature from other disciplines such as Jurisprudence and philosophy in order to see how they conceptualize this ambiguous notion of "causality". In the process, I began asking: What would count as evidence when inferring singular causation? If we are not seeking to generalize from repeated events, how can we validate if something was the cause of another thing? Is "cause" inherently coupled with philosophical necessity? Are causal feedback loops somehow different than linear causation (this is particularly interesting since within the Rubin framework causality is exogenous, endogenous relationships are not causal)? How do we demarcate the boundaries of our causal explanation without resulting in an infinite regress? These sorts of questions might not be explicitly addressed in an empirical discipline but they nevertheless are something to consider when learning and applying empirical methodologies to real world data generating processes.

In my studies I came across three related but distinct conceptions of causality that opened my eyes to alternative explanatory frameworks. I first came across J.L. Mackie's paper Causes and Conditions. This lead me to Richard W. Wright's paper Causation in Tort Law. Both share a common inspiration; the legal scholars Hart and Honore authored Causation in the Law which addresses questions of causality in the fields of torts and criminal law. From what I can gather, Mackie and Wright attempt to construct a generalized conception of causality from the work of Hart and Honore. As a side note, H.L.A. Hart was incredibly influential within Anglo-American jurisprudence; his book The Concept of Law is a must read, in my opinion, for anyone interested in the structure of legal systems. Anyway, if you are unfamiliar with tort law, a tort is a civil wrong that causes the claimant to suffer loss or harm, resulting in liability for the person who committed the tortious act. On the surface, this sounds trivial, but determining legal responsibility has been, and still is, a matter of debate in which there hasn't been any clear decision criteria proposed that can definitively state X rather than Y caused Z. Wright attempts to explicate a set of criteria which can unambiguously distinguish between competing possible causal explanations in the field of torts. Mackie's work also provides criteria for determining "the cause" in a given situation that extends beyond the legal framework. Both of these criteria provide accounts of singular causation. This makes sense in the context of a tort since we are not seeking to extrapolate a causal generalization from a finite sample, but rather we want to ascertain responsibility. Both of these papers exposed me to a rich plethora of research in jurisprudence and evidence based reasoning; I really could dedicate an entire blog to this work. For the sake of brevity, we will focus on the latter two papers and one final paper from Jonathan Schaffer titled Contrastive Causation. This paper extends and alleviates some of the problems in the former two. I will not go too far in depth in any of these papers. I just want to highlight some of the major concepts from each.

Let's start with Causes and Conditions. Mackie wants to extend the traditional conception of "cause" beyond merely something that is a necessary and sufficient preceding condition. He begins by providing an example of a house fire. Investigators conclude that "the cause" of the fire was an electrical short circuit in a certain place. Mackie asks "What is the exact force of their statement?". Clearly, the short circuit is not a necessary condition for a house fire. You can imagine many situations where house fires were not caused by a short circuit. The investigator is also not claiming the short circuit to be a sufficient condition for the house catching fire; if a short circuit occurred in an area without flammable material presumably we can conclude the house would not catch fire. So the short-circuit is neither a necessary or sufficient condition for the house catching fire. Then in what sense is it "the cause"? Mackie responds to this question by stating:

At least part of the answer is that there is a set of conditions (of which some are positive and some are negative), including the presence of inflammable material, the absence of a suitably placed sprinkler, and no doubt quite a number of others, which combined with the short-circuit constituted a complex condition that was sufficient for the house's catching fire - sufficient, but not necessary, for the fire could have started in other ways. Also, of this complex condition, the short-circuit was an indispensable part: the other parts of this condition, conjoined with one another in the absence of the short-circuit, would not have produced the fire. The short-circuit is said to have caused the fire in thus an indispensable part of a complex sufficient (but not necessary) condition of the fire.

From this analysis Mackie derives his famous INUS condition: an insufficient but necessary part of a condition which is itself unnecessary but sufficient for the result. When we speak of causation for particular events, this is the condition that Mackie has in mind. If you are unfamiliar with this, I will provide a visual of what I think Mackie is referring to when formalizing this condition.

Consider the visualization above. Each circle represents an INUS set. In other words, each box, if instantiated, can bring about an effect. When Mackie says "a cause is an insufficient but necessary part", he is referring to one element of each set. If the element is not present, the effect will not be brought about; this is just what it means for something to be necessary. However, just because the necessary condition is present, this does not mean the effect will be brought about if the sufficient conditions (remaining letters) are not instantiated for that particular INUS set under consideration. When Mackie mentions that the condition is "itself unnecessary but sufficient for the result", I think he is referring to the fact that just because one of these sets of conditions is not present, does not mean it can't be brought about by another set. Take for example if the first two are not instantiated; the effect can still be brought about if the third is instantiated. So, each circle represents an INUS set, and each star represents the necessary element of each set. And crucially, none of these sets are by themselves "necessary" for the effect to be brought about. In the case of the house fire, the investigator discovers that the short circuit occurred and that other conditions were present; the conjunction of every condition including the short-circuit was sufficient for the event to occur.

Mackie provides a more technical definition of the INUS condition. Let A stand for the INUS condition, let B and ¬C stand for other conditions, positive and negative, which were needed in conjunction with A to form the sufficient condition. The conjunct AB(¬C) represents a sufficient condition for the outcome, and contains no redundant factors, in other words, this conjunction is minimally sufficient. DEF, GHI, etc. are all of the other minimally sufficient conditions for the outcome. The disjunction of all the minimally sufficient conditions for the outcome constitute a necessary and sufficient condition for the outcome; this means "ABC" or "DEF" or "GHI" or etc. all represent necessary and sufficient conditions, and each disjunct represent a minimally sufficient condition. Each conjunct in each minimally sufficient condition represents the INUS condition. More formally, let X stand for B(¬C) in AB(¬C), and Y stand for all other possible minimally sufficient conditions (DEF,GHI,JKF...), then the INUS condition is defined as follows:

A is an INUS condition of a result P if and only if, for some X and for some Y, (AX or Y) is a necessary and sufficient condition of P, but A is not a sufficient condition of P and X is not a sufficient condition of P.

If A is present in each set of minimally sufficient conditions, then A will still be referred to as the INUS condition. Furthermore, if AX is a minimally sufficient condition, there is no need for X to be bound to A; in other words KX can be a minimally sufficient condition as well (AX or KX can bother cause the event); a common set of factors can be coupled with different INUS conditions. In the case that A is minimally sufficient to bring about the outcome, Mackie states that A is "at least an INUS condition".

To summarize it, Mackie states that when someone says "A caused P", there are implicitly saying the following claims:

A is at least an INUS condition of P, that is, there is a necessary and sufficient condition of P which has one of these forms: (AX or Y), (A or Y), AX, A.
A was present on the occasion in question
The factors represent in X, if any, in the formula for the necessary and sufficient condition were present on the occasion in question
Every disjunct in Y which does not contain A as a conjunct was absent on the occasion in question

Falsifying any of these conditions would refute the condensed statement "A caused P". Essentially, these are the assumptions we implicitly rely upon when asserting singular causation. Mackie does not claim these to be exhaustive but nevertheless they are an essential feature of what causation is supposed to mean. In the context of argumentation schemas (presented by Douglas Walton elsewhere), you could probably formulate a set of critical questions that probe whether these assumptions hold in a particular situation. Take number 4 as an example; we could ask the question "are there other minimally sufficient conditions that have been proven to be ruled out on this occasion?". This means this framework is defeasible, like the others we will encounter shortly.

Next Mackie introduces the notion of a Causal Field. This is used to augment his earlier analysis of causal questions. A causal field is a field in which you are looking for the cause of an event. It represents the region that is to be divided by the cause. For example, if you ask, “What causes influenza?”, you may mean in all organisms susceptible to types of the disease, or you may mean to restrict your focus to human beings alone. These represent two different causal fields. A causal field is similar to our sample space; it is our domain of interest. To say that Z is a necessary and sufficient condition for P in F is equivalent to ‘All FP are Z and all FZ are P’, where ‘F’ refers to the causal field. Obviously there is still indeterminacy in the choice of the causal field. Mackie recognizes this problem; essentially it is a question of relevance, which I take to be equivalent to the formalization of a causal model in Judea Pearl's framework. The causal field is the model of potential interacting components of a causal model. Mackie recognizes that in general this is rather arbitrary, but I think when considered in light of Pearl's framework, you can select more reasonable causal fields over potentially inferior causal fields and make explicit causal assumptions. After introducing the notion of causal field, Mackie restates the phrase "A caused P" to actually mean "A caused P in relation to field F", leading us to the more general statement: "A is at least an INUS condition of P in the field F, that is, there is a condition which, given the presence of whatever features characterize F throughout, is necessary and sufficient for P, and which is of one of these forms, (AX or Y), (A or Y), AX, or A". Causal fields are normally taken for granted, unless explicitly modeled in Pearl's framework.

I think Mackie and I diverge in his analysis of identifying relevant and irrelevant causal fields. According to Mackie, "the field in relation to which we are looking for a cause of this effect, or saying that such-and-such is a cause, may be definite enough for us to be able to say that certain facts or possibilities are irrelevant to the particular causal problem under consideration, because they would constitute a shift from the intended field to a different one"; on the surface I agree with this statement, but as he continues "thus if we are looking for the cause or causes of influenza, meaning its cause in relation to the field human beings, we may dismiss as not directly relevant, evidence which shows that some proposed cause fails to produce influenza in rats". I think it's quite bold to assert that potentially analogous causal fields are irrelevant when considering the question at hand; particularly in medical trials, it is common to test some intervention on some animal with comparable biological mechanisms to humans, extrapolating the effects to the intended unit of analysis. The role of analogy in identifying the causal field plays a prominent role in causal reasoning. I don't think this is the main thrust of Mackie's argument; rather he is just using this as an example to demonstrate that we engage in some process of eliminating irrelevant causal factors from our causal field or causal model. What I am stating is that, analogical arguments appear to be a component of the process by which we reduce or expand the causal field to eliminate irrelevant factors or include relevant factors. This is just something to consider. Mackie modifies the INUS condition to include a causal field to eliminate potential for an infinite regress. For example, in the case of the house fire, one can include in the set of negative conjuncts "an item such as the earth's not being destroyed by a nuclear explosion"; but obviously this seems irrelevant to the case in point and wouldn't provide any utility in determining the INUS condition, essentially it is outside of our causal field. I'll talk a bit more about these "negative conditions" later when we introduce Schaffer's work. Just remember that In all causal statements, we need to specify the causal field. The cause is required to differentiate, within a wider region in which the effect sometimes occurs and sometimes does not, the sub-region in which it occurs: this wider region is the causal field. Conceptually, I think this is similar to the reference class problem in statistics; when estimating the probability of some event X, it is not always clear what should be the reference class by which you calculate the probability. For example, when estimating the probability of a car crash, you could use as your reference class the number of accidents in all of history, the number of accidents occurring in with cars of this particular make, or the number of accidents occurring in some particular geographical region. In statistical analyses, there is normally a reason for selecting one reference class over another. This is particularly similar to how we determine the causal field in a given situation. It seems to be primarily driven by the application and subject matter knowledge.

Mackie claims this analysis extends to causal generalizations. When we say that "Eating sugar causes cavities" we are expressing some INUS condition with respect to a causal field. This extends to functional relationships, in which we take into consideration the magnitude of the effect on P by some causal factor, and where concepts such as necessity and sufficient are special cases. Essentially, in the case of functional dependence, there are combinations of negative and positive factors that can bring about some effect, with respect to some causal field (reference class). There are different complex combinations containing an INUS condition that causal effect an outcome. In this context, we speak of the total cause, the complete set of factors on whose individual magnitudes combine to yield the magnitude of P, given the field F. According to Mackie "that is, variations of P in F are completely covered by a formula which is a function of the factors in this 'complete set', and of these alone. The total cause is analogous to a necessary and sufficient set". So in the situation where eating sugar causes cavities, sugar acts as an INUS condition among a set of factors, positive and negative, effecting the outcome P. Identifying the causal forces "on a given occasion" can be found by method of partial derivatives ceteris paribus.

I mentioned earlier that in Pearl's framework we are able, in principle, to determine directionality given the do calculus operator. Mackie was obviously unaware of this development in mathematics since he wrote his paper before Pearl's book on Causality. Given a directed acyclic graphical representation of some causal system, or causal field in Mackie's terms, we can infer whether X -> Y; not merely correlation since we aren't working with equality operators any longer. Direction of causation is mentioned in Mackie's paper so I'd like to touch on it briefly before moving onto the next account of causation. Mackie frames the problem as a problem with determining the direction of time, however, which I am unaware whether Pearl explicitly acknowledges this in his analysis. I know in the Rubin framework, manipulations precede their effects, and we are in control of these interventions so there is no ambiguity as to what comes first (cause or effect). I will list out Mackie's concerns:

Intuitively, there is a relation that may be called "causal priority" implicit in the phrase "A caused P" that is unidirectional. For this relation to hold, A -> P; there is an asymmetric relation between cause and effect. This is exemplified in econometrics with notions such as Granger Causality, although "cause" in this context is meant to be predictive causality which is different than the notion of control and manipulation.
The relation above is not identical with temporal priority. This can be quite confusing for some since it would imply backwards causation. Our intuitions of causation are inherently fused with temporality, so a future cause effecting a present outcome seems mind boggling. However, it should be conceivable that there should be backwards causation (also known as Retrocausality); so we need not limit the possibilities of causal order. The possibility of backwards causation is best seen in physical models of the universe, so if you're interested definitely refer to that literature. Going down this path however becomes very metaphysical so be aware.
Mackie's analysis does not explicitly reference causal ordering. Our intuitive understanding of necessity and sufficiency includes such a reference. Causal priority is separate from the notion of necessary and sufficient condition.
The statement "A is at least an INUS condition of P is not synonymous with "P is at least an INUS condition of A"; this is just a reiteration of the fact that the causal relation is asymmetric.
Causal priority is partly based on the direction of explanation. Causal priority is also linked with controllability. If there is a relation between A and B, and we control A, and the relation between A and B still holds, we determine that A is causally prior to B. Our rejection of B being causally prior to A rests on our knowledge that our action is causally prior to A; thus shifting the question of causal priority. It has yet to be answered. In other words, we assume A is causally prior to B because we know we manipulated A before B, but we still have not explained what causal priority is. In my view as a non-philosopher, I would simply ignore this and fall back on a Rubin framework. What I mean to say is that causal priority does not need an explanation at this level. I am sure there are counterexamples. You can imagine a situation where you were not in complete control, assuming you were, mistaking your causal action as the "causally prior" intervention on the system without considering there was a confounder. Mackie mentions something like this. He says that if we randomize the effect of A on B we can argue that A is causally prior to B. In essence this is what econometrics is all about, especially with models like Instrument variables estimation. Mackie says that "it is true that our knowledge of the direction of causation in ordinary cases is thus based on what we find to be controllable", and on what we either find to be random or find that we can randomize; but this cannot without circularity be taken as providing a full account either of what we mean by causal priority or how we know about it".

The last part, number 5, is very interesting from an inferential standpoint. Mackie mentions it by referring to something Popper mentioned in "The Arrow of Time". Mackie states that "If a stone is dropped into a pool, the entry of the stone will explain the circular waves. But the reverse process, with contracting circular waves, 'would demand a vast number of distant coherent generators of waves the coherence of which, to be explicable would have to be shown ... as originating from one center.' That is, if B is an occurrence which involves a certain sort of 'coherence' between a large number of separated items, whereas A is a single event, and A and B are causally connected, A will explain B in a way in which B will not explain A unless some other single event, say C, first explains the coherence of B". Think about it this way; suppose you observe an effect and are asked what the cause was, the potential number of causal explanations, explodes. When you tinker with something in a laboratory, you can control variables in interest and observe that the stone caused the waves. Without direct intervention, observing the waves and inferring a cause leads to a vast number of potential explanations; were the waves caused by a fish? Were they caused by a human? You see the point. Causation in this case (more formally known as causal discovery) is an Inverse Problem. Imagine you walk into a kitchen and see a puddle of water on the floor, and I ask you to infer the shape of the ice that melted resulting in the puddle. Inverse problems are truly amazing; there are a ton of Bayesian approaches. In econometrics, the goal is to do exactly what Popper recommends; identify a proper Instrumental variable C that can explain both A and B. If you are interested in more of these methods I recommend reading Mostly Harmless Econometrics.

Ok so this blog is turning out to be much longer that "brief". I just wanted to show Mackie's framework for determining singular causation because RCT's cannot infer from the population to the individual; something people seem to literally ignore and make this unfounded leap in inference anyway. Actually, I should be more clear; statistical methods allow you to infer about an individual data point the expected value, given assumptions about the distribution, data generating process, and functional form. In essence, it is a best guess about an observation, given information about the population or sample. Nevertheless, you do not have case-specific information allowing you to infer singular causality. The next account of singular causation comes from legal scholar Richard Wright in his paper Causation in Tort Law. Wright is identifying issues with the "But-For" test of causality in cases involving tortuous behavior. He offers a solution that seems a lot like Mackie's. His paper is really long, 100 pages actually, about 10 times longer that Mackie's paper; so I am not going to dive super in depth.

Remember, Tort law is primarily concerned with identifying whether the tortuous aspect of the defendants behavior caused some sort of negative outcome to the prosecution. This is all about liability; who ought to be responsible for something that happened. Legal scholars have written about this in depth; for example Michael S. Moore has written a book Causation and Responsibility where he discusses the inherent coupling between causation and ascription of moral or legal responsibility for events. He addresses issues such as foreseeability of the harm, complicity, and the use of risk analysis to attribute responsibility. I think people normally conceive of causation as something scientific and disconnected from concepts such as morality, but there are obvious links between our conception of causality and moral responsibility. We should also note that responsibility surely entails causation; but it should also be clear that causation does not entail responsibility. Does "responsibility for" and "caused" mean the same thing? If you caused something, does that mean you should be responsible for something? Intuitively, we can agree that directly causing harm to someone requires accountability. But what about scenarios where you are part of a collective? Or what about situations where there is not immediate harm? Or if it is undetectable, but cumulatively leads to gradual harm, such that after approaching a threshold the harm becomes apparent? These sort of scenarios require that we are able to correctly identify causation with the given relevant facts pertinent to the situation. This is a lot easier said than done.

We should consider a few more examples to illustrate the need of an effective criteria for determining causation. How we conceive of causality will directly effect who we decide is liable in situations concerning torts. There are obviously clear cases where someone pulls the trigger of a gun causing the death of someone else. But how about a case where a car manufacturer guarantees 99% reliability of their brake system but someone gets in a collision due to faulty design? Or how about if someone you are responsible for hurts or damages someone else or their belongings; does responsibility transfer to the care taker? How about a case of a negligent city official who fails to warn their citizens of unsafe drinking water, as in the case of Flint Michigan (this is a case of distributed responsibility)? Or what about the difficult problem of identifying a very delayed causal effect after consuming some product? What about something accidental, how do we prove that something was an accident; consider a situation where someone mops a floor with very slippery soap, leaving the general conditions hazardous to anyone walking in the area. They fail to prop up a sign indicating a warning. You walk into the building with some scissors and a friend, but both of you slip, and you accidently stab your friend. Are you to blame? You can think of other situations where an increase of hazard lead to more negative outcomes that wouldn't have happened under safer conditions. What if you have a child that does something bad, and you just happen to be a dead beat parent, but in this particular instance your child's behavior was the result of some external factor outside of your control; should we hold the parent liable on the basis of their poor character or should we identify another cause? What if you poison someone, assuming the effect will complete within 48 hours, but the person you poisoned got in a car crash and died prior to the 48 hours completing; you technically did not cause their death but should you be held accountable (was the poison actually going to kill them)? What if you engineer a new financial product and mislead customers about its true risk profile, culminating in a financial crash that (unknowingly to you) cascades throughout the economy because of predatory lending at the mortgage level, resulting in the decimation of retirement funds in a matter of minutes? It sure seems someone should be held accountable, but who and to what extent? What if you have insurance, so you technically are not liable under the conditions of your contract with the insurer, except under very particular conditions that violate the terms of your agreement; should the insurance agency be able to decide whether to insure you or not? Tort law is very interesting; we are concerned with responsibility, accountability, and repercussions. If we have a faulty understanding of causation, we might be unfairly or unjustly punishing someone. Or we might fail to hold those to account who have negatively effected someone else. A good test will not be overly permissive in what it counts as a causal condition. It will also not be under permissive, failing to account for actual instances of causation. You can think about it in terms of sensitivity and specificity of a statistical test; your causal test should minimize the number of false positives and false negatives, and maximize the number of true positives and true negatives. In other words, if you hypothesize that some tortuous behavior was the cause of some harm, you want to maximize the probability that your categorization was indeed true.

Very briefly, the but-for test can be expressed as "but for the action, the result would not have happened". Wright introduces the NESS test, which states that a particular condition was a cause of a specific consequence if and only if it was a necessary element of a set of antecedent actual conditions that was sufficient for the occurrence of the consequence. Wright contrasts his test with the traditional "but-for" (necessary condition) test, identifying two types of overdetermined causation cases it fails to handle: duplicative causation and preemptive causation. The but-for test states "that an act (omission, condition, etc.) was a cause of an injury, if and only if, but for the act, the injury would not have occurred. That is, the act must have been a necessary condition for the occurrence of the injury". Wright correctly notes that this intuitive conception fails to account for cases where we assert no causation, when there clearly was an identifiable cause. Overdetermined causation are cases where a factor other than the specified action would have been sufficient to produce an injury in the absence of the specified act, but "its effects either (1) were preempted by the more immediately operative effects of the specified act or (2) combined with or duplicated those of the specified act to jointly produce the inquiry" (Wright 1175). In other words, Wright is arguing that overdetermined causation results in a false negative under the but-for (necessity) test.

Wright provides an example of preemptive causation and duplicative causation with the following examples. "For example, D shoots and kills P just as P was about to drink a cup of tea that was poisoned by C. D's shot was a preemptive cause of P's death, C's poisoning of the tea was not a cause because its potential effects were preempted"; this is the case of preemption. In the case of duplication, as an example "C and D independently start separate fires, each of which would have been sufficient to destroy P's house. The fires converge and burn down the house". An application of the but-for test would show that no one caused the house to burn down, and that D's shot did not kill P; obviously both conclusions are absurd. Wright argues that the but-for test is too restrictive in its application when identifying factual causation. Necessity by itself is not sufficient for identifying all instances of causation.

Throughout pages 1777-1788 Wright rebuts most of the attempts to modify the but-for test. I am not going to cover all of these, since this post is supposed to be short. Instead I am going to jump into his analysis of the NESS test and compare it to Mackie's test for singular causation. As part of the NESS test, Wright begins by describing Hume's analysis that causation is about the instantiation of antecedent conditions that are sufficient for bringing about an effect, but we nevertheless cannot directly observe causal relations themselves, rather we observe the through repetition or regularity of following one another, forming causal laws or generalizations thereof. Like Mackie's notion of a causal field, there is a point in which we must differentiate between antecedent conditions that are irrelevant from those that are relevant. This allows us to state which conditions invariably lead to the effect. Hume sees causation as some set of conditions that lead to an invariant relation between effect and that causal set, something of necessity, John Stuart Mill disagreed. Mill saw causation as something pluralistic. This appears to be similar to Mackie's account of a causally sufficient set, and flows directly from Hart and Honore's account of causation in which the NESS test is based. The essence of the concept under this account is that "a particular condition was a cause of (condition contributing to) a specific consequence if and only if it was a necessary element of a set of antecedent actual conditions that was sufficient for the occurrence of the consequence". This is in line with Mill, because we are now allowing for a plurality of sufficient sets. This is obviously similar to an INUS condition; perhaps they are isomorphic.

My point in bringing the NESS test into discussion is that it's generated to determine singular causation. Consider the scenarios listed above for determining legal responsibility; none of these situations are about statistical generalizations and yet these are some of the most practical and consequential situations where we need to reason correctly about causation. In law, the notion of "justice" appears to be something consistent with the correct application of rules that distinguish deserved punishment from undeserved punishment for some behavior. If you do not have coherent, correct, or valid rules for making necessary distinctions, your legal system diverges from the ideal "just system" we are striving for. But consider these rules for your own personal life. Are you identifying causation based on a neutral framework that minimizes the misclassification error rate? Anyways, the paper is interesting. Wright dives into specific details about how the NESS test generalizes beyond the but-for test to account for problem cases. Take a moment to think about how NESS is better because it is less restrictive in identifying causes that are more in line with our intuition. Should causality necessarily be intuitive?

The last paper I want to discuss is Contrastive Causation. Let's begin with a quote:

Causation is widely assumed to be a binary relation: c causes e. I will
argue that causation is a quaternary, contrastive relation: c rather than C*
causes e rather than E, where C and E* are nonempty sets of contrast
events.

Contrastivism is an epistemological theory proposed by Jonathan Schaffer. Simply stated, it holds that knowledge statements have the structure "S knows that P rather than Q". His analysis of causation draws on this notion of "rather than Q", which I take to be very similar to Mackie's INUS conditions and the NESS account of causality. If we consider the INUS account, each disjunct in the set of all possible INUS conditions represents the contrast class in which we compare the actual cause. Consider an example Schaffer uses:

One might argue that binarity reflects the surface form of causal ascriptions. But surface form is equivocal. There are indeed binary ascriptions, such as ‘Pam’s throwing the rock caused the window to shatter’. But there are also contrastive ascriptions, such as ‘Pam’s throwing the rock rather than the pebble caused the window to shatter rather than crack’.

This "rather than" formulation, to me, seems like we are contrasting the "cause" against a set of possible causes. However, I do think that Schaffer is intending on making a stronger claim; the effect "shattering the window" is contrasted with "cracking the window", in other words "effects" aren't just considered "in the abstract" isolated from other ways an effect could have manifested, rather they are also comparative, relative to similar "effects". Also consider that the contrastive account would help us understand responsibility in tort law by identifying intention; suppose the effect "window shattering" occurred when someone threw a pebble, rather than a rock. We might be inclined to describe this scenario as "accidental" and therefore, responsibility will be assigned and evaluated differently, given our contrastive assumptions that someone throwing a large rock would have intended the effect because it was foreseeable given common knowledge.

The contrastive account assumes causal relata are events, it expresses causal relatedness using counterfactual dependence; In particular, by using the difference making test: c rather than c* causes e rather than e* if and only if O(c) > O(e) (in words: if c* had occurred, then e* would have occurred), and lastly assumes that the contrast sets C* and E* are fixed by context. There are about six paradoxes that this account allegedly resolves, but we will only consider a few of them since this was supposed to be a short post. Before we begin though I want to make a few remarks on the first assumption about causation being about events. Defining an event is not trivial. Consider the case of a house fire used to demonstrate INUS conditions. How exactly are we defining the outcome of interest? Is the house completely burnt down or only partially burnt down? Presumably, whatever the INUS condition is, depends on how we describe the outcome. Upon analysis, I can claim that AX was an INUS condition for Y, and that had AX or any other INUS condition not been instantiated, Y would not have occurred. However, slightly modify the outcome to be "partially burnt down" and the set of possible INUS conditions seems to necessarily change, in order to account for the fire somehow stopping, leaving the house intact. In other words, Mackie's account is only contrastive on the c side; c rather than C* causes e. If you modify the outcome then you have to consider a new set of INUS conditions. This obviously complicates the situation a bit. Given the counterfactual dependence criteria Schaffer uses, it only accounts for alternative outcomes in the set of outcomes E. It does not express that some other cause, or some other INUS condition, will bring about that identical outcome E. Accordingly, c rather than C* causes e rather than E*, means that the alternative outcomes are actually different events; this account cannot explain how different sets of causes bring about one particular event. Anyway, this is why I mentioned that defining your event space can be problematic. Should "house burning down" and "house partially burning down" be considered the same event? How do we demarcate the boundaries between events to make causal analysis even possible? The thing is, I think Schaffer's analysis is distinct from Mackie's, despite being very similar, in that it is answering a different kind of causal question. Think about it this way. Mackie and the NESS account are creating a mapping, a many to one relationship between the set of possible causes and one outcome; they are showing how one decides which one of these potential mappings from cause to effect can be identified for a particular causal outcome or event under consideration. Schaffer's approach seems to be like a many-to-many mapping between the contrast sets C* and E*; C1 maps to E1, C2 maps to E2, C3 maps to E3. So by saying C caused E we are saying that, the C rather than C* (the closest or more "similar" causal factor being the more relevant contrast, perhaps C2) causes E rather than E* (the contrast event being the most similar, or whatever context dependent event we are interested in, perhaps E2). Now I don't think Schaffer explicitly mentions whether C1 and C2 can both map to the same E, which is what Mackie's account actually solves for. Schaffer's account seems to implicitly rely upon possible-world semantics. The idea of a possible world is that things could have been otherwise; there are many alternative scenarios that could have manifested other than the current one. Possible worlds are an attempt to make sense of counterfactual dependence: one event E counterfactually depends on another event C just in case if C had not occurred then E would not have occurred. C not occurring would imply that another event E could have occurred; another possible world.

Now, in the this philosophical theory, there is a notion of "closeness" between possible worlds. I find this incredibly problematic. Consider a metric space in which we define some measure to estimate distance between two points in Euclidian space or Phase Space; Manhattan distance, Jaccard distance, whatever you want. In empirical disciplines, these actually measure distance between data points instantiated in the real world; there is no hypothetical object that are measuring against. Similarity in the machine learning literature is defined by one of these metrics in hyperspace, it is how we estimate the "similarity" between two vectors (photographs, collections of text, etc.). This has problems of its own, but closeness and similarity, within empirical disciplines, means there has to be an actualization of something that can be measured. You can construct hypothetical datasets to test the predictive quality of the metric you select; but the hypothetical data is still actualized in the real world. We have no way of estimating something "in the real world" to something that has not been actualized. It seems necessary that The idea of "similar possible worlds" must be entrenched fully within thought experiments, dependent on analogy and metaphor. There is no measurable way to estimate the closeness of some alternative possible world. David Lewis talks about this idea of closeness; but in empirical disciplines we need an actual measure or reference class. Claiming counterfactual states of the world are more or less similar seems arbitrary at best; relying heavily on the intuition of the linguistic community. So in Schaffer's account, when we say "c rather than C* causes e rather than E*", it seems to me we are relying on some assumption about the contrast class which depends on this "similarity" between possible worlds; something we can construct ad-hoc. I am not saying this is incorrect, I am just saying in practical situations where we must construct a counterfactual state of the world to determine causation or causal effect, we must be very weary of the criterion we select to estimate "similarity"; are we referring to qualitative similarity, quantitative similarity, or something else? Lewis provides a set of criteria. A "similar possible world" is defined as one that satisfies these four criteria:

It is of the first importance to avoid big, widespread, diverse violations of law.
It is of the second importance to maximize the spatiotemporal region throughout which perfect match of particular fact prevails.
It is of the third importance to avoid even small, localized, simple violations of law.
It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly.

Does it have to satisfy all four exhaustively? How do we know if we have satisfied one? This framework also seems dependent on the way in which we understand "scientific laws"; I am guessing physical laws, but what about statistical laws and principles? What about Stylized Facts and concepts such as Economic Laws? Many of these are dependent on a Ceteris Paribus clause, which to me seems to beg the very question of "what is causation" to begin with (see the paper There is No Such Thing as a Ceteris Paribus Law). I should note too that these criteria are not agreed upon either. For example, Jonathan Bennett (2003) notes that:

when the antecedent of a conditional is not about a particular event, Lewis’s conditions provide the wrong results. For instance, if the antecedent is of the form If one of these events had not happened, then Lewis’s rules say that the nearest world where the antecedent is true is always the world where the most recent such event did not happen. But this does not seem to provide intuitively correct truth conditions for such conditionals.

I don't want to continue diving down this metaphysical rabbit hole. I just want to note that the criteria we select is going to depend on the differences we think are important; meaning the selection is highly contextual. It will be driven by pragmatic factors relevant to the decision makers, reasoners, and parties involved within an ongoing discussion. Selecting the contrast class seems equivalent to selecting the nearest possible world, but we ought to think critically about how we select the contrast class.

Moving on; It seems to me that if Mackie's framework accounts for many to one relations and Schaffer's framework accounts for many-to-many relations, we could generalize beyond both by means of Multigraphs and Directed Acyclic Graphs. Pearl's approach uses DAG's but Stephen Wolframs Multiway Causal Graphs seem to be a useful tool for generalizing possible causal relations and their counterfactual conditionals. The ancestor node on a path along the causal graph that has been actualized is "the cause", and alternative paths represent the contrastive states of the world that could have been instantiated, given the instantiation of the ancestor node. This is what a Multiway graph looks like; each path represents potential causal histories and the nodes at the end represent potential outcomes. This mathematical structure is a multiway system. These are a kind of substitution system in which multiple states are permitted at any stage. This accommodates rule systems in which there is more than one possible way to perform an update. Alternative ways the system can update refers to these counterfactual scenarios or contrastive events. Of course, figuring out the state transition rules is a complicated matter.

Anyway, I don't want to digress down this rabbit hole. Let's move along and take a look at the paradoxes Schaffer's framework resolves and wrap this up. We will look at the problem of whether absences are causal and whether causation is transitive.

Are absences causal? I will list out the pro and contra reasons Schaffer lists and touch briefly on why we may think they are or aren't.

(+) Absence causation is intuitive: intuition accepts some absences as causal. The gardener not watering the lawn seems to cause the law dying.
(+) Absences play the predictive and explanatory roles of causes and effects. The car driver not slamming on the brakes helps us predict whether there will be an accident.
(+) Absences play the moral and legal roles of causes and effects. For example, negligent parents who don't feed their children seem morally responsible.
(+) Absences mediate causation by disconnection. For example, severing the connection between two wire components will cause a power outage.
(-) Absence causation is counterintuitive: intuition rejects some absences as causal. For example, saying the queen of England not watering your flowers caused my flowers death.
(-) Absence causation is theoretically problematic. What does a negative statement denote? There are three options: (i) a nonactual event, (ii) an actual fact and (iii) an actual event. The first two conflict with the idea of causation involving actual events. One and three conflict with the idea of counterfactual dependence.
(-) Absence causation is metaphysically abhorrent. The gardener not watering the flowers implies there is no energy momentum flow or other physical processes connecting this absence. It violates spatiotemporal continuity; "not watering my flowers" could be said to occur literally anywhere.

Schaffer claims the paradoxes are resolved by contrastivity:

The reconciliation strategy is as follows: (i) treat negative nominals as denoting actual events, and (ii) treat absence-talk as tending to set the associated contrast to the possible event said to be absent. For instance, given that the gardener napped and my flowers wilted, ‘The gardener’s not watering my flowers caused my flowers not to blossom’, is to be interpreted as: the gardener’s napping rather than watering my flowers caused my flowers to wilt rather than blossom. Beginning with (6), the contrastive strategy takes the third view of negative nominals, as denoting actual events. This third view no longer conflicts with counterfactual dependence given contrastivity since the contrast turns the counterfactual antecedent down the right path: to O(c*) instead of ~O(c). Thus, the gardener’s napping rather than watering my flowers did cause the flowers to wilt rather than blossom, whereas the gardener’s napping rather than watching the news did not. Contrastivity thus reconciles absence causation with a counterfactual, event-based framework. Turning to (1)–(4), the contrastive strategy allows for absence-citing ascriptions to turn out true. This explains their intuitive acceptability. The negligent father’s moral and legal responsibility is grounded in the truth of: the father’s slurping gin rather than feeding his child caused the child to starve rather than be nourished. The predictive and explanatory role of causes is respected: the pilot’s fiddling with his cap rather than lowering the landing gear serves to predict that, and explain why, the plane crashed rather than landed safely. Causation by disconnection is mediated: the executioner’s beheading the prisoner rather than burying the hatchet caused the prisoner’s brain cells to starve rather than be oxygenated, which in turn caused the prisoner to die rather than survive. Contrastivity thus preserves all the plusses of absence causation. Moving to (7), the contrastive strategy locates the “oomph” in
absence causation. Where c and e are not actually connected, the members of C* and E* would have been connected. Or in more complex cases, there is a continuous chain composed of connections and wouldbe-connections.9 Thus, when the gardener naps rather than watering my flowers, there is a would-be-connection from the causal contrast of the watering to the effectual contrast of my flowers blooming.
Schaffer p. 5-6

Schaffer recommends resolving the fifth conflict on pragmatic grounds. The queen not watering the flowers is simply not a relevant alternative. I find this very similar to the pragmatic notion of "closeness" with respect to similar worlds.

Is causation transitive? Transitivity would mean that if C causes D, and D causes E, does it follow that C causes E? Schaffer provides the pro and contra arguments:

(+) Transitivity is intuitive.
(+) Transitivity links causal histories. This depends on number one. If there is no transitivity, then there is no causal sequence leading to a conclusion which seems counterintuitive.
(-) Transitivity fails in the dog-bite case; the case in which a terrorist is about to detonate a bomb with his right finger, when a dog runs by and bites off that finger. The dog's bye causes the terrorist to press with the left finger, causing an explosion. But intuitively, the dog obviously does not cause the explosion.
(-) Transitivity fails in the boulder case; a boulder is rolling down a hill towards a hiker, when the hiker sees it and ducks out of the way. The boulder's rolling causes the hikers ducking, and the hikers ducking causes his survival, but intuitively the bounders rolling does not cause survival.
(-) Transitivity fails in the nudgings case; a speck of dust nudges the rock off some trajectory onto another trajectory, which causes the rock to reach trajectory 2's midpoint, causes a window to shatter. But obviously this is counterintuitive, the specks nudge does not cause the window to shatter, "To hold otherwise is to miscount such paradigmatic non-causes as preempted backups, innocent bystanders, and hounds baying in the distance as causes since their presence contributes photons and sound-waves that nudge the causal process".

Schaffer claims that paradoxes of transitivity are resolved as follows: (i) treat causation as differentially transitive: if c rather than C* causes d rather than D*, and d rather than D* causes e rather than E*, then c rather than C* causes e rather than E; and (ii) reveal the counterexamples to require illicit shifts in D.

For the the claim of intuition, Schaffer claims that fixing the contrast class throughout the utterance resolves any paradox because the statements will conform to differential transitivity. He doesn't really elaborate so I can't comment much. This flows into the second the second notion of causal history; fixing the contrast classes for all set of potential causal explanatory factors leads to proper inference. For the cases where transitivity seems to be violated, Schaffer states:

Bounding over to the counterexamples of (3)–(4), none of these involve differential transitivity. All require illicit shifts in the value of D. That is, all the counterexamples have the form: c rather than c* causes d rather than d1*, and d rather than d2* causes e rather than e*, where d1* != d2*. In (3), Dog’s biting off Terrorist’s right forefinger rather than barking causes Terrorist’s pressing with his left forefinger rather than his right forefinger: c rather than c* causes d rather than d1*. But Terrorist’s pressing with his left forefinger rather than his right forefinger does not cause the bomb to explode rather than remain intact. The bomb would explode either way. So d rather than d1* does not cause e rather than e*. What does cause e rather than e* is Terrorist’s pressing with his left forefinger rather than walking away (d rather than d2*), and Dog’s biting off Terrorist’s right forefinger rather than barking does not cause that. There is no differential chain. The value of D has illicitly shifted from {Terrorist’s pressing with his right forefinger} to {Terrorist’s walking away}.
Schaffer P. 15

The other alleged failures of transitivity will fall short for similar reasons. The rest of his paper dives into his defense for the assumptions of this framework that I listed earlier. It becomes more esoteric and metaphysical so we will skip that.

I just want to end with a few comments. I started by differentiating causal generalizations from singular causation, spoke about how people incorrectly infer from a causal generalization case specific details relevant to the causal investigation of an individual within that population, and showed a couple of philosophical accounts that describe singular causation for every day reasoning. Causality has literally been discusses for thousands of years so you can obviously go very deep. I mostly think about causation in terms of systems and controllability; the notion of control reflects the fact I've been trained on the Rubin causal model, intervention on a system seems self-evident. However, we need not necessarily have direct control over a system to infer causality, as the research from Joshua Angrist on Natural Experiments has demonstrated. What we are truly interested in is identifying the exogenous variables that interact causally with our system of endogenous variables. If we have direct control over the exogenous variables, that is an added benefit; then we can empirically test hypotheses about say, the effects of a drug, effectiveness of a policy, or quality of a program, given our manipulation of the variable. This also applies to dynamical systems. In the real world, time components are essential features of many systems. Dynamical systems are widely used in science and engineering to model systems consisting of several interacting components. Often, they can be given a causal interpretation in the sense that they not only model the evolution of the states of the system's components over time, but also describe how their evolution is affected by external interventions (exogenous "shocks" in the economic jargon) on the system that perturb the dynamics. This "shock" is known as an Impulse Response in control theory; given some level or value of an external impulse, we can trace out the dynamics of the system over time, estimating the trajectory of a measure and equilibrium behavior. Why do I like this formulation? By distinguishing between exogenous shocks and endogenous variables, we can have a better description of system performance. Imagine your car breaks down; was it due to an endogenous failure (like a part being deficient) or some external factor causing stress on the system leading to collapse? Dynamical causal modeling allows you to understand the causal influence of something on a system with interacting components. Take an example from economics; does poverty cause illiteracy or does illiteracy cause poverty? Well, from a dynamical systems perspective, this is not a well formed question because the two could be interacting in a causal loop due to self reinforcing feedback mechanisms; understanding these two phenomenon requires us to understand their dynamics within a broader system that causally effects both of them. The dynamical systems perspective also gives you insight into lagged causal effects that would go unobserved in a mere cross sectional analysis. In practice, controllability of social systems is very difficult; in econometrics we always wonder whether something can truly be exogenous, Nevertheless, I think these approaches capture our intuitive notions of causation, manipulation, and control; despite singular causation could still be somewhat of a problem. For more on this see Causal Analysis Based On System Theory.

Lastly, I just want to note that causal explanation is but one type of explanation. Not all explanatory frameworks rely on causal chains of manipulation. An obvious example of a non-causal explanation that people mistake for causation is that of planetary motion. These equations are non-causal; they are descriptive. Do not make the mistake of thinking they are. Another, although contentious, example is that of Evolutionary theory; Evolution does not "cause" things to be, it is a description of various biological processes and stochastic dynamics. The level of causal description is also relevant for us to understand "what the cause" was in a particular situation. Depending on the specific scale in which you observe a system, you can have vastly different understandings of what that thing is. For example, zoom in on my biological system, and you will see a vastly different organism composed of interconnected subsystems than you would if you zoomed out and observed me interacting in my environment. The same idea is true of causation; we say something is the cause of something else, given a background assumption about the particular spatial scale we are interested. So if I say "So and So got Covid and died as a result of bad policy", I am obviously looking at it from a different scale than if I said "So and So got Covid and died because they had hypertension". Spatial scales are not only important, but also temporal scales. Something may appear to be "causal" on one time scale, but zoom out and it appears to be part of a larger process. Temporal and spatial considerations are essential to clarify when imposing causal interpretations on an event.

I found this great flowchart for books on causal inference: Which causal inference book you should read...

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