1 Introduction

The concept of cause (as used in sentences of the form ‘A caused B’, or its converse ‘B is an effect of A’) is quite often explained as ‘If A had not been the case (or occurred), then B would not have been the case (or happened)’. It is a counterfactual (or contrary-to-fact) statement since A and B in fact both occurred.

How, then, do we know what would have happened if A had in fact not happened? It is obvious that our justification for saying what would have happened, if the stream of events had differed from what actually occurred, must be based on some inference from actually occurring and observed regularities to unobserved events. How do we ascertain that?

This is an instance of the problem of induction, the problem to state under what conditions an inductive inference can be relied upon.Footnote 1 If we have a trustworthy and exception-less law at our hand, the problem is solved. Instead of using observed states as initial conditions in calculating future states of affairs, we can put in non-actual values and use the strict law to calculate what would have happened in such a non-actual case. But in most situations we have no strict law at our disposal. We will further discuss the connections between laws and causation in Chap. 7.

This will not work if we use non-strict laws, i.e., laws with so called ceteris paribus clauses (‘All else being the same’). This is so because when we imagine a non-actual initial condition we do not know whether other relevant circumstances also would differ from the actual observed situation. When we suspect that our law is not strict we add the clause ’ceteris paribus’ just because we do not know which factors we need to take into account or not.

Can one say something about counterfactuals and causation without using strict laws? Empiricists are sceptical. Stating truth conditions for counterfactuals has proven to be a deep problem and there is no agreement about its solution. It appears, if anything, more difficult than stating truth conditions for statements about causal relations.

2 Goodman on Counterfactuals

The seminal paper in the discussion about counterfactuals was Nelson Goodman’s The Problem of Counterfactual Conditionals (Goodman, 1946). In that paper Goodman observed that there is a profound semantic difference between counterfactual and indicative conditionals.

Counterfactual and indicative conditionals differ in verb form; counterfactuals are expressed using the subjunctive mood, whereas indicative conditionals are expressed in indicative mood, and this difference indicates a semantic difference. The following example may illustrate:

  • Indicative conditional: If it is raining right now, then Sally is inside.

  • Counterfactual conditional: If it were raining right now, then Sally would be inside.

These two sentences differ in meaning, hence there is a difference in their truth conditions. In order to analyse this difference, we start with the truth table for the indicative conditional (also called the material conditional):

A

B

If A, then B

True

True

True

True

False

False

False

True

True

False

False

True

We can formulate the content of this table as that an indicative conditional is true whenever the antecedent is false or the consequent is true.Footnote 2

If we apply this truth table to the counterfactual conditional it will come out true, since the verb form ‘were’ means that what follows is in fact not the case. Hence ‘were’ tells us that the antecedent is false, it is not raining. So far, so good. But according to the truth table it doesn’t matter whether Sally is inside or not, both #1 and #2 come out true:

  1. #1.

    If it were raining right now, then Sally would be inside.

  2. #2.

    If it were raining right now, then Sally would not be inside.

That cannot be correct, they contradict each other. Hence, the truth conditional analysis where antecedent and consequent are evaluated separately must be wrong in the case of counterfactual statements. No truth table can account for the semantics of counterfactual sentences.

As Goodman observed, it is some sort of connection between the events described in the antecedent and the consequent that determines the truth value of a counterfactual statement. This connection is not, and cannot be, reflected in any merely logical connection between antecedent and consequent, since logic concerns the forms of sentences and formal relations between sentences.

Goodman next observed that the difference between true and false counterfactuals is that true counterfactuals are connected to laws, while false are connected to true, accidental generalisations. Here is one example of a contrast between a law and an accidental generalisation (not discussed by Goodman, but by several later philosophers):

  1. #3.

    All spheres of gold are less than 1 km in diameter.

  2. #4.

    All spheres of U-235 are less than 1 km in diameter.

#3 and #4 have the same logical form and presumably are they both true. (If one would find somewhere in the universe a really big lump of gold, we could have a longer diameter.) But there is a difference; #4 is a consequence of fundamental laws of nuclear physics, and therefore itself a law, which means that it is not merely true but necessarily so; it is impossible to assemble an amount of U-235 bigger than the critical mass (52 kg, a sphere with diameter = 17 cm) of this radioactive isotope.Footnote 3 By contrast, #3 is accidentally true; it just happens to be no big lumps of gold in the universe.

Based on #3 and #4 we can now construct two counterfactuals, one true and one false:

  1. #5.

    If x were a sphere of gold, it would be less than 1 km i diameter.

  2. #6.

    If x were a sphere of U-235, it would be less than 1 km in diameter.

As far as we know, #5 is false while #6 is certainly true. So it seems reasonable to say that true counterfactuals are based on laws whereas false counterfactuals are based on accidental generalisations.

This would be a real step forward, if we had a clear explanation of the difference between laws and accidental generalisations. But we have not; many people have strong intuitions about the difference, but so far no generally accepted analysis is in sight.Footnote 4

Goodman concluded his paper by admitting that he had no solution to the problem with counterfactual conditionals, since he had no suggestion of how to distinguish between laws and accidental generalisations.

The discussion about causation and counterfactuals has been intense and one may discern two main strategies: either to analyse causation in terms of counterfactuals, or the other way round. This choice is guided by ones metaphysical views: David Lewis (1973) and many others think that a semantics of counterfactuals in terms of possible worlds is satisfactory and taking that as a firm ground one can then define causation in terms of counterfactuals. Those sceptical about the existence of possible worlds, or even the intelligibility of this notion, (How do you identify a possible but not actual world?) hold that the explanation should go from causation to counterfactuals.

We belong to this latter camp. Counterfactual statements belong both to our vernacular and to scientific discourse; they are widely used and there is no reason to assume that users of this idiom tacitly or explicitly delve into deep metaphysics concerning possible worlds and our access to them. Hence, we think counterfactuals should be explained in terms of more basic concepts such as causes or perhaps laws. As we showed in Chap. 3, causal talk belongs to our very basic vocabulary, learned already when first learning to talk our mother tongue. This means that explanations of the meanings of less basic expressions should be done in terms of the basic vocabulary.

In 1940s, when Goodman’s paper was published, there were little discussion about the concept of a law of nature. The received view was that when a hypothetical general statement was supported by a sufficient number of observations and no counter instances were observed, one had reason to believe that the general conclusion was correct, and it was then elevated to being a law. Those who elaborated the details of this line of thought argued that probability arguments could be used. But this idea met, justifiably, devastating criticism. Having high probability is not the same as certainty, and strict laws are certain. Furthermore, as Goodman pointed out, there is a profound difference between laws, which are necessary, and other true general statementsof the same logical form, which are not necessary, and this difference could not be analysed using only empirical arguments.

What does this mean for the counterfactual analysis of causation? Our view is that in so far as we are unclear about the meaning of ‘cause’, giving this concept a definition in terms of counterfactuals is no step forward; counterfactuals are strongly related to laws, and both the notions of counterfactual and law are less clear than that of cause. So what to do? James Woodward has suggested a way out.

3 Woodward’s Account of Causation

James Woodwardhas discussed the counterfactual analysis of causation in several papers (Woodward, 1997, 2002, 2003, 2008, 2016). One might guess that Woodward was inspired by Goodman’s observation of the strong connection between true counterfactuals and laws, although he made no references to Goodman’s paper. Taking into account that the concept of a law of nature is as much in dispute as are counterfactuals, Woodward’s step forward was to base true counterfactuals on what he called ‘invariances’.

An invariance is an observed regularity, although not one elevated to the status of being a law. Thus Woodward was able to avoid the metaphysical jungle of necessities. Neither is an invariance merely an accidental generalisation. Woodward’s idea is that an observed regularity which has been used in several successful predictions may be labelled an invariance, which tells us that it is a weaker concept than that of a natural law. But what, more precisely, is the difference?

Woodward intended ‘invariances’ to refer to regularly occurring phenomena restricted to some region in space and time. One could for example say that it is an invariance (or ‘restricted regularity’) that almost all people who has spent 10 years or more in Sweden understand, to some degree, Swedish, while hesitating to call this regularity a ‘law’. But it depends on what we mean by ‘law’.

In any case, if we accept this regularity we are prone to accept as true the counterfactual ‘If NN had been living in Sweden for 15 years, she would understand Swedish’, said about a certain person that only understands her mother tongue, say Swahili, and has never been in Sweden.

This type of local and restricted invariances differ from laws in that they are not exception-less. Observing such an exception we are prone to ask for an explanation, i.e., a causal explanation. We are back to causes.

One further difference between laws and invariances is that laws properly so called are integrated into a theory consisting of a number of laws logically related to each other.

The question is: can one refer to an invariance as evidence for a counterfactual? It seems that the answer is no, unless we know the causal mechanism producing the invariance. For how could an invariance observed in a number of cases be known to be valid also in a non-observed case? Invariances may have exceptions, they are not strict laws, and how do we know that an unobserved case is not an exception?

It seems that an ‘explanation’ of the concept of cause in terms of counterfactual dependence is no step forward. It is much more reasonable to say that we can explain ‘counterfactual dependence’ in terms of causes. The word ‘cause’ and it’s synonyms (‘bring about’, ‘lead to’, ‘produce’, etc.) belong to common language and is much easier to understand than any technical term.

4 Potential Outcomes Instead of Counterfactuals?

Instead of analysing causation in terms of counterfactuals, Rubin, following Neyman (1923) and Fisher (1925), uses the concept of potential outcomes. Here is how he motivates it:

Some authors (e.g. Greenland et al., 1999; Dawid, 2000) call the potential outcomes “counterfactuals”, borrowing the term from philosophy (e.g. Lewis, 1973). I much prefer Neyman’s implied term ‘potential outcomes’ because these values are not counterfactual until after treatments are assigned, and calling all potential outcomes ‘counterfactuals’ certainly confuses quantities that can never be observed (e.g. your height at the age of 3 if you were born in the Arctic) and so are truly a priori counterfactual, with unobserved potential outcomes that are not a priori counterfactual (see Frangakis and Rubin (2002), Rubin (2004); and the discussion and reply for more on this point.) (Rubin, 2005, 325)

Here is a simple illustration of how to use the concept of potential outcomes. Suppose we have randomly divided a test sample, taken from some population, into two groups, one consisting of those being treated in some way, the rest is the control group. For each unit in the treatment group one can only observe its actual state after the treatment and similarly for the control group; in this group one can only observe the state of a unit after not being treated during the experiment. Both the actual state and the non-actual possible state of a unit are observable, although only one state is actually observed. Hence the term ‘potential outcome’. We can now compare the observed outcomes in the two groups. We can calculate the conditional probability for the outcome B, conditioned on the intervention A, p(B—A) and compare with the marginal probability p(B). If prob(B —A) > prob(B), we have strong reason to believe that A is a cause of B. (N.B. the indefinite ‘a cause’; there may be more causes!) The intervention A may be a intentional manipulation or an intervention not planned by the experimenter, i.e., a so called ‘natural experiment’.

Replacing the concept of potential outcome for counterfactual in discussions about causation is a significant step towards a more empirical approach. Moreover, it connects to the manipulability account of causation, see Sects. 3.4 and 6.1. It is useful when making inferences about causation from observed results of experiments, and also in making inferences from so called ‘natural experiments’, see Sect. 6.2.

From a philosophical point of view this is a significant improvement as compared with the counterfactual analysis. The semantics of counterfactuals in terms of possible worlds faces two obstacles: (1) how do we identify a possible world and (2) which possible worlds should we take into account when describing the semantics of causation? One needs to impose restrictions on what to count as a possible world, which is usually made in terms of similarity to our actual world. This can be made formally stringent, but it is not helpful for the empirical researcher, since it leaves the notion of similarity with the actual world undefined. The crucial question is ‘Similar in what respect?’

The terminology of potential outcomes is, in comparison, applicable to actual experiments and observations. The set of potential outcomes are defined in the experimental design. We perform an experiment and explicitly state the set of possible outcomes, of which one is actualised. For further discussion see e.g., (Menzies and Beebee, 2020) and (Rubin, 2005).

5 Summary

In ordinary parlance we take it for granted that the sentence ‘A is the cause of B’ is more or less synonymous with ‘If A had not occurred, B would not have occurred.’ This assumption is then used for explaining causation in terms of counterfactuals. But on second thoughts one may reasonably conclude that this is not of much value as an explanation; the meaning of counterfactuals is much more foggy than the meaning of ‘cause’. How do we know what would have happened if the course of events had been different from what actually happened?

It seems that only if we have a strict scientific law at our disposal can we know with some certainty what would have happened, if the conditions had been different from what they actually were. But in very many cases we know no strict laws, so an analysis of causation in terms of counterfactuals is no step forward.

Woodward has suggested to use term ‘invariance’ instead of laws for explaining causation; Invariances are inferred from observed regularities in some local setting and believed to be true also in unobserved cases in the same type of settings, while not being elevated to the status of law and not integrated into a theoretical structure of laws related to each other.

A somewhat similar approach is taken by e.g. Rubin, who has suggested replacing the concept of potential outcome for counterfactual. Given a dynamical equation we can calculate what would be the outcome for any chosen initial condition, actual or not. This equation guides the time evolution from the initial situation and we can map the set of selected initial conditions onto the set of potential outcomes. The dynamics may be strict (mapping one initial state onto one final state), or probabilistic, mapping a set of possible outcomes from each initial state). The list of alternative outcomes are clearly stated and all are observable; but only one will be observed, that which is actualised.

Discussion Questions

  1. 1.

    The truth table for the indicative conditional (displayed in Sect. 4.2.) are by many students not immediately accepted. Many often wonder about the combinations where the antecedent is false. Why is it correct to say that a conditional statement is true whenever the antecedent is false?

  2. 2.

    It is not possible to construct a truth table for the counterfactual conditional. Why?

  3. 3.

    Are there any laws that are not necessary? If there is any such example, why, then, is it said to be a law?

  4. 4.

    What is the difference between an accidental generalisation (a ‘regularity’) and an ‘invariance’ in the sense of Woodward?

  5. 5.

    Why is an analysis of causes in terms of potential outcomes to be preferred over an analysis in terms of counterfactuals?