Counterfactual conditional


Counterfactual conditionals are a class of conditional sentence which allow speakers to discuss what would or could have been true under potentially different circumstances, e.g. "If it was raining right now, then Sally would be inside." Counterfactuals are characterized grammatically by their use of "fake past" marking, which some languages use in combination with other kinds of morphology such as subjunctive or conditional mood. They are contrasted with indicative conditionals, which do not bear this extra marking and are generally restricted to discussing what might actually be.
The term was coined by Nelson Goodman in 1947, extending Roderick Chisholm's earlier notion of a contrary-to-fact conditional. Since then, they have become one of the central phenomena in formal semantics, philosophy of language, and philosophical logic. They have also played a large role in other sub-fields of philosophy, human geography, psychology, cognitive psychology, history, political science, economics, social psychology, law, organizational theory, marketing, and epidemiology.

Examples

The difference between indicative and counterfactual conditionals can be illustrated by the following contrast:
  1. Indicative: If it is raining right now, then Sally is inside.
  2. Counterfactual: If it was raining right now, then Sally would be inside.
These conditionals differ in both form and meaning. The indicative conditional uses the present tense form "is" in both the "if" clause and the "then" clause. As a result, it conveys that the speaker is agnostic about whether it is raining. The counterfactual example uses the past tense form "was" in the "if" clause and the modal "would" in the "then" clause. As a result, it conveys that the speaker does not believe that it is raining. This use of past tense is often referred to as fake past since it does not convey its usual temporal meaning.
The difference in meaning between indicative and counterfactual sentences can be seen starkly in the Jackson Pair, where the indicative example is formed with a non-fake past tense, and the counterfactual example is formed with a pluperfect.
  1. Indicative: If Oswald did not shoot Kennedy, then someone else did.
  2. Counterfactual: If Oswald had not shot Kennedy, then someone else would have.
The antecedent of the first sentence may or may not be true according to the speaker, so the consequent also may or may not be true. The consequent is asserted by the speaker to be true if the antecedent is true. In the counterfactual example, the speaker is speaking with a certainty that Oswald did shoot Kennedy. According to the speaker, the if clause is false, so the then clause deals with the counterfactual result, i.e., what would have happened.

Terminology

The term counterfactual conditional was coined by Nelson Goodman in 1947, extending Roderick Chisholm's earlier notion of a contrary-to-fact conditional. Present day linguists and philosophers of language sometimes avoid the term counterfactuals because not all examples express counterfactual meanings. For instance, the classic example known as the "Anderson Case" has the characteristic grammatical form of a counterfactual conditional, but does not convey that its antecedent is false or unlikely.
The term subjunctive conditional has been used as an alternative, even though it is likewise acknowledged as a misnomer. Many languages do not have a morphological subjunctive and many that do have it don’t use it for this sort of conditional. Moreover, languages that do use the subjunctive for such conditionals only do so if they have a specific past subjunctive form.
Recently the term X-Marked has been proposed as a replacement, evoking the extra marking that these conditionals bear. Those adopting this terminology refer to indicative conditionals as O-Marked conditionals, reflecting their ordinary marking.

Psychology

People engage in counterfactual thinking frequently. Experimental evidence indicates that people's thoughts about counterfactual conditionals differ in important ways from their thoughts about indicative conditionals.

Comprehension

Participants in experiments were asked to read sentences, including counterfactual conditionals, e.g., ‘If Mark had left home early, he would have caught the train’. Afterwards, they were asked to identify which sentences they had been shown. They often mistakenly believed they had been shown sentences corresponding to the presupposed facts, e.g., ‘Mark did not leave home early’ and ‘Mark did not catch the train’. In other experiments, participants were asked to read short stories that contained counterfactual conditionals, e.g., ‘If there had been roses in the flower shop then there would have been lilies’. Later in the story, they read sentences corresponding to the presupposed facts, e.g., ‘there were no roses and there were no lilies’. The counterfactual conditional primed them to read the sentence corresponding to the presupposed facts very rapidly; no such priming effect occurred for indicative conditionals. They spent different amounts of time 'updating' a story that contains a counterfactual conditional compared to one that contains factual information and focused on different parts of counterfactual conditionals.

Reasoning

Experiments have compared the inferences people make from counterfactual conditionals and indicative conditionals. Given a counterfactual conditional, e.g., 'If there had been a circle on the blackboard then there would have been a triangle', and the subsequent information 'in fact there was no triangle', participants make the modus tollens inference 'there was no circle' more often than they do from an indicative conditional. Given the counterfactual conditional and the subsequent information 'in fact there was a circle', participants make the modus ponens inference as often as they do from an indicative conditional. See counterfactual thinking.

Psychological accounts

argues that people construct mental representations that encompass two possibilities when they understand, and reason from, a counterfactual conditional, e.g., 'if Oswald had not shot Kennedy, then someone else would have'. They envisage the conjecture 'Oswald did not shoot Kennedy and someone else did' and they also think about the presupposed facts 'Oswald did shoot Kennedy and someone else did not'. According to the mental model theory of reasoning, they construct mental models of the alternative possibilities.

The logic of counterfactuals

Counterfactuals were first discussed by Nelson Goodman as a problem for the material conditional used in classical logic. Because of these problems, early work such as that of W.V. Quine held that counterfactuals aren't strictly logical, and do not make true or false claims about the world. However, in the 1970s, David Lewis showed that these problems are surmountable given an appropriate logical framework. Work since then in formal semantics, philosophical logic, philosophy of language, and cognitive science has built on Lewis's insight, taking it in a variety of different directions.

Problems for semantic accounts

The problem of counterfactuals

According to the material conditional analysis, a natural language conditional, a statement of the form ‘if P then Q’, is true whenever its antecedent, P, is false. Since counterfactual conditionals are those whose antecedents are false, this analysis would wrongly predict that all counterfactuals are vacuously true. Goodman illustrates this point using the following pair in a context where it is understood that the piece of butter under discussion had not been heated.
  1. If that piece of butter had been heated to 150º, it would have melted.
  2. If that piece of butter had been heated to 150º, it would not have melted.
More generally, such examples show that counterfactuals are not truth-functional. In other words, knowing whether the antecedent and consequent are actually true is not sufficient to determine whether the counterfactual itself is true.

Context dependence and vagueness

Counterfactuals are context dependent and vague. For example, either of the following statements can be reasonably held true, though not at the same time:
  1. If Caesar had been in command in Korea, he would have used the atom bomb.
  2. If Caesar had been in command in Korea, he would have used catapults.

    Non-monotonicity

Counterfactuals are non-monotonic in the sense that their truth values can be changed by adding extra material to their antecedents. This fact is illustrated by Sobel sequences such as the following:
  1. If Hannah drinks coffee she'll be happy.
  2. If Hannah drinks coffee and the coffee has gasoline in it, she'll be sad.
  3. If Hannah drinks coffee and the coffee has gasoline in it and Hannah is a gasoline-drinking robot, she'll be happy.
One way of formalizing this fact is to say that the principle of Antecedent Strengthening should not hold for any connective > intended as a formalization of natural language conditionals.

Strict conditional

The strict conditional analysis treats natural language counterfactuals as being equivalent to the modal logic formula. In this formula, expresses necessity and is understood as material implication. This approach was first proposed in 1912 by C.I. Lewis as part of his axiomatic approach to modal logic. In modern relational semantics, this means that the strict conditional is true at w iff the corresponding material conditional is true throughout the worlds accessible from w. More formally:
Unlike the material conditional, the strict conditional is not vacuously true when its antecedent is false. To see why, observe that both and will be false at if there is some accessible world where is true and is not. The strict conditional is also context-dependent, at least when given a relational semantics. In the relational framework, accessibility relations are parameters of evaluation which encode the range of possibilities which are treated as "live" in the context. Since the truth of a strict conditional can depend on the accessibility relation used to evaluate it, this feature of the strict conditional can be used to capture context-dependence.
The strict conditional analysis encounters many known problems, notably monotonicity. In the classical relational framework, when using a standard notion of entailment, the strict conditional is monotonic, i.e. it validates Antecedent Strengthening. To see why, observe that if holds at every world accessible from, the monotonicity of the material conditional guarantees that will be too. Thus, we will have that.
This fact led to widespread abandonment of the strict conditional, in particular in favor of Lewis's variably strict analysis. However, subsequent work has revived the strict conditional analysis by appealing to context sensitivity. This approach was pioneered by Warmbrōd, who argued that Sobel sequences don't demand a non-monotonic logic, but in fact can rather be explained by speakers switching to more permissive accessibility relations as the sequence proceeds. In his system, a counterfactual like "If Hannah had drunk coffee, she would be happy" would normally be evaluated using a model where Hannah's coffee is gasoline-free in all accessible worlds. If this same model were used to evaluate a subsequent utterance of "If Hannah had drunk coffee and the coffee has gasoline in it...", this second conditional would come out as trivially true, since there are no accessible worlds where its antecedent holds. Warmbrōd's idea was that speakers will switch to a model with a more permissive accessibility relation in order to avoid this triviality.
Subsequent work by Kai von Fintel, Thony Gillies, and Malte Willer has formalized this idea in the framework of dynamic semantics, and given a number of linguistic arguments in favor. One argument is that conditional antecedents license negative polarity items, which are thought to be licensed only by monotonic operators.
  1. If Hannah had drunk any coffee, she would be happy.
Another argument in favor of the strict conditional comes from Irene Heim's observation that Sobel Sequences are generally infelicitous in reverse.
  1. If Hannah had drunk coffee with gasoline in it, she would not be happy. But if she had drunk coffee, she would be happy.
Sarah Moss and Karen Lewis have responded to these arguments, showing that a version of the variably strict analysis can account for these patterns, and arguing that such an account is preferable since it can also account for apparent exceptions. As of 2020, this debate continues in the literature, with accounts such as Willer arguing that a strict conditional account can cover these exceptions as well.

Variably strict conditional

, Robert Stalnaker, and Donald Nute modeled counterfactuals using the possible world semantics of modal logic. In order to distinguish counterfactual conditionals from material conditionals, they used a new logical connective '>', where A > B read as "If it were the case that A, then it would be the case that B."
In these approaches, the semantics of a conditional A > B is given by some function on the relative closeness of worlds where A is true and B is true, on the one hand, and worlds where A is true but B is not, on the other.
On Lewis's account, A > C is vacuously true if and only if there are no worlds where A is true ; non-vacuously true if and only if, among the worlds where A is true, some worlds where C is true are closer to the actual world than any world where C is not true; or false otherwise. Although in Lewis's Counterfactuals it was unclear what he meant by 'closeness', in later writings, Lewis made it clear that he did not intend the metric of 'closeness' to be simply our ordinary notion of overall similarity.
Example:
On Lewis's account, the truth of this statement consists in the fact that, among possible worlds where he ate more for breakfast, there is at least one world where he is not hungry at 11 am and which is closer to our world than any world where he ate more for breakfast but is still hungry at 11 am.
Stalnaker's account differs from Lewis's most notably in his acceptance of the limit and uniqueness assumptions. The uniqueness assumption is the thesis that, for any antecedent A, among the possible worlds where A is true, there is a single one that is closest to the actual world. The limit assumption is the thesis that, for a given antecedent A, if there is a chain of possible worlds where A is true, each closer to the actual world than its prececessor, then the chain has a limit: a possible world where A is true that is closer to the actual worlds than all worlds in the chain. On Stalnaker's account, A > C is non-vacuously true if and only if, at the closest world where A is true, C is true. So, the above example is true just in case at the single, closest world where he ate more breakfast, he does not feel hungry at 11 am. Although it is controversial, Lewis rejected the limit assumption because it rules out the possibility that there might be worlds that get closer and closer to the actual world without limit. For example, there might be an infinite series of worlds, each with a coffee cup a smaller fraction of an inch to the left of its actual position, but none of which is uniquely the closest.
One consequence of Stalnaker's acceptance of the uniqueness assumption is that, if the law of excluded middle is true, then all instances of the formula ∨ are true. The law of excluded middle is the thesis that for all propositions p, p ∨ ¬p is true. If the uniqueness assumption is true, then for every antecedent A, there is a uniquely closest world where A is true. If the law of excluded middle is true, any consequent C is either true or false at that world where A is true. So for every counterfactual A > C, either A > C or A > ¬C is true. This is called conditional excluded middle. Example:
On Stalnaker's analysis, there is a closest world where the fair coin mentioned in and is flipped and at that world either it lands heads or it lands tails. So either is true and is false or is false and true. On Lewis's analysis, however, both and are false, for the worlds where the fair coin lands heads are no more or less close than the worlds where they land tails. For Lewis, "If the coin had been flipped, it would have landed heads or tails" is true, but this does not entail that "If the coin had been flipped, it would have landed heads, or: If the coin had been flipped it would have landed tails."

Other accounts

Ramsey

Counterfactual conditionals may also be evaluated using the so-called Ramsey test: A > B holds if and only if the addition of A to the current body of knowledge has B as a consequence. This condition relates counterfactual conditionals to belief revision, as the evaluation of A > B can be done by first revising the current knowledge with A and then checking whether B is true in what results. Revising is easy when A is consistent with the current beliefs, but can be hard otherwise. Every semantics for belief revision can be used for evaluating conditional statements. Conversely, every method for evaluating conditionals can be seen as a way for performing revision.

Ginsberg

Ginsberg has proposed a semantics for conditionals which assumes that the current beliefs form a set of propositional formulae, considering the maximal sets of these formulae that are consistent with A, and adding A to each. The rationale is that each of these maximal sets represents a possible state of belief in which A is true that is as similar as possible to the original one. The conditional statement A > B therefore holds if and only if B is true in all such sets.

Within empirical testing

The counterfactual conditional is the basis of experimental methods for establishing causality in the natural and social sciences, e.g., whether taking antibiotics helps cure bacterial infection. For every individual, u, there is a function that specifies the state of u's infection under two hypothetical conditions: had u taken antibiotic and had u not taken antibiotic. Only one of these states can be observed in any instance, since they are mutually exclusive. The overall effect of antibiotic on infection is defined as the difference between these two states, averaged over the entire population. If the treatment and control groups are selected at random, the effect of antibiotic can be estimated by comparing the rates of recovery in the two groups.
In 1748, when defining causation, David Hume referred to a counterfactual case:

Pearl

The tight connection between causal and counterfactual relations has prompted Judea Pearl to reject both the possible world semantics and those of Ramsey and Ginsberg. The latter was rejected because causal information cannot be encoded as a set of beliefs, and the former because it is difficult to fine-tune Lewis's similarity measure to match causal intuition. Pearl defines counterfactuals directly in terms of a "structural equation model" – a set of equations, in which each variable is assigned a value that is an explicit function of other variables in the system. Given such a model, the sentence "Y would be y had X been x" is defined as the assertion: If we replace the equation currently determining X with a constant X = x, and solve the set of equations for variable Y, the solution obtained will be Y = y. This definition has been shown to be compatible with the axioms of possible world semantics and forms the basis for causal inference in the natural and social sciences, since each structural equation in those domains corresponds to a familiar causal mechanism that can be meaningfully reasoned about by investigators.

Footnotes