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.^[19]

1. If that piece of butter had been heated to 150°, it would have melted.
2. If that piece of butter had been heated to 150°, 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.^[18]

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:^[20]

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:^[19][21][22]

1. If Hannah had drunk coffee, she would be happy.
2. If Hannah had drunk coffee and the coffee had gasoline in it, she would be sad.
3. If Hannah had drunk coffee and the coffee had gasoline in it and Hannah were a gasoline-drinking robot, she would 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.

• Antecedent Strengthening: ${\displaystyle P>Q\models (P\land R)>Q}$

Strict conditional

The strict conditional analysis treats natural language counterfactuals as being equivalent to the modal logic formula ${\displaystyle \Box (P\rightarrow Q)}$. In this formula, ${\displaystyle \Box }$ expresses necessity and ${\displaystyle \rightarrow }$ 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.^[18] 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:

• Given a model ${\displaystyle M=\langle W,R,V\rangle }$, we have that ${\displaystyle M,w\models \Box (P\rightarrow Q)}$ iff ${\displaystyle M,v\models P\rightarrow Q}$ for all ${\displaystyle v}$ such that ${\displaystyle Rwv}$

Unlike the material conditional, the strict conditional is not vacuously true when its antecedent is false. To see why, observe that both ${\displaystyle P}$ and ${\displaystyle \Box (P\rightarrow Q)}$ will be false at ${\displaystyle w}$ if there is some accessible world ${\displaystyle v}$ where ${\displaystyle P}$ is true and ${\displaystyle Q}$ is not. The strict conditional is also context-dependent, at least when given a relational semantics (or something similar). 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 ${\displaystyle P\rightarrow Q}$ holds at every world accessible from ${\displaystyle w}$, the monotonicity of the material conditional guarantees that ${\displaystyle P\land R\rightarrow Q}$ will be too. Thus, we will have that ${\displaystyle \Box (P\rightarrow Q)\models \Box (P\land R\rightarrow Q)}$.

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 (1981), 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 had 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 (2001), Thony Gillies (2007), and Malte Willer (2019) 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 (i.e. sound strange) 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 (2012) and Karen Lewis (2018) 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 (2019) arguing that a strict conditional account can cover these exceptions as well.^[18]

Variably strict conditional

In the variably strict approach, 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 (a) vacuously true if and only if there are no worlds where A is true (for example, if A is logically or metaphysically impossible); (b) 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 (c) 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:

If he had eaten more at breakfast, he would not have been hungry at 11 am.

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 (unique) 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 predecessor, 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. (The uniqueness assumption entails the limit assumption, but the limit assumption does not entail the uniqueness assumption.) 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 (and therefore the uniqueness 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. (See Lewis 1973: 20.)

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 (A > C) ∨ (A > ¬C) 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 (CEM). Example:

(1) If the fair coin had been flipped, it would have landed heads.

(2) If the fair coin had been flipped, it would have landed tails (i.e. not heads).

On Stalnaker's analysis, there is a closest world where the fair coin mentioned in (1) and (2) is flipped and at that world either it lands heads or it lands tails. So either (1) is true and (2) is false or (1) is false and (2) true. On Lewis's analysis, however, both (1) and (2) 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."

Causal models

The causal models framework analyzes counterfactuals in terms of systems of structural equations. In a system of equations, 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" (formally, X = x > Y = y ) 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. This approach was developed by Judea Pearl (2000) as a means of encoding fine-grained intuitions about causal relations which are difficult to capture in other proposed systems.^[23]

Belief revision

In the belief revision framework, counterfactuals are treated using a formal implementation of the Ramsey test. In these systems, a counterfactual 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 (1986) 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.^[24]

Description

In many languages, counterfactuality is marked by past tense morphology.^[25] Since these uses of the past tense do not convey their typical temporal meaning, they are called fake past or fake tense.^[26][27][28] English is one language which uses fake past to mark counterfactuality, as shown in the following minimal pair.^[29] In the indicative example, the bolded words are present tense forms. In the counterfactual example, both words take their past tense form. This use of the past tense cannot have its ordinary temporal meaning, since it can be used with the adverb "tomorrow" without creating a contradiction.^{[25][26][27][28]}

1. Indicative: If Natalia leaves tomorrow, she will arrive on time.
2. Counterfactual: If Natalia left tomorrow, she would arrive on time.

Modern Hebrew is another language where counterfactuality is marked with a fake past morpheme:^[30]

	im	Dani	haya	ba-bayit	maχa ɾ	hayinu	mevakRim	oto
	if	Dani	be.pst.3sm	in-home	tomorrow	be.pst.1pl	visit.ptc.pl	he.acc

'If Dani had been home tomorrow, we would’ve visited him.'

Palestinian Arabic is another:^[30]

iza	kaan	fi	l-bet	bukra	kunna	zurna-a
if	be.pst.3sm	in	the-house	tomorrow	be.pst.1pl	visit.pst.pfv.1pl-him

'If he had been home tomorrow, we would’ve visited him.'

Fake past is extremely prevalent cross-linguistically, either on its own or in combination with other morphemes. Moreover, theoretical linguists and philosophers of language have argued that other languages' strategies for marking counterfactuality are actually realizations of fake tense along with other morphemes. For this reason, fake tense has often been treated as the locus of the counterfactual meaning itself.^[26][31]

Formal analyses

In formal semantics and philosophical logic, fake past is regarded as a puzzle, since it is not obvious why so many unrelated languages would repurpose a tense morpheme to mark counterfactuality. Proposed solutions to this puzzle divide into two camps: past as modal and past as past. These approaches differ in whether or not they take the past tense's core meaning to be about time.^[32][33]

In the past as modal approach, the denotation of the past tense is not fundamentally about time. Rather, it is an underspecified skeleton which can apply either to modal or temporal content.^[26][32][34] For instance, the particular past as modal proposal of Iatridou (2000), the past tense's core meaning is what's shown schematically below:

1. The topic x is not the contextually-provided x

Depending on how this denotation composes, x can be a time interval or a possible world. When x is a time, the past tense will convey that the sentence is talking about non-current times, i.e. the past. When x is a world, it will convey that the sentence is talking about a potentially non-actual possibility. The latter is what allows for a counterfactual meaning.

The past as past approach treats the past tense as having an inherently temporal denotation. On this approach, so-called fake tense isn't actually fake. It differs from "real" tense only in how it takes scope, i.e. which component of the sentence's meaning is shifted to an earlier time. When a sentence has "real" past marking, it discusses something that happened at an earlier time; when a sentence has so-called fake past marking, it discusses possibilities that were accessible at an earlier time but may no longer be.^[35][36][37]