Counterfactual conditional

Counterfactual conditionals (also subjunctive or X-marked) are conditionals which discuss what would have been true under 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 tense marking, which some languages use in combination with other kinds of morphology including aspect and mood.[1][2][3] Counterfactuals are contrasted with indicatives, which have no extra grammatical marking and are generally restricted to discussing what might actually be. Counterfactual conditionals are one of the most studied phenomena in formal semantics, philosophy of language, and philosophical logic. They have also played a role in other fields including psychology,[4][5][6][7] history,[8] political science,[9] economics,[10] law,[11] marketing,[12] and epidemiology.[13]

"Counterfactual" redirects here. For other uses, see Counterfactual (disambiguation).

Overview

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 fake 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.

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 John F. Kennedy|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,[14] extending Roderick Chisholm's earlier notion of a contrary-to-fact conditional.[15] 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.[16][17]

Anderson Case: If the patient had taken arsenic, he would have blue spots.[18]

The term subjunctive conditional has been used as an alternative, even though it is likewise acknowledged as a misnomer.[19] Many languages do not have a morphological subjunctive (e.g. Danish and Dutch) and many that do have it don’t use it for this sort of conditional (e.g. French, Swahili, all Indo-Aryan languages that have a subjunctive). Moreover, languages that do use the subjunctive for such conditionals only do so if they have a specific past subjunctive form. [20][21][22]

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.[23][24][25]

The antecedent of a conditional is sometimes referred to as its "if"-clause or protasis. The consequent of a conditional is sometimes referred to as a "then"-clause or as an apodosis.

Logic and semantics

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.[26]

Classic puzzles

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.[27]

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.[28]

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:[29]

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:[30] [31][32]

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 was 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: ${\displaystyle P>Q\models (P\land R)>Q}$

Possible worlds accounts

The most common logical accounts of counterfactuals are couched in the possible world semantics. Broadly speaking, these approaches have in common that they treat a counterfactual A > B as true if B holds across some set of possible worlds where A is true. They vary mainly in how they identify the set of relevant A-worlds.

David Lewis's variably strict conditional is considered the classic analysis within philosophy. The closely related premise semantics proposed by Angelika Kratzer is often taken as the standard within linguistics. However, there are numerous possible worlds approaches on the market, including dynamic variants of the strict conditional analysis originally dismissed by Lewis.

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.[33] 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.[34]

Variably strict conditional

David Lewis developed the standard version of the variably strict analysis in his 1973 book Counterfactuals.

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."

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 (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. [35]

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:

"… we may define a cause to be an object, followed by another, and where all objects, similar to the first, are followed by objects similar to the second. Or in other words, where, if the first object had not been, the second never had existed …" — David Hume, An Enquiry Concerning Human Understanding.[36]

Pearl

The tight connection between causal and counterfactual relations has prompted Judea Pearl (2000) 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" (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.[37]

The grammar of counterfactuality

Languages use different strategies for expressing counterfactuality. Languages may have a dedicated counterfactual morphemes, or they may recruit some combination of tense, aspect, and mood morphemes. Since the early 2000s, linguists, philosophers of language, and philosophical logicians have intensely studied the nature of this grammatical marking, and it continues to be an active area of study.

Fake tense

Many languages indicate counterfactuality using past tense morphology.[38] When the past tense is used this way, it is called fake past or fake tense since it does not seem to contribute its typical temporal meaning.[39][40][41] Fake past marking is sometimes erroneously referred to as "subjunctive", even though it is not the subjunctive mood.[40][41] One handbook article cautions that "it is a mistake to confuse this correlation of form and function with the subjunctive mood".[40]

English is one language which uses fake past to mark counterfactuality, as evidenced by the following contrast. In the indicative example, the bolded words are in the present tense. In the counterfactual example, both words are in the past tense. This past tense cannot have its ordinary temporal meaning, since it does not clash with the adverb "tomorrow".[38][39][40][41]

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:[42]

	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:[42]

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.[39][43]

In theoretical linguistics, 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 approaches and past as past approaches.[44][45] These approaches differ in what they take the core meanings of past tense morphemes to be. In the past as modal approach, past tense isn't inherently temporal. Rather, its core meaning is an underspecified skeleton which can be applied either in the temporal or modal domains.[39][44][46] For instance, in Iatridou's (2000) analysis, the past tense's core meaning requires that the topic x is not the contextually-provided x. When x is a point in time, this means that the sentence is talking about the past. When x is a possible world, this means that the sentence is talking about non-actual situations. On the other hand, the past as past approach treats the past tense as having an inherently temporal core meaning. On this approach, so-called fake tense isn't actually fake. It differs from "real" tense only in the way 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 fake past marking, it discusses things that were possibilities at an earlier time.[47][48][49]

Fake aspect

Fake aspect often accompanies fake tense in languages that mark aspect. In some languages (e.g. Modern Greek, Zulu, and the Romance languages) this fake aspect is imperfective. In other languages (e.g. Palestinian Arabic) it is perfective. However, in other languages including Russian and Polish, counterfactuals can have either perfective or imperfective aspect.[43]

Fake imperfective aspect is demonstrated by the two Modern Greek sentences below. These examples form a minimal pair, since they are identical except that the first uses past imperfective marking where the second uses past perfective marking. As a result of this morphological difference, the first has a counterfactual meaning, while the second does not.[39]

An	eperne	afto	to	sirpoi	θa	γinotan	kala
if	take.PST.IMPV	this		syrup	FUT	become.PST.IMPV	well

'If he took this syrup, he will get better'

An	ipχe	afto	to	sirpoi	θa	eγine	kala
if	take.PST.PFV	this		syrup	FUT	become.PST.PFV	well

'If he took this syrup, he must be better'

This imperfective marking has been argued to be fake on the grounds that it is compatible with completive adverbials such as "in one month":[39]

An

eχtizes

to

spiti

(mesa)

se

ena

mina

θa

prolavenes

na

to

pulisis

prin

to

kalokeri

if

build.IMPV

the

house

in

one

month

FUT

have-time-enough.IMPV

to

it

sell

before

the

summer

'If you built this house in a month, you would be able to sell it before the summer.'

In ordinary non-conditional sentences, such adverbials are compatible with perfective aspect but not with imperfective aspect:[39]

Eχtise	afto	to	spiti	(mesa)	se	ena	mina
build.PFV	this		house	in		one	month

'She built this house in one month'

*Eχtize	afto	to	spiti	(mesa)	se	ena	mina
build.IMPV	this		house	in		one	month

'She was building this house in one month'

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’.[5] 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.[50] They spent different amounts of time 'updating' a story that contains a counterfactual conditional compared to one that contains factual information[51] and focused on different parts of counterfactual conditionals.[52]

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.[53] 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

Byrne 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'.[54] According to the mental model theory of reasoning, they construct mental models of the alternative possibilities.[55]

See also

• Conditional sentence
• Indicative conditional
• Modal logic
• Causality
• David Lewis
• Angelika Kratzer
• Sabine Iatridou

Footnotes

1. Iatridou, Sabine (2000). "The grammatical ingredients of counterfactuality" (PDF). Linguistic Inquiry. 31 (2): 231–270. doi:10.1162/002438900554352.
2. Kaufmann, Stefan (2005). "Conditional predictions". Linguistics and Philosophy. 28 (2). 183-184. doi:10.1007/s10988-005-3731-9. S2CID 60598513.
3. Egré, Paul; Cozic, Mikaël (2016). "Conditionals". In Aloni, Maria; Dekker, Paul (eds.). Cambridge Handbook of Formal Semantics. Cambridge University Press. p. 515. ISBN 978-1-107-02839-5.
4. Roese, N.J. & Olson, J.M. (eds.), What Might Have Been: The Social Psychology of Counterfactual Thinking, Lawrence Erlbaum Associates, (Mahwah), 1995;
   • Sanna, L.J., "Defensive Pessimism, Optimism, and Simulating Alternatives: Some Ups and Downs of Prefactual and Counterfactual Thinking", Journal of Personality and Social Psychology, Vol. 71, No. 5, (November 1996), pp. 1020–36;
   • Roese, N.J., "Counterfactual Thinking", Psychological Bulletin, Vol. 121, No. 1, (January 1997), pp. 133–48;
   • Sanna, L.J., "Defensive Pessimism and Optimism: The Bitter-Sweet Influence of Mood on Performance and Prefactual and Counterfactual Thinking", Cognition and Emotion, Vol. 12, No. 5, (September 1998), pp. 635–65;
   • Sanna, L.J. & Turley-Ames, K.J., "Counterfactual Intensity", European Journal of Social Psychology, Vol. 30, No. 2, (March/April 2000), pp. 273–96;
   • Sanna, L.J., Parks, C.D., Meier, S., Chang, E.C., Kassin, B.R., Lechter, J.L., Turley-Ames, K.J. & Miyake, T.M., "A Game of Inches: Spontaneous Use of Counterfactuals by Broadcasters During Major League Baseball Playoffs", Journal of Applied Social Psychology, Vol. 33, No. 3, (March 2003), pp. 455–75, etc.
5. Fillenbaum, Samuel (1974). "Information amplified: Memory for counterfactual conditionals". Journal of Experimental Psychology. 102 (1): 44–49. doi:10.1037/h0035693.
6. Crawford, M.T. & McCrea, S.M., "When Mutations meet Motivations: Attitude Biases in Counterfactual Thought", Journal of Experimental Social Psychology, Vol. 40, No. 1, (January 2004), pp. 65–74, etc.
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8. Greenberg, M. (ed.), The Way It Wasn’t: Great Science Fiction Stories of Alternate History, Citadel Twilight, (New York), 1996;
   • Dozois, G. & Schmidt, W. (eds.), Roads Not Taken: Tales of Alternative History, The Ballantine Publishing Group, (New York), 1998;
   • Sylvan, D. & Majeski, S., "A Methodology for the Study of Historical Counterfactuals", International Studies Quarterly, Vol. 42, No. 1, (March 1998), pp. 79–108;
   • Ferguson, N., (ed.), Virtual History: Alternatives and Counterfactuals, Basic Books, (New York), 1999;
   • Cowley, R. (ed.), What If?: The World’s Foremost Military Historians Imagine What Might have Been, Berkley Books, (New York), 2000;
   • Cowley, R. (ed.), What If? 2: Eminent Historians Imagine What Might have Been, G.P. Putnam’s Sons, (New York), 2001, etc.
9. Fearon, J.D., "Counterfactuals and Hypothesis Testing in Political Science", World Politics, Vol. 43, No. 2, (January 1991), pp. 169–95;
   • Tetlock, P. E. & Belkin, A. (eds.), Counterfactual Thought Experiments in World Politics, Princeton University Press, (Princeton), 1996;
   • Lebow, R.N., "What’s so Different about a Counterfactual?", World Politics, Vol. 52, No. 4, (July 2000), pp. 550–85;
   • Chwieroth, J.M., "Counterfactuals and the Study of the American Presidency", Presidential Studies Quarterly, Vol. 32, No. 2, (June 2002), pp. 293–327, etc.
10. DN McCloskey (1987) “Counterfactuals” J Eatwell et al., The New Palgrave: A Dictionary of Economics (London: Macmillan), Vol. 1, pp. 701–03;
    • J Heckman (2001) “Econometrics, Counterfactuals and Causal Models.” Keynote Address International Statistical Institute. Seoul, Korea;
    • R Cowan & R Foray 2002) "Evolutionary Economics and the Counterfactual Threat: On the Nature and Role of Counterfactual History as an Empirical Tool in Economics" Journal of Evolutionary Economics, Vol. 12, No. 5 (December), pp. 539–62;
    • JG Hülsmann (2003) “Facts and Counterfactuals in Economic Law” Journal of Libertarian Studies, Vol. 17, No. 1, pp. 57–102;
    • N Cartwright (2007) “Counterfactuals in Economics: A Commentary” Hunting Causes and Using Them – Approaches in Philosophy and Economics (Cambridge, Cambridge University Press), pp. 236ff;
    • M Hashem Pesaran & RP Smith (2012) “Counterfactual Analysis in Macroeconometrics: An Empirical Investigation into the Effects of Quantitative Easing” Research in Economics, Vol. 70, No. 2 (2016), pp. 262–80.
11. Strassfeld, R.N., "If...: Counterfactuals in the Law", George Washington Law Review, Volume 60, No. 2, (January 1992), pp. 339–416;
    • Spellman, B.A. & Kincannon, A., "The Relation between Counterfactual (“but for”) and Causal reasoning: Experimental Findings and Implications for Juror’s Decisions", Law and Contemporary Problems, Vol. 64, No. 4, (Autumn 2001), pp. 241–264;
    • Prentice, R.A. & Koehler, J.J., "A Normality Bias in Legal Decision Making", Cornell Law Review, Vol. 88, No. 3, (March 2003), pp. 583–650, etc.
12. Hetts, J.J., Boninger, D.S., Armor, D.A., Gleicher, F. & Nathanson, A., "The Influence of Anticipated Counterfactual Regret on Behavior", Psychology & Marketing, Vol. 17, No. 4, (April 2000), pp. 345–68;
    • Landman, J. & Petty, R., "“It Could Have Been You”: How States Exploit Counterfactual Thought to Market Lotteries", Psychology & Marketing, Vol. 17, No. 4, (April 2000), pp. 299–321;
    • McGill, A.L., "Counterfactual Reasoning in Causal Judgements: Implications for Marketing", Psychology & Marketing, Vol. 17, No. 4, (April 2000), pp. 323–43;
    • Roese, N.J., "Counterfactual Thinking and Marketing: Introduction to the Special Issue", Psychology & Marketing, Vol. 17, No. 4, (April 2000), pp. 277–80;
    • Walchli, S.B. & Landman, J., "Effects of Counterfactual Thought on Postpurchase Consumer Affect", Psychology & Marketing, Vol. 20, No. 1, (January 2003), pp. 23–46, etc.
13. Randerson, J., "Fast action would have saved millions", New Scientist, Vol. 176, No. 2372, (7 December 2002), p. 19;
    • Haydon, D.T., Chase-Topping, M., Shaw, D.J., Matthews, L., Friar, J.K., Wilesmith, J. & Woolhouse, M.E.J., "The Construction and Analysis of Epidemic Trees With Reference to the 2001 UK Foot-and-Mouth Outbreak", Proceedings of the Royal Society of London Series B: Biological Sciences, Vol. 270, No. 1511, (22 January 2003), pp. 121–27, etc.
14. Goodman, N., "The Problem of Counterfactual Conditionals", The Journal of Philosophy, Vol. 44, No. 5, (27 February 1947), pp. 113–28.
15. Chisholm, R.M., "The Contrary-to-Fact Conditional", Mind, Vol. 55, No. 220, (October 1946), pp. 289–307.
16. von Fintel, Kai (1998). "The Presupposition of Subjunctive Conditionals" (PDF). In Sauerland, Uli; Percus, Oren (eds.). The Interpretive Tract. Cambridge University Press. pp. 29–44.
17. Egré, Paul; Cozic, Mikaël (2016). "Conditionals". In Aloni, Maria; Dekker, Paul (eds.). Cambridge Handbook of Formal Semantics. Cambridge University Press. p. 515. ISBN 978-1-107-02839-5.
18. Anderson, Alan (1951). "A Note on Subjunctive and Counterfactual Conditionals". Analysis. 12 (2): 35–38. doi:10.1093/analys/12.2.35.
19. See for instance Ippolito, Michela (2013). Subjunctive conditionals. MIT Press. ISBN 9780262019484.: "I use the traditional term "subjunctive" to refer to [these conditionals] even though there isn’t anything clearly subjunctive about the verbal morphology in its clauses... by labeling a conditional "subjunctive", I indicate the presence of temporal morphology that is not interpreted as locating the eventuality described in the antecedent clause in time. "
20. Iatridou, Sabine (2000). "The grammatical ingredients of counterfactuality" (PDF). Linguistic Inquiry. 31 (2): 231–270. doi:10.1162/002438900554352.
21. Kaufmann, Stefan (2005). "Conditional predictions". Linguistics and Philosophy. 28 (2). 183-184. doi:10.1007/s10988-005-3731-9. S2CID 60598513.
22. Egré, Paul; Cozic, Mikaël (2016). "Conditionals". In Aloni, Maria; Dekker, Paul (eds.). Cambridge Handbook of Formal Semantics. Cambridge University Press. p. 515. ISBN 978-1-107-02839-5.
23. von Fintel, Kai; Iatridou, Sabine. Prolegomena to a theory of X-marking Unpublished lecture slides.
24. von Fintel, Kai; Iatridou, Sabine. X-marked desires or: What wanting and wishing crosslinguistically can tell us about the ingredients of counterfactuality Unpublished lecture slides.
25. Schulz, Katrin (2017). Proceedings from Semantics and Linguistic Theory. Semantics and Linguistic Theory. Linguistic Society of America. pp. 547–570. doi:10.3765/salt.v27i0.4149.
26. Starr, Will (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
27. Goodman, N., "The Problem of Counterfactual Conditionals", The Journal of Philosophy, Vol. 44, No. 5, (27 February 1947), pp. 113–28.
28. Starr, Will (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
29. Lewis, David (1979). "Counterfactual dependence and time's arrow". Noûs. 13 (4): 455–476. doi:10.2307/2215339. JSTOR 2215339. Counterfactuals are infected with vagueness, as everyone agrees.
30. Goodman, N., "The Problem of Counterfactual Conditionals", The Journal of Philosophy, Vol. 44, No. 5, (27 February 1947), pp. 113–28.
31. Lewis, David (1973). "Counterfactuals and Comparative Possibility". Journal of Philosophical Logic. 2 (4). doi:10.2307/2215339. JSTOR 2215339.
32. Lewis, David (1973). Counterfactuals. Cambridge, MA: Harvard University Press. ISBN 9780631224952.
33. Starr, Will (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
34. Starr, Will (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
35. "Review of the paper: M. L. Ginsberg, "Counterfactuals," Artificial Intelligence 30 (1986), pp. 35–79", Zentralblatt für Mathematik, FIZ Karlsruhe – Leibniz Institute for Information Infrastructure GmbH, pp. 13–14, 1989, Zbl 0655.03011.
36. Hume, D. (Beauchamp, T.L., ed.), An Enquiry Concerning Human Understanding, Oxford University Press, (Oxford), 1999, (7), p. 146: emphasis in the original).
37. Pearl, Judea (2000). Causality. Cambridge University Press.
38. Palmer, Frank Robert (1986). Mood and modality. Cambridge University Press.
39. Iatridou, Sabine (2000). "The grammatical ingredients of counterfactuality" (PDF). Linguistic Inquiry. 31 (2): 231–270. doi:10.1162/002438900554352.
40. Portner, Paul (2009). Modality. Oxford University Press. ISBN 0199292426.
41. von Fintel, Kai; Iatridou, Sabine (2020). Prolegomena to a Theory of X-Marking. Manuscript.
42. Karawani, Hadil (2014). The Real, the Fake, and the Fake Fake in Counterfactual Conditionals, Crosslinguistically (PDF) (Thesis). Universiteit van Amsterdam.
43. Bjorkman, Bronwyn; Halpert, Claire (2013). "In search of (im)perfection: the illusion of counterfactual aspect" (PDF). In Keine, Stefan; Sloggett, Shayne (eds.). Proceedings of NELS 42. NELS. 42. UMass Amherst GLSA.
44. Schulz, Katrin (2014). "Fake tense in conditional sentences: A modal approach". Natural Language Semantics. 22 (2).
45. Starr, Will (2019). "Supplement to "Counterfactuals": Indicative and Subjunctive Conditionals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
46. Mackay, John (2019). "Modal interpretation of tense in subjunctive conditionals". Semantics and Pragmatics. 12 (2).
47. Arregui, Ana (2007). "When aspect matters: the case of would-conditionals". Natural Language Semantics. 15 (3).
48. Ippolito, Michela (2003). "Presuppositions and implicatures in counterfactuals". Natural Language Semantics. 11 (2).
49. Khoo, Justin (2015). "On Indicative And Subjunctive Conditionals" (PDF). Philosophers' Imprint. 15.
50. Santamaría, Carlos; Espino, Orlando; Byrne, Ruth M. J. (2005). "Counterfactual and Semifactual Conditionals Prime Alternative Possibilities" (PDF). Journal of Experimental Psychology: Learning, Memory, and Cognition. 31 (5): 1149–1154. doi:10.1037/0278-7393.31.5.1149. PMID 16248757.
51. De Vega, Manuel; Urrutia, Mabel; Riffo, Bernardo (2007). "Canceling updating in the comprehension of counterfactuals embedded in narratives" (PDF). Memory & Cognition. 35 (6): 1410–1421. doi:10.3758/BF03193611. PMID 18035637. S2CID 26161334.
52. Ferguson, Heather; Sanford, Anthony (2008). "Anomalies in real and counterfactual worlds: An eye-movement investigation" (PDF). Journal of Memory and Language. 58 (3): 609–626. doi:10.1016/j.jml.2007.06.007.
53. Byrne, Ruth M. J.; Tasso, Alessandra (1999). "Deductive reasoning with factual, possible, and counterfactual conditionals". Memory & Cognition. 27 (4): 726–740. doi:10.3758/BF03211565. PMID 10479830.
54. Byrne, Ruth M. J. (2005). The Rational Imagination. doi:10.7551/mitpress/5756.001.0001. ISBN 9780262269629.
55. Johnson-Laird, Philip Nicholas; Byrne, Ruth M. J. (1991). Deduction. ISBN 9780863771491.

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