反事实

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

Counterfactual conditionals (also subjunctive or X-marked) are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood.

反事实条件句 Counterfactual conditionals（虚拟条件或X标记的）是用来讨论在不同情况下什么为真的的 条件句 conditional sentence。例如：如果彼得相信鬼魂的存在，他就会害怕来到这里。反事实句与指示句形成对比，后者一般只限于讨论开放的可能性。反事实动词的语法特征是使用虚拟时态语法，这种虚拟时态语法与时态和语态等其他语法结合使用。

Counterfactuals are one of the most studied phenomena in philosophical logic, formal semantics, and philosophy of language. They were first discussed as a problem for the material conditional analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic possible world approach, in which a counterfactual's truth hinges on its consequent holding at certain possible worlds where its antecedent holds. More recent formal analyses have treated them using tools such as causal models and dynamic semantics. Other research has addressed their metaphysical, psychological, and grammatical underpinnings, while applying some of the resultant insights to fields including history, marketing, and epidemiology.

反事实是哲学逻辑、形式语义学和语言哲学中研究最多的现象之一。它们首先作为条件句的实质条件分析的问题被讨论，条件句把它们都当作是显而易见的事实。从20世纪60年代开始，哲学家和语言学家发展出现在经典的可能世界方法，在这种方法中，反事实的真理取决于它在某些可能世界中的后果，而这些可能世界的前因是成立的。最近的形式化分析使用因果模型和动态语义等工具对它们进行了处理。其他研究已经解决了他们的形而上学、心理学和语法基础，同时将一些结果的见解应用到包括历史，市场营销和流行病学领域。

Overview

Examples

The difference between indicative and counterfactual conditionals can be illustrated by the following minimal pair:

指示条件句和反事实条件句之间的区别可以用下面的简单例子来说明：

1. Indicative Conditional: If it is raining right now, then Sally is inside.

1. 指示条件句： 如果现在正在下雨，那么Sally 就在里面（If it is raining right now, then Sally is inside）。

1. Simple Past Counterfactual: If it was raining right now, then Sally would be inside.

1. 一般过去时的反事实：如果现在正在下雨，那么Sally就会在里面（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.

这些条件句在形式和意义上都不同。指示条件在“如果(if)”和 “那么(then)”两个从句中都使用现在时态is。因此，它传达了这样一种信息：说话人对是否下雨是不可知的。反事实的例句在“如果”从句中使用虚拟时态“was”，在“ then”句中使用情态“ would”。因此，它传达的信息是，说话人并不相信天在下雨。

English has several other grammatical forms whose meanings are sometimes included under the umbrella of counterfactuality. One is the past perfect counterfactual, which contrasts with indicatives and simple past counterfactuals in its use of pluperfect morphology:

英语还有其他几种语法形式，其含义有时被包括在反事实的范畴内。其中一个是过去完成时的反事实，在使用过去完成时态时，它与指示词和一般完成时的反事实形成对比：

1. Past Perfect Counterfactual: If it had been raining yesterday, then Sally would have been inside.

1. 过去完成时的反事实：如果昨天下了雨，那么Sally就会在里面(If it had been raining yesterday, then Sally would have been inside)。

Another kind of conditional uses the form "were", generally referred to as the irrealis or subjunctive form.

Another kind of conditional uses the form "were", generally referred to as the irrealis or subjunctive form.

另一种条件句的用法是“were”，一般称为“非现实（irrealis）”或“虚拟（subjunctive）”。

1. Irrealis Counterfactual: If it were raining right now, then Sally would be inside.

1. 非现实反事实（Irrealis Counterfactual）：如果现在正在下雨，那么Sally应该在里面(If it were raining right now, then Sally would be inside)。

Past perfect and irrealis counterfactuals can undergo conditional inversion:

过去完成时和非现实的反事实可以进行条件倒置：

1. Were it raining, Sally would be inside.

1. 如果下雨的话，Sally就会在里面（Were it raining, Sally would be inside）。

1. Had it rained, Sally would be inside.

1. 那时如果下雨的话，Sally就会在里面（Had it rained, Sally would be inside）。

Terminology

<！——鉴于术语上的巨大但往往是细微的差异，本节必须经过仔细的编辑。在点击“发布更改”之前，请考虑结果文本是否有助于读者理解这些术语是如何使用的。如果结果文本读起来像是“热狗是三明治的辩论吗?”删除所有字符提示后，请不要点击”发布更改”。特别是，请确保(1)明确区分事实主张和术语定义(2)记住，不同的来源可以以不同的方式使用单一术语(3)对术语的每个术语或用法进行不偏不倚的框架性解释。-->

The term counterfactual conditional is widely used as an umbrella term for the kinds of sentences shown above. However, not all conditionals of this sort express contrary-to-fact 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 counterfactual conditional is widely used as an umbrella term for the kinds of sentences shown above. However, not all conditionals of this sort express contrary-to-fact 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.

“反事实条件（counterfactual conditional）”这一术语被广泛用作上述各类句子的总称。然而，并非所有这类条件句都表达与事实相反的意思。例如，被称为“ Anderson 案例”的经典例子具有反事实条件的典型语法形式，但是并不表明它的先行词是假的或不可能的。

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

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

1. Anderson案例：如果病人服用了砒霜，他会长出蓝斑（If the patient had taken arsenic, he would have blue spots）。

Such conditionals are also widely referred to as subjunctive conditionals, though this term is likewise acknowledged as a misnomer even by those who use it. 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. Thus, subjunctive marking is neither necessary nor sufficient for membership in this class of conditionals.

Such conditionals are also widely referred to as subjunctive conditionals, though this term is likewise acknowledged as a misnomer even by those who use it. 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. Thus, subjunctive marking is neither necessary nor sufficient for membership in this class of conditionals.

这种条件句也被广泛地称为虚拟条件句（subjunctive conditionals），尽管这个术语同样被使用者认为是用词不当。许多语言都没有虚拟语气（如丹麦语和荷兰语)，许多有从句的语言也不把它用于这种条件句（如法语、斯瓦希里语、所有有从句的印度-雅利安语）。此外，只有将虚拟语气用于此类条件的语言才具有特定的过去虚拟语气形式。因此，虚拟标记既不是必要的，也不是充分的。

The terms counterfactual and subjunctive have sometimes been repurposed for more specific uses. For instance, the term "counterfactual" is sometimes applied to conditionals that express a contrary-to-fact meaning, regardless of their grammatical structure. Along similar lines, the term "subjunctive" is sometimes used to refer to conditionals that bear fake past or irrealis marking, regardless of the meaning they convey.

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.

反事实（counterfactual） and 从句（subjunctive）这两个术语有时被重新用于更具体的用途。例如，不管其语法结构如何，"反事实"这个术语有时被用于表达与事实相反的意思的条件语。按照类似的思路，不管其表达的意思如何，"从句"这个术语有时被用于指带有虚拟过去或非现实标记的条件语。

最近有人提出用术语 X-Marked这个词来替代，以概括这些条件语所带有的额外标记。采用这个术语的人把指示性条件语称为O-Marked条件语，反映了它们的普通标记。

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.

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.

一个条件的 前件（antecedent）有时被称为 "如果"从句或条件子句。条件的结果有时被称为"那么"子句或结论子句。

Logic and semantics

经典问题（Classic puzzles）

反事实的问题（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.

根据实质条件的分析，自然语言条件句即“如果p，那么q（if P then Q）”的陈述，只要其前件p为假就是真。由于反事实条件句是那些前置假设的条件句，这种分析会错误地预测所有反事实条件句都是虚假的。Goodman在理解到正在讨论的那块黄油没有被加热的情况下，用下面的一对例子来说明这一点。

If that piece of butter had been heated to 150º, it would have melted.

如果那块黄油被加热到150度，它就会融化。

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.

If that piece of butter had been heated to 150º, it would not have melted.

如果那块黄油被加热到150度，它就不会融化。

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.

更一般地说，这些例子表明反事实不具备真理功能。换句话说，知道前件和结果是否为真并不足以确定反事实本身是否为真。

上下文依赖和含糊不清（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: 反事实是依赖于上下文且含糊不清的。例如，以下任一陈述都可以合理地成立，但不能同时成立：

If Caesar had been in command in Korea, he would have used the atom bomb.

1. 如果凯撒（Caesar）当时在朝鲜指挥，他会使用原子弹。

If Caesar had been in command in Korea, he would have used catapults.

1. 如果凯撒在朝鲜指挥，他会使用弹弓。

非单调性（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:

反事实是非单调的，因为它们的真值可以通过在其前件中添加额外的信息而改变。这一事实可以通过 Sobel 序列得到说明，例如：

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

1. 如果汉娜喝了咖啡，她会很高兴。
2. 如果汉娜喝了咖啡，而且咖啡里有汽油，她会很伤心。
3. 如果汉娜喝了咖啡，咖啡里有汽油，而汉娜是一个喝汽油的机器人，她会很高兴。

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.

对此事实进行形式化的一种方法是说，前件增强（Antecedent Strengthening）原则不适用于任何旨在作为自然语言条件句形式化的连接词>。

• Antecedent Strengthening: [math]\displaystyle{ P \gt Q \models (P \land R) \gt Q }[/math]

• 前件增强: [math]\displaystyle{ P \gt Q \models (P \land R) \gt Q }[/math]

考虑可能存在的世界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.

反事实的最常见的逻辑解释是可能世界语义学。一般来说，这些方法的共同点是，如果B在A成立的某些可能世界中成立，那么它们就认为反事实 A > B为真。它们的主要区别在于如何确定相关A世界集的方式。

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.

大卫·刘易斯（David Lewis）严格可变的条件被认为是哲学中的经典分析。安吉利卡·克拉策（Angelika Kratzer）提出的紧密相关的前提语义常常被视为语言学中的标准。然而，市场上有许多可能世界的方法，包括最初被Lewis摒弃的严格条件分析的动态变体。

严格的条件Strict conditional

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

严格条件分析将自然语言反事实视为等同于模态逻辑公式[math]\displaystyle{ \Box(P \rightarrow Q) }[/math]。在这个公式中， [math]\displaystyle{ \Box }[/math]表示必要性，[math]\displaystyle{ \rightarrow }[/math]被理解为实质条件。这种方法最早是在1912年由C.I. Lewis提出的，作为他对模态逻辑的公理化方法的一部分。

• Given a model [math]\displaystyle{ M = \langle W,R,V \rangle }[/math], we have that [math]\displaystyle{ M,w \models \Box(P \rightarrow Q) }[/math] iff [math]\displaystyle{ M, v \models P \rightarrow Q }[/math] for all [math]\displaystyle{ v }[/math] such that [math]\displaystyle{ Rwv }[/math]
• 给定一个模型 [math]\displaystyle{ M = \langle W,R,V \rangle }[/math], 对于所有 [math]\displaystyle{ v }[/math] 使得 [math]\displaystyle{ Rwv }[/math]， 当且仅当[math]\displaystyle{ M, v \models P \rightarrow Q }[/math] ，我们有 [math]\displaystyle{ M,w \models \Box(P \rightarrow Q) }[/math]。

Unlike the material conditional, the strict conditional is not vacuously true when its antecedent is false. To see why, observe that both [math]\displaystyle{ P }[/math] and [math]\displaystyle{ \Box(P \rightarrow Q) }[/math] will be false at [math]\displaystyle{ w }[/math] if there is some accessible world [math]\displaystyle{ v }[/math] where [math]\displaystyle{ P }[/math] is true and [math]\displaystyle{ Q }[/math] 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.

与实质条件不同，严格条件在其前件为假时严格为真。要知道为什么，请观察，如果有一些可能世界[math]\displaystyle{ v }[/math]，其中[math]\displaystyle{ P }[/math]为真，[math]\displaystyle{ Q }[/math]为假，那么[math]\displaystyle{ P }[/math]和 [math]\displaystyle{ \Box(P \rightarrow Q) }[/math]在[math]\displaystyle{ w }[/math]处都为假。严格条件也是依赖于上下文的，至少在给定关系语义（或类似的东西）时是如此。在关系框架中，可及性关系是评价的参数，它编码了在上下文中被视为 "活 "的可能性范围。由于严格条件的真实性可能取决于用来评价它的可及性关系，所以严格条件的这一特征可以用来捕捉上下文的依赖性。

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 [math]\displaystyle{ P \rightarrow Q }[/math] holds at every world accessible from [math]\displaystyle{ w }[/math], the monotonicity of the material conditional guarantees that [math]\displaystyle{ P \land R \rightarrow Q }[/math] will be too. Thus, we will have that [math]\displaystyle{ \Box(P \rightarrow Q) \models \Box(P \land R \rightarrow Q) }[/math].

严格条件分析遇到了许多已知的问题，特别是单调性。在经典的关系框架中，当使用标准的蕴涵概念时，严格条件是单调的，也就是说，它验证了前件增强。要知道为什么，观察一下，如果[math]\displaystyle{ P \rightarrow Q }[/math]在每个来自[math]\displaystyle{ w }[/math]的世界上成立。那么物质条件的单调性保证了 [math]\displaystyle{ P \land R \rightarrow Q }[/math] 也将是如此。因此，我们将有[math]\displaystyle{ \Box(P \rightarrow Q) \models \Box(P \land R \rightarrow Q) }[/math]。

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.

这一事实导致了对严格条件的广泛放弃，特别是支持刘易斯的可变严格分析。然而，随后的工作通过对语境敏感性的诉求恢复了严格条件分析。这种方法是由Warmbrōd（1981）开创的，他认为Sobel序列 并不要求非单调逻辑，而事实上，随着序列的进行，说话人可以切换到更宽松的可及性关系来解释。在他的系统中，像“如果Hannah喝了咖啡，她会很高兴”这样的反事实，通常会用Hannah的咖啡在所有可及世界中不含汽油的模型进行评价。如果这个模型被用来评估随后的“如果汉娜喝了咖啡，而咖啡里有汽油……”的话语，这个第二个条件就会被认为是微不足道的真实，因为没有任何可访问的世界的前件是成立的。Warmbrōd的想法是，说话人将转向一个具有更宽松的可及性关系的模型，以避免这种琐碎性。

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.

Kai von Fintel（2001）、Thony Gillies（2007）和Malte Willer（2019）的后续工作在动态语义学的框架内将这一想法正式化，并给出了一些支持的语言学论据。其中一个论点是，条件前置词许可否定性词语，而这些词被认为只能由单调性运算符许可。

 If Natalia leaves tomorrow, she will arrive on time.

1. 如果Natalia明天离开，她会准时到达。

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.

If Hannah had drunk coffee with gasoline in it, she would not be happy. But if she had drunk coffee, she would be happy.

1. 如果Hannah喝了含有汽油的咖啡，她就不会高兴。但如果她喝了咖啡，她就会高兴。

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.

Sarah Moss（2012）和Karen Lewis（2018）对这些论点做出了回应，表明一个版本的可变严格分析可以解释这些模式，并认为这样的解释是可取的，因为它也可以解释明显的例外情况。截至2020年，这一争论在文献中仍在继续，Willer（2019）等人认为，严格条件账户也可以涵盖这些例外情况。

可变严格条件 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.

在可变严格方法中，条件A > B的语义是由一些函数给出的，一方面是A为真、B为真的世界，另一方面是A为真、B为假的世界的相对接近程度。

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.

在刘易斯的论述中，A > C 是(a)空洞的真实，只有在没有A为真的世界时（例如，如果A在逻辑上或形而上学上是不可能的）；(b)非空洞的真实，只有在A为真的世界中，一些C为真的世界比任何C不为真的世界更接近实际世界；或者(c)虚假，在其他世界里。尽管在刘易斯的《反事实》中，他对“接近性（closeness）”的意思并不清晰，但在后来的著作中，刘易斯明确表示，他并不打算将“接近性”的尺度简单地作为我们对整体相似性的普通概念。

Example:

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

例子：

如果他在早餐时吃多一点，他在上午11点就不会饿。

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.

根据刘易斯的说法，这个陈述的真理在于：在他早餐吃得更多的可能世界中，至少有一个他在上午11点不饿的世界比任何他早餐吃得更多但在上午11点仍然饿的世界更接近我们的世界。

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. For instance, the particular past as modal proposal of Iatridou (2000), the past tense's core meaning is what's shown schematically below:

过去式作为情态动词的方法，过去时的外延从根本上说不是关于时间的。相反，它是一个未指定的框架，既可以应用于模态内容，也可以应用于时态内容。例如，Iatridou (2000)的特殊过去式作为情态提议，过去式的核心含义是下面的图示:

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.)

Stalnaker的论述与Lewis的论述最明显的不同在于，他接受了“极限（limit）”和“唯一性假设（uniqueness assumptions）”。唯一性假设的论点是：对于任何前件A，在A为真的可能世界中，有一个最接近实际世界的单一（唯一）世界。极限假设的论点是，对于一个给定的前件A，如果存在一个A为真的可能世界链，每个世界都比它的前一个世界更接近实际世界，那么这个链就有一个极限：一个A为真的可能世界比这个链中的所有世界更接近实际世界。（唯一性假设包含了极限假设，但极限假设并不包含唯一性假设）。根据Stalnaker的观点，当且仅当在最接近A为真的世界中，C为真时，A>C才是非空洞的真。因此，上面的例子是真的，只是在他吃了更多早餐的唯一最接近的世界中，他在上午11点不觉得饿。虽然有争议，但Lewis拒绝了极限假设（因此也拒绝了唯一性假设），因为它排除了这样一种可能性，即可能存在着越来越接近实际世界的世界，而没有极限。例如，可能会有一系列无限的世界，每个世界的咖啡杯都在其实际位置的左边小几分之一英寸，但其中没有一个是唯一最接近的。（见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:

Stalnaker接受唯一性假设的一个结果是，如果排除中间律是真的，那么公式(A>C)∨(A>¬C)的所有实例都是真的。排他性中间律的论题是：对于所有命题p，p∨¬p都是真的。如果唯一性假设为真，那么对于每一个前件A，都有一个唯一最接近的世界，其中A为真。如果排除中间法则是真的，任何结果C在A为真的那个世界里要么是真，要么是假。所以对于每一个反事实A>C，要么A>C，要么A>¬C为真。这就是所谓的条件排除中间法（CEM）。例子：

(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).

(1) 如果公平的硬币被抛出，它将会正面朝上。

(2) 如果公平的硬币被抛出，它将会反面朝上（即不是正面朝上）。

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."

根据Stalnaker的分析，存在一个最接近的世界，在这个世界里，(1)和(2)中提到的公平的硬币被抛出，硬币要么正面朝上，要么反面朝上。因此，要么（1）是真，（2）是假，要么（1）是假，（2）是真。然而，根据Lewis的分析，(1)和(2)都是假的，因为公平的硬币正面朝上的世界并不比反面朝上的世界更接近或更远离。对Lewis来说，“如果硬币被抛出，它将正面朝上或反面朝上”是真的，但这并不意味着“如果硬币被抛出，它将落在正面”，或“如果硬币被抛出，它就会反面朝上”。

其他考虑 Other accounts

因果模型 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.

因果模型框架从结构方程（structural equations）系统的角度分析反事实。在一个方程系统中，每个变量都被分配了一个值，这个值是系统中其他变量的显式函数。给定这样一个模型，“如果X是X，Y就会是Y（Y would be y had X been x）”这个句子 (形式上为 X = x > Y = y )被定义为断言。如果我们用一个常数X = x取代当前决定 X的方程，并求解变量Y的方程组，得到的解将是Y = y。这个定义已被证明与可能世界语义学的公理兼容，并构成自然科学和社会科学中因果推理的基础。因为这些领域的每个结构方程都对应于一个熟悉的因果机制，这个因果机制可以被研究者进行有意义地推理。这种方法是由Judea Pearl(2000)提出的，作为编码关于因果关系的细粒度直觉的手段，这些直觉在其他提议的系统中难以捕捉。

信念修正 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.

在信念修正框架中，反事实是用 Ramsey检验的形式化实现来处理的。在这些系统中，当且仅当在当前的知识体系添加 A后得到的结果是B时，反事实A > B成立。这个条件将反事实条件与信念修正联系起来，因为对A > B的评价可以通过首先用A修正当前的知识，然后检查B在什么结果中是否为真。当A与当前的信念一致时，修正是很容易的，但在其他情况下可能会很难。每一个用于信念修正的语义都可以用于评价条件语句。反过来说，每一种评价条件语句的方法都可以被看作是一种执行修正的方法。

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.

Ginsberg(1986)提出了一种条件句的语义，它假定当前的信念形成了一组命题公式，考虑这些公式中与A一致的最大集合，并在每个集合中加入A。其理由是，这些最大集合中的每一个都代表了一种可能的信念状态，在这种状态下，A为真，且与原始状态尽可能相似。因此，当且仅当B在所有这些集合中都为真时，条件陈述句A > B才成立。