counterfactual
Смотреть что такое «counterfactual» в других словарях:
Counterfactual — may refer to: Counterfactual conditional, a grammatical form (which also relates to philosophy and logic) Counterfactual subjunctive, grammatical forms which in English are known as the past and pluperfect forms of the subjunctive mood… … Wikipedia
counterfactual — counterfactual, counterfactual conditional A proposition which states what would have followed had the actual sequence of events or circumstances been different. Thus, to claim that the Battle of Alamein altered the outcome of the Second World… … Dictionary of sociology
counterfactual — adj. contrary to fact; of assertions, ideas, assumptions. [WordNet 1.5] … The Collaborative International Dictionary of English
counterfactual — [kount΄ər fak′cho͞o əl] adj. contrary to the facts of an event, situation, etc. n. a counterfactual idea, assumption, or argument … English World dictionary
counterfactual — 1946, from COUNTER (Cf. counter ) + FACTUAL (Cf. factual) … Etymology dictionary
counterfactual — adjective Date: 1946 contrary to fact … New Collegiate Dictionary
counterfactual — counterfact, n. counterfactually, adv. /kown teuhr fak chooh euhl/, n. Logic. a conditional statement the first clause of which expresses something contrary to fact, as If I had known. [1945 50; COUNTER + FACTUAL] * * * … Universalium
counterfactual — 1. adjective /ˌkaʊntɚˈfæktʃuəl,ˌkaʊn.tə(ɹ)ˈfæk.tʃu.əl/ Contrary to the facts; untrue. 2. noun /ˌkaʊntɚˈfæktʃuəl,ˌkaʊn.tə(ɹ)ˈfæk.tʃu.əl/ a) A claim, hypothesis, or other belief that is contrary to the facts. In recent years there has been… … Wiktionary
counterfactual — (Roget s Thesaurus II) adjective Devoid of truth: false, specious, spurious, truthless, untrue, untruthful, wrong. See TRUE … English dictionary for students
counterfactual — n. statement which expresses what could or would happen under different circumstances … English contemporary dictionary
counterfactual — coun·ter·factual … English syllables
counterfactual
1 counterfactual
2 counterfactual
3 counterfactual
4 counterfactual
См. также в других словарях:
Counterfactual — may refer to: Counterfactual conditional, a grammatical form (which also relates to philosophy and logic) Counterfactual subjunctive, grammatical forms which in English are known as the past and pluperfect forms of the subjunctive mood… … Wikipedia
counterfactual — counterfactual, counterfactual conditional A proposition which states what would have followed had the actual sequence of events or circumstances been different. Thus, to claim that the Battle of Alamein altered the outcome of the Second World… … Dictionary of sociology
counterfactual — adj. contrary to fact; of assertions, ideas, assumptions. [WordNet 1.5] … The Collaborative International Dictionary of English
counterfactual — [kount΄ər fak′cho͞o əl] adj. contrary to the facts of an event, situation, etc. n. a counterfactual idea, assumption, or argument … English World dictionary
counterfactual — 1946, from COUNTER (Cf. counter ) + FACTUAL (Cf. factual) … Etymology dictionary
counterfactual — adjective Date: 1946 contrary to fact … New Collegiate Dictionary
counterfactual — counterfact, n. counterfactually, adv. /kown teuhr fak chooh euhl/, n. Logic. a conditional statement the first clause of which expresses something contrary to fact, as If I had known. [1945 50; COUNTER + FACTUAL] * * * … Universalium
counterfactual — 1. adjective /ˌkaʊntɚˈfæktʃuəl,ˌkaʊn.tə(ɹ)ˈfæk.tʃu.əl/ Contrary to the facts; untrue. 2. noun /ˌkaʊntɚˈfæktʃuəl,ˌkaʊn.tə(ɹ)ˈfæk.tʃu.əl/ a) A claim, hypothesis, or other belief that is contrary to the facts. In recent years there has been… … Wiktionary
counterfactual — (Roget s Thesaurus II) adjective Devoid of truth: false, specious, spurious, truthless, untrue, untruthful, wrong. See TRUE … English dictionary for students
counterfactual — n. statement which expresses what could or would happen under different circumstances … English contemporary dictionary
counterfactual — coun·ter·factual … English syllables
Counterfactuals
In philosophy, counterfactual modality has given rise to difficult semantic, epistemological, and metaphysical questions:
These questions have attracted significant attention in recent decades, revealing a wealth of puzzles and insights. While other entries address the epistemic—the epistemology of modality—and metaphysical questions—possible worlds and actualism—this entry focuses on the semantic question. It will aim to refine this question, explain its central role in certain philosophical debates, and outline the main semantic analyses of counterfactuals.
Section 1 begins with a working definition of counterfactual conditionals (§1.1), and then surveys how counterfactuals feature in theories of agency, mental representation, and rationality (§1.2), and how they are used in metaphysical analysis and scientific explanation (§1.3). Section 1.4 then details several ways in which the logic and truth-conditions of counterfactuals are puzzling. This sets the stage for the sections 2 and 3, which survey semantic analyses of counterfactuals that attempt to explain this puzzling behavior.
Section 2 focuses on two related analyses that were primarily developed to study the logic of counterfactuals: strict conditional analyses and similarity analyses. These analyses were not originally concerned with saying what the truth-conditions of particular counterfactuals are. Attempts to extend them to that domain, however, have attracted intense criticism. Section 3 surveys more recent analyses that offer more explicit models of when counterfactuals are true. These analyses include premise semantics (§3.1), conditional probability analyses (§3.2) and structural equations/causal models (§3.3). They are more closely connected to work on counterfactuals in psychology, artificial intelligence, and the philosophy of science.
Sections 2 and 3 of this entry employ some basic tools from set theory and logical semantics. But these sections also provide intuitive characterizations alongside formal definitions, so familiarity with these tools is not a pre-requisite. Readers interested in more familiarity with these tools will find basic set theory, as well as Gamut (1991) and Sider (2010) useful.
1. Counterfactuals and Philosophy
This section begins with some terminological issues (§1.1). It then provides two broad surveys of research that places counterfactuals at the center of key philosophical issues. Section 1.2 covers the role of counterfactuals in theories of rational agency, mental representation, and knowledge. Section 1.3 focuses on the central role of counterfactuals in metaphysics and the philosophy of science. Section 1.4 will then bring a bit of drama to the narrative by explaining how counterfactuals are deeply puzzling from the perspective of classical and modal logics alike.
1.1 What are Counterfactuals?
In philosophy and related fields, counterfactuals are taken to be sentences like:
This entry will follow this widely used terminology to avoid confusion. However, this usage also promotes a confusion worth dispelling. Counterfactuals are not really conditionals with contrary-to-fact antecedents. For example (2) can be used as part of an argument that the antecedent is true (Anderson 1951) :
(2) If there had been intensive agriculture in the Pre-Columbian Americas, the natural environment would have been impacted in specific ways. That is exactly what we find in many watersheds.
On these grounds, it might be better to speak instead of subjunctive conditionals, and reserve the term counterfactual for subjunctive conditionals whose antecedent is assumed to be false in the discourse. [1] While slightly more enlightened, this use of the term does not match the use of counterfactuals in the sprawling philosophical and interdisciplinary literature surveyed here, and has its own drawbacks that will be discussed shortly. This entry will use counterfactual conditional and subjunctive conditional interchangeably, hoping to now have dispelled the suggestion that all counterfactuals, in that sense, have contrary-to-fact antecedents.
The terminology of indicative and subjunctive conditionals is also vexed, but it aims to get at a basic contrast which begins between two different forms of conditionals that can differ in truth value. (3) and (4) can differ in truth-value while holding fixed the world they are being evaluated in. [2]
(3) If Oswald didn’t kill Kennedy, someone else did. (Indicative) (4) If Oswald hadn’t killed Kennedy, someone else would’ve. (Subjunctive)
It is easy to imagine a world where (3) is true, and (4) false. Consider a world like ours where Kennedy was assassinated. Further suppose Oswald didn’t do it, but some lone fanatic did for deeply idiosyncratic reasons. Then (3) is true and (4) false. Another aspect of the contrast between indicative and subjunctive conditionals is illustrated in (5) and (6).
(5) # Bob never danced. If Bob danced, Leland danced. (6) Bob never danced. If Bob had danced, Leland would have danced. (7) Bob never danced. If Bob were to dance, Leland would dance.
Indicatives like (5) are infelicitous when their antecedent has been denied, unlike the subjunctives like (6) and (7) ( Stalnaker 1975; Veltman 1986 ).
The indicative and subjunctive conditionals above differ from each other only in particular details of their linguistic form. It is therefore plausible to explain their contrasting semantic behavior in terms of the semantics of those linguistic differences. Indicatives, like (3) and (5), feature verbs in the simple past tense form, and no modal auxiliary in the consequent. Subjunctives, like (4) and (6), feature verbs in the past perfect (or “pluperfect”) with a modal would in the consequent. Something in the neighborhood of these linguistic and semantic differences constitutes the distinction between indicative and subjunctive conditionals—summarized in Figure 1. [3]
| Examples | Antecedents | Consequents | Deny Antecedent? | |
|---|---|---|---|---|
| Indicative | (3), (5) | V-ed, … | V-ed, … | Not felicitous |
| Subjunctive | (4), (6) | had V-ed, were to V, V-ed, … | would have V, would V, would have V-ed, … | Can be felicitous |
Figure 1: Rough Guide to Indicative and Subjunctive Conditionals
As with most neighborhoods, there are heated debates about the exact boundaries and the names—especially when future-oriented conditionals are included. These debates are surveyed in the supplement Indicative and Subjunctive Conditionals. The main entry will rely only on the agreed-upon paradigm examples like (3) and (4). The labels indicative and subjunctive are also flawed since these two kinds of conditionals are not really distinguished on the basis of whether they have indicative or subjunctive mood in the antecedent or consequent. [4] But the terminology is sufficiently entrenched to permit this distortion of linguistic reality.
Much recent work has been devoted to explaining how the semantic differences between indicative and subjunctive conditionals can be derived from their linguistic differences—rather than treating them as semantically unrelated. Much of this work has been done in light of Kratzer’s ( 1986, 2012 ) general approach to modality according to which all conditionals are treated as two-place modal operators. This approach is also discussed in the supplement Indicative and Subjunctive Conditionals. [5] This entry will focus on the basic logic and truth-conditions of subjunctive conditionals as a whole, and will use the following notation for them (following Stalnaker 1968 ). [6]
This project and notation has an important limitation that should be highlighted: it combines the meaning of the modal would and if…then… into a single connective “\(>\)”. This makes it difficult to adequately represent subjunctive conditionals like:
(8) a. If Maya had run, she might have been elected. b. If Maya had run, she might have been elected and would have been an excellent Senator. c. “Mr. Taft never asked my advice in the matter, but if he had asked it, I should have emphatically advised him against thus stating publicly his religious belief.” (Theodore Roosevelt) d. If Maya had run, she probably would have won and she might have won big.
Conditionals like (8a) have figured in debates about the semantics of counterfactuals and have been modeled either as a related connective (D. Lewis 1973b: §1.5) or a normal would-subjunctive conditional embedded under might ( Stalnaker 1980, 1984: Ch.7 ). But the more complex examples (8b)–(8d) highlight the need for a more refined compositional analysis, like those surveyed in Indicative and Subjunctive Conditionals. So, while this notation will be used in §1.4 and throughout §2 and §3, it should be regarded as an analytic convenience rather than a defensible assumption.
1.2 Agency, Mind, and Rationality
Counterfactuals have played prominent and interconnected roles in theories of rational agency. They have figured prominently in views of what agency and free will amount to, and played important roles in particular theories of mental representation, rational decision making, and knowledge. This section will outline these uses of counterfactuals and begin to paint a broader picture of how counterfactuals connect to central philosophical questions.
1.2.1 Agency, Choice, and Free Will
A defining feature of agents is that they make choices. Suppose a citizen votes, and in doing so chooses to vote for X rather than Y. It is hard to see how this act can be a choice without a corresponding counterfactual being true:
(9) If the citizen had wanted to vote for Y, they could have.
The idea that choice entails the ability to do otherwise has been taken by many philosophers to underwrite our practice of holding agents responsible for their choices. But understanding the precise meaning of the counterfactual could have claim in (9) requires navigating the classic problem of free will: if we live in a universe where the current state of the universe is determined (or near enough) by the prior state of the universe and the physical laws, then it seems like every action of every agent, including their “choices”, are predetermined. So interpreting this intuitively plausible counterfactual (9) leads quite quickly to a deep philosophical dilemma. One can maintain, with some Incompatibilists, that (9) is a false claim about what’s physically possible, and revisit the understanding of agency, choice, and responsibility above—the entry incompatibilist theories of free will explores this further. [7] Alternatively, one can maintain that (9) is a true claim about some non-physical sense of possibility, and explain how that is appropriate to our understanding of choice and responsibility—the entry compatibilism explores this further. It is wrong to construe debates about free will as just debates about the meaning of counterfactuals. But, the semantics of counterfactuals can have a substantive impact on delimiting the space of possible solutions, and perhaps even deciding between them. The same is true for research on counterfactual thinking in psychology.
1.2.2 Rationality
1.2.3 Mental Representation, Content, and Knowledge
In a Bayesian framework, probabilities are real numbers between 0 and 1 assigned to propositional variables A, B, C,…. These probabilities reflect an agent’s subjective credence, e.g., \(P(A)=0.6\) reflects that they think A is slightly more likely than not to be true. [10] At the heart of Bayesian Networks are the concepts of conditional probability and two variables being probabilistically independent. \(P(B \mid A)\) is the credence in B conditional on A being true and is defined as follows:
Conditional probabilities allow one to say when B is probabilistically independent of A: when an agent’s credence in B is the same as their credence in B conditional on A and conditional on \(\neg A\).
Bayesian networks represent relations of probabilistic dependence. For example, an agent’s knowledge about a system containing eight variables could be represented by the directed acyclic graph and system of structural equations between those variables in Figure 2.
Figure 2: Bayesian Network and Structural Equations. [An extended description of figure 2 is in the supplement.]
As Sloman (2005: 177) highlights, this form of representation fits well with a guiding idea of embodied cognition: mental representations in biological agents are constrained by the fact that their primary function is to facilitate successful action despite uncertain information and bounded computational resources. Bayesian networks have also been claimed to address a deep and central issue in artificial intelligence called the frame problem (e.g., Glymour 2001: Ch. 3 ). For the purposes of this entry, it is striking how fruitful this approach to mental representation has been, since counterfactual dependence is at its core.
This approach also appeals to laws, which are another key philosophical concept connected to counterfactuals—see §1.3 below.
Counterfactuals are not just used to analyze how a given mental state represents reality, but also when a mental state counts as knowledge. Numerous counterexamples, like Gettier cases, make the identification of knowledge with justified true belief problematic—for further details see the analysis of knowledge. But some build on this analysis by proposing further conditions to address these counterexamples. Two counterfactual conditions are prominent in this literature:
Both concepts are ways of articulating the idea that S’s beliefs must be formed in a way that is responsive to p being true. The semantics of counterfactuals have interacted with this project in a number of ways: in establishing their non-equivalence, refining them, and adjudicating putative counterexamples.
1.3 Metaphysical Analysis and Scientific Explanation
Counterfactuals have played an equally central role in metaphysics and the philosophy of science. They have featured in metaphysical theories of causation, supervenience, grounding, ontological dependence, and dispositions. They have also featured in issues at the intersection of metaphysics and philosophy of science like laws of nature and scientific explanation. This section will briefly overview these applications, largely linking to related entries that cover these applications in more depth. But, this overview is more than just a list of how counterfactuals have been applied in these areas. It helps identify a cluster of inter-related concepts (and/or properties) that are fruitfully studied together rather than in isolation.
(10) A caused C. (11) If A had not occurred, C would not have occurred.
This basic idea has been elaborated and developed in several ways. D. Lewis (1973a, c) refines it using his similarity semantics for counterfactuals—see §2.3. The resulting counterfactual analysis of causation faces a number of challenges—see counterfactual theories of causation for discussion and references. But this has simply inspired a new wave of counterfactual analyses that use different tools.
Recently, Schaffer (2016) and Wilson (2018) have also used structural equations to articulate a counterfactual theory of metaphysical grounding. [14] Metaphysical grounding is a concept widely employed in metaphysics throughout its history, but has been the focus of intense attention only recently—see entry metaphysical grounding for further details. As Schaffer (2016) puts it, the fact that Koko the gorilla lives in California is not a fundamental fact because it is grounded in more basic facts about the physical world, perhaps facts about spacetime and certain physical fields. Statements articulating these grounding facts constitute distinct metaphysical explanations. So conceived, metaphysical grounding is among the most central concepts in metaphysics. The key proposals in Schaffer (2016) and Wilson (2018) are to use structural equations to model grounding relations, and not just causal relations, and in doing so capture parallels between causation and grounding. Indeed, they define grounding in terms of structural equations in the same way as the authors above defined causation in terms of structural equations. The key difference is that the equations articulate what grounds what. While this approach to grounding has its critics (e.g., Koslicki 2016 ), it is worth noting here since it places counterfactuals at the center of metaphysical explanations. [15] Counterfactuals have been implicated in other key metaphysical debates. Work on dispositions is a prominent example. A glass’s fragility is a curious property: the glass has it in virtue of possibly shattering in certain conditions, even if those conditions are never manifested in the actual world, unlike say, the glass’s shape. This dispositional property is quite naturally understood in terms of a counterfactual claim:
(12) A glass is fragile if and only if it would break if it were struck in the right way.
It is not just metaphysical explanation where counterfactuals have been central. They also feature prominently in accounts of scientific explanation and laws of nature. Strict empiricists have attempted to characterize scientific explanation without reliance on counterfactuals, despite the fact that they tend to creep in—for further background on this see scientific explanation. Scientific explanations appeal to laws of nature, and laws of nature are difficult to separate from counterfactuals. Laws of nature are crucially different from accidental generalizations, but how? One prominent idea is that they “support counterfactuals”. As Chisholm (1955: 97) observed, the counterfactual (14) follows from the corresponding law (13) but the counterfactual (16) does not follow from the corresponding accidental generalization (15).
(13) All gold is malleable. (14) If that metal were gold, it would be malleable. (15) Every Canadian parent of quintuplets in the first half of the 20th century is named “Dionne”. (16) If Jones, who is Canadian, had been parent of quintuplets during the first half of the 20th century, he would have been named “Dionne”.
A number of prominent views have emerged from pursuing this connection. Woodward (2003) argues that the key feature of an explanation is that it answers what-if-things-had-been-different questions, and integrates this proposal with a structural equations approach to causation and counterfactuals. [16] Lange (1999, 2000, 2009) proposes an anti-reductionist account of laws according to which they are identified by their invariance under certain counterfactuals. Maudlin (2007: Ch.1) also proposes an anti-reductionist account of laws, but instead uses laws to define the truth-conditions of counterfactuals relevant to physical explanations. For more on these views see laws of nature.
1.4 Semantic Puzzles
It should now be clear that a wide variety of central philosophical topics rely crucially on counterfactuals. This highlights the need to understand their semantics: how can we systematically specify what the world must be like if a given counterfactual is true and capture patterns of valid inference involving them? It turns out to be rather difficult to answer this question using the tools of classical logic, or even modal logic. This section will explain why.
Logical semantics (Frege 1893; Tarski 1936; Carnap 1948) provided many useful analyses of English connectives like and and not using Boolean truth-functional connectives like \(\land\) and \(\neg\). Unfortunately, such an analysis is not possible for counterfactuals. In truth-functional semantics, the truth of a complex sentence is determined by the truth of its parts because a connective’s meaning is modeled as a truth-function—a function from one or more truth-values to another. Many counterfactuals have false antecedents and consequents, but some are true and others false. (17a) is false—given Joplin’s critiques of consumerism—and (17b) is true.
(17) a. If Janis Joplin were alive today, she would drive a Mercedes-Benz. b. If Janis Joplin were alive today, she would metabolize food.
It may be useful to state the issue a bit more precisely.
In truth-functional semantics, the truth-value (True/False: 1/0) of a complex sentence is determined by the truth-values of its parts and particular truth-function expressed by the connective. This is illustrated by the truth-tables for negation \(\neg\), conjunction \(\land\), and the material conditional \(\supset\) in Figure 3.
Figure 3: Negation (\(\neg\)), Conjunction (\(\land\)), Material Conditional (\(\supset\))
Truth-functional logic is inadequate for counterfactuals not just because the material conditional \(\supset\) does not capture the fact that some counterfactuals with false antecedents like (17a) are false. It is inadequate because there is, by definition, no truth-functional connective whatsoever that simultaneously combines two false sentences to make a true one like (17b) and combines two false ones to make a false one like (17a). In contemporary philosophy, this is overwhelmingly seen as a failing of classical logic. But there was a time at which it fueled skepticism about whether counterfactuals really make true or false claims about the world at all. Quine ( 1960: §46, 1982: Ch.3 ) voices this skepticism and supports it by highlighting puzzling pairs like (18) and (19):
(18) a. If Caesar had been in charge [in Korea], he would have used the atom bomb. b. If Caesar had been in charge [in Korea], he would have used catapults. (19) a. If Bizet and Verdi had been compatriots, Bizet would have been Italian. b. If Bizet and Verdi had been compatriots, Verdi would have been French.
(20) a. If I had struck this match, it would have lit.
\(\mathsf L>\) b. If I had struck this match and done so in a room without oxygen, it would have lit.
\(\mathsf <(S\land \neg O)>L>\)
Lewis ( 1973c: 419; 1973b: 10 ) dramatized the problem by considering sequences such as (21), where adding more information to the antecedent repeatedly flips the truth-value of the counterfactual.
(21) a. If I had shirked my duty, no harm would have ensued.
\(\mathsf \neg H>\) b. Though, if I had shirked my duty and you had too, harm would have ensued.
\(\mathsf <(I \land U)>H>\) c. Yet, if I had shirked my duty, you had shirked your duty and a third person done more than their duty, then no harm would have ensued.
\(\mathsf <(I\land U\land T) >\neg H>\) \(\vdots\)
The English discourse (21) is clearly consistent: it is nothing like saying I shirked my duty and I did not shirk my duty. This property of counterfactual antecedents is known by a technical name, non-monotonicity, and is one of the features all contemporary accounts are designed to capture. As will be discussed in §2.2, even modal logic does not have the resources to capture semantically non-monotonic operators.
The potential circularity for background conditions takes a bit more explanation. Suppose one claims to have specified all of the background conditions relevant to the truth of (20a), as in (22a). Then it is tempting to say that (20a) is true because (22c) follows from (22a), (22c), and the physical laws.
(22) a. The match was dry, oxygen was present, wind was below a certain threshold, the potential friction between the striking surface and the match was sufficient to produce heat, that heat was sufficient to activate the chemical energy stored in the match head … b. The match was struck. c. The match lit.
But now suppose there is an agent seeing to it that a fire is not started, and will only strike the match if it is wet. In this case the counterfactual (20a) is intuitively false. However, unless one adds the counterfactual, if the match were struck, it would have to be wet, to the background conditions, (22c) still follows from (22a), (22c), and the physical laws. That would incorrectly predict the counterfactual to be true. In short, it seems that the background conditions must themselves consist of counterfactuals. Any analysis of counterfactuals that captures their sensitivity to background facts must either eliminate these appeals to counterfactuals, or show how this appeal is non-circular, e.g., part of a recursive, non-reductive analysis.
To summarize, this section has identified three key theses about the semantics of counterfactuals and a central problem:
These theses, along with Goodman’s Problem, were once grounds for skepticism about the coherence of counterfactual discourse. But with advances in semantics and pragmatics, they have instead become the central features of counterfactuals that contemporary analyses aim to capture.
2. The Logic of Counterfactuals
This section will survey two semantic analyses of counterfactuals: the strict conditional analysis and the similarity analysis. These conceptually related analyses also have a shared explanatory goal: to capture logically valid inferences involving counterfactuals, while treating them non-truth-functionally, leaving room for their context dependence, and addressing the non-monotonic interpretation of counterfactual antecedents. Crucially, these analyses abstract away Goodman’s Problem because they are not primarily concerned with the truth-conditions of particular counterfactuals—just as classical logic does not take a stand on which atomic sentences are actually true. Instead, they say only enough about truth-conditions to settle matters of logic, e.g., if \(\phi\) and \(\phi>\psi\) are true, then \(\psi\) is true. Sections 2.5 and 2.6 will revisit questions about the truth-conditions of particular counterfactuals, Goodman’s Problem and the philosophical projects surveyed in §1.
The following subsections will detail strict conditional and similarity analyses. But it is useful at the outset to consider simplified versions of these two analyses alongside each other. This will clarify their key differences and similarities. Both analyses are also stated in the framework of possible world semantics developed in Kripke (1963) for modal logics. The following subsection provides this background and an overview of the two analyses.
2.1 Introducing Strict and Similarity Analyses
The two key concepts in possible worlds semantics are possible worlds and accessibility spheres (or relations). Intuitively, a possible world w is simply a way the world could be or could have been. Formally, they are treated as primitive points in the set of all possible worlds W. But their crucial role comes in assigning truth-conditions to sentences: a sentence \(\phi\) can only said to be true given a possible world w, but since w is genuinely possible, it cannot be the case that both \(\phi\) and \(\neg\phi\) are true at w. Accessibility spheres provide additional structure for reasoning about what’s possible: for each world w, \(R(w)\) is the set of worlds accessible from w. [18] This captures the intuitive idea that given a possible world w, a certain range of other worlds \(R(w)\) are possible, in a variety of senses. \(R_1(w)\) might specify what’s nomologically possible in w by including only worlds where w’s natural laws hold, while \(R_2(w)\) specifies what’s metaphysically possible in w.
These tools furnish truth-conditions for a formal language including non-truth-functional necessity (\(<<\medsquare>>\)) and possibility (\(<<\meddiamond>>\)) operators: [19]
In classical logic, the meaning of \(\phi\) is simply its truth-value. But in modal logic, it is the set of possible worlds where \(\phi\) is true: \(<\llbracket>\phi<\rrbracket>\). So \(\phi\) is true in w, relative to v and R, just in case \(w\in<\llbracket>\phi<\rrbracket>^R_v\):
Only clauses 6 and 7 rely crucially on this richer notion of meaning. \(<<\medsquare>>\phi\) says that in all accessible worlds \(R(w)\), \(\phi\) is true. \(<<\meddiamond>>\phi\) says that there are some accessible worlds where \(\phi\) is true. Logical concepts like consequence are also defined in terms of relations between sets of possible worlds. The intersection of the premises must be a subset of the conclusion (i.e., every world where the premises are true, the conclusion is true):
Given this framework, the strict analysis can be formulated very simply: \(\phi > \psi\) should be analyzed as \(<<\medsquare>>(\phi\supset\psi)\). This says that all accessible \(\phi\)-worlds are \(\psi\)-worlds. This analysis can be depicted as in Figure 4. [20]
Figure 4: Truth in \(w_0\) relative to R. [An extended description of figure 4 is in the supplement.]
The red circle delimits the worlds accessible from \(w_0\), the x-axis divides \(\phi\) and \(\neg\phi\)-worlds, and the y-axis \(\psi\) and \(\neg\psi\)-worlds. \(<<\medsquare>>(\phi\supset\psi)\) says that there are no worlds in the blue shaded region.
It is crucial to highlight that this semantics does not capture the non-monotonic interpretation of counterfactual antecedents. For example, \(<\llbracket>\mathsf\land \mathsf<\rrbracket>^R_v\) is a subset of \(<\llbracket>\mathsf<\rrbracket>\), and this means that any time \(<<\medsquare>>(\mathsf
On the similarity analysis, \(\phi >\psi\) is true in \(w_0\), roughly, just in case all the \(\phi\)-worlds most similar to \(w_0\) are \(\psi\)-worlds. To model this notion of similarity, one needs more than a simple accessibility sphere. One way to capture it is with with a nested system of spheres \(\mathcal
Figure 5: Truth in \(w_0\) relative to \(\mathcal
The most similar \(\phi\)-worlds are in the innermost gray region. So, this analysis excludes any worlds from being in the shaded innermost blue region. Comparing Figures 4 and 5, one difference stands out: the similarity analyses does not require that there be no \(\phi\land\neg\psi\)-worlds in any sphere, just in the innermost sphere. For example, world \(w_1\) does not prevent the counterfactual \(\phi >\psi\) from being true. It is not in the \(\phi\)-sphere most similar to w. This is the key to semantically capturing the non-monotonic interpretation of antecedents. The truth of \(\mathsf C>\) does not guarantee the truth of \(\mathsf <(A\land B)>C>\) precisely because the most similar \(\mathsf\)-worlds may be in the innermost sphere, and the most similar \(\mathsf
Since antecedent monotonicity is the key division between strict and similarity analyses, it is worthwhile being a bit more precise about what it is, and what its associated inference patterns are.
The crucial patterns associated with antecedent monotonicity are:
AS and SDA clearly follow from antecedent monotonicity. By contrast, Transitivity and a plausible auxiliary assumption entail antecedent monotonicity, [22] and the same is true for Contraposition. [23] With these basics in place, it is possible to focus in on each of these analyses in more detail. In doing so, it will become clear that there are important differences even among variants of the similarity analysis and variants of the strict analysis. This entry will focus on what these analyses predict about valid inferences involving counterfactuals.
2.2 Strict Conditional Analyses
The strict conditional analysis has a long history, but its contemporary form was first articulated by Peirce: [24]
“If A is true then B is true”… is expressed by saying, “In any possible state of things, [w], either \([A]\) is not true [in w], or \([B]\) is true [in w]”. (Peirce 1896: 33)
Just as the logic of \(<<\medsquare>>\) will vary with constraints that can be placed on R, so too will the logic of strict conditionals. [26] For example, if one does not assume that \(w\in R(w)\) then modus ponens will not hold for the strict conditional: \(\psi\) will not follow from \(\phi\) and \(<<\medsquare>>(\phi\supset\psi)\). But even without settling these constraints, some basic logical properties of the analysis can be established. The discussion to follow is by no means exhaustive. [27] Instead, it will highlight the logical patterns which are central to the debates between competing analyses.
The core idea of the basic strict analysis leads to the following validities.
(23) a. JFK couldn’t have passed universal healthcare. b. If JFK had passed universal healthcare, he would have granted insects coverage.
Contrary to pattern 3, the false (23b) does not intuitively follow from the true (23a). Similarly, for pattern 4. Suppose one’s origin from a particular sperm and egg is an essential feature of oneself. Then (24a) is true.
(24) a. Joplin had to have come from the particular sperm and egg she in fact came from. b. If there had been no life on Earth, then Joplin would have come from the particular sperm and egg she in fact came from.
And, yet, many would hesitate to infer (24b) on the basis of (24a). Each of these patterns follow from the core idea of the strict analysis. While these counterexamples may not constitute a conclusive objection, they do present a problem for the basic strict analysis. The second wave strict analyses surveyed in §2.2.1 are designed to solve it, however. They are also designed to address another suite of validities that are even more problematic.
The strict analysis is widely criticized for validating antecedent monotonic patterns. It is worth saying a bit more precisely, using Definition 9 and Figure 6, why antecedent monotonicity holds for the strict conditional.
Figure 6: Strict Conditionals are Antecedent Monotonic. [An extended description of figure 6 is in the supplement.]
If \(\phi_1\strictif\psi\) is true, then the shaded blue region is empty, and the position of \(\phi_2\) reflects the fact that \(<\llbracket>\phi_2<\rrbracket>^R_v\subseteq<\llbracket>\phi_1<\rrbracket>^R_v\)—recall that all worlds above the x-axis are \(\phi_1\)-worlds. Since the shaded blue region within \(\phi_2\) is also empty, all \(\phi_2\) worlds in \(R(w)\) are \(\psi\)-worlds. That is, \(\phi_2\strictif \psi\) is true.
Recall that Transititivity and Contraposition entail antecedent monotonicity, so it remains to show that both hold for the strict conditional. To see why Contraposition holds for the strict conditional, note again that if \(\phi\strictif\psi\) is true in w, then all \(\phi\)-worlds in \(R(w)\) are \(\psi\)-worlds, as depicted in the left Venn diagram in Figure 7. Now suppose w is a \(\neg\psi\)-world in \(R(w)\). As the diagram makes clear, w has to be a \(\neg\phi\)-world, and so \(\neg\psi\strictif\neg\phi\) must be true in w. Similarly, if \(\neg\psi\strictif\neg\phi\) is true in w, then all \(\neg\psi\)-worlds in \(R(w)\) are \(\neg\phi\)-worlds, as depicted in the right Venn diagram in Figure 7. Now suppose w is a \(\phi\)-world in \(R(w)\). As depicted, w has to be a \(\psi\)-world, and so \(\phi\strictif\psi\) must be true in w.
The validity of Transititivity for the strict conditional is also easy to see with a Venn diagram.
The premises guarantee that all \(\phi_2\)-worlds in \(R(w)\) are \(\phi_1\)-worlds, and that all \(\phi_1\)-worlds in \(R(w)\) are \(\psi\)-worlds. That gives one the relationships depicted in Figure 8. To show that \(\phi_2\strictif\psi\) follows, suppose that w is a \(\phi_2\)-world in \(R(w)\). As Figure 8 makes evident, w must then be a \(\psi\)-world.
Antecedent monotonic patterns are an ineliminable part of a strict conditional logic. Examples of them often sound compelling. For example, the transitive inference (25) sounds perfectly reasonable, as does the antecedent strengthening inference (26).
(25) a. If the switch had been flipped, the light would be on. b. If the light had been on, it would not have been dark. c. So, if the switch had been flipped, it would not have been dark. (26) a. If the switch had been flipped, the light would be on. b. So, if the switch had been flipped and I had been in the room, the light would be on.
Similar examples for SDA are easy to find. However, counterexamples to each of the four patterns have been offered.
Counterexamples to Antecedent Strengthening were already discussed back in §1.4. Against Transititivity, Stalnaker (1968: 48) points out that (27c) does not intuitively follow from (27a) and (27b).
(27) a. If J. Edgar Hoover were today a communist, then he would be a traitor. b. If J. Edgar Hoover had been born a Russian, then he would today be a communist. c. If J. Edgar Hoover had been born a Russian, he would be a traitor.
Contra Contraposition, D. Lewis (1973b: 35) presents (28).
(28) a. If Boris had gone to the party, Olga would still have gone. b. If Olga had not gone, Boris would still not have gone.
Suppose Boris wanted to go, but stayed away to avoid Olga. Then (28b) is false. Further suppose that Olga would have been even more excited to attend if Boris had. In that case (28a) is true. Against SDA, Mckay & van Inwagen (1977: 354) offer:
(29) a. If Spain had fought for the Axis or the Allies, she would have fought for the Axis. b. If Spain had fought for the Allies, she would have fought for the Axis.
(29b) does not intuitively follow from (29a).
These counterexamples have been widely taken to be conclusive evidence against the strict analysis (e.g., D. Lewis 1973b; Stalnaker 1968 ), since they follow from the core assumptions of that analysis. As a result, D. Lewis (1973b) and Stalnaker (1968) developed similarity analyses which build the non-monotonicity of antecendents into the semantics of counterfactuals—see §2.3. However, there was a subsequent wave of strict analyses designed to systematically address these counterexamples. In fact, they do so by unifying two features of counterfactuals: the non-monotonic interpretation of their antecedents and their context-sensitivity.
2.2.1 Second Wave Strict Conditional Analyses
The key idea in Warmbrōd (1981a,b) is that the accessibility sphere in the basic strict analysis should be viewed as a parameter of the context. Roughly, the idea is that \(R(w)\) corresponds to background facts assumed by the participants of a discourse context. For example, if they are assuming propositions (modeled as sets of possible worlds) A, B, and C then \(R(w)=A\cap B\cap C\). The other key idea is that trivial strict conditionals are not pragmatically useful in conversation. If a strict conditional \(\mathsf
(P) If the antecedent \(\phi\) of a conditional is itself consistent, then \(R(w)\cap<\llbracket>\phi<\rrbracket>^R_v\) should be consistent.
On this view, \(R(w)\) may very well change over the course of a discourse as a result of conversationalists adhering to (P). This part of the view is central to explaining away counterexamples to antecedent monotonic validities.
Consider again the example from Goodman 1947 that appeared to be a counterexample to Antecedent Strengthening.
(30) a. If I had struck this match, it would have lit. b. If I had struck this match and done so in a room without oxygen, it would have lit.
Now note that if (30a) is going to come out true, the proposition that there is oxygen in the room O must be true in all worlds in the initial accessibility sphere \(R_0(w)\). However, if (30b) is interpreted against \(R_0(w)\), the antecedent will be inconsistent with \(R_0(w)\) and so express a trivial, uninformative proposition. Warmbrōd (1981a,b) proposes that in interpreting (30b) we are forced by to adopt a new, modified accessibility sphere \(R_1(w)\) where O is no longer assumed. But if this is right, (30a) and (30b) don’t constitute a counterexample to Antecedent Strengthening because they are interpreted against different accessibility spheres. It’s like saying All current U.S. presidents are intelligent doesn’t entail All current U.S. presidents are unintelligent because this sentence before Donald Trump was sworn in was true, but uttering it afterwards was false. There is an equivocation of context, or so Warmbrōd (1981a,b) contends.
Fintel (2001) and Gillies (2007) propose analyses where counterfactuals have strict truth-conditions, but they also have a dynamic meaning which effectively changes \(R(w)\) non-monotonically. They argue that such a theory can better explain particular phenomena. Chief among them is reverse Sobel sequences. Recall the sequence of counterfactuals (21) presented by Lewis ( 1973b, 1973c: 419 ), and attributed to Howard Sobel. Reversing these sequences is not felicitous:
(31) a. If I had shirked my duty and you had too, harm would have ensued.
\(\mathsf <(I \land U)>H>\) b. # If I had shirked my duty, no harm would have ensued.
\(\mathsf \neg H>\)
2.3 Similarity Semantics
When we allow for the possibility of the antecedent’s being true in the case of a counterfactual, we are hypothetically substituting a different world for the actual one. It has to be supposed that this hypothetical world is as much like the actual one as possible so that we will have grounds for saying that the consequent would be realized in such a world. (Todd 1964: 107)
Recall the major difference between this proposal and the basic strict analysis: the similarity analysis uses a graded notion of similarity instead of an absolute notion of accessibility. It also allows most similar worlds to vary between counterfactuals with different antecedents. These differences invalidate antecedent monotonic inference patterns. This section will introduce similarity analyses in a bit more formal detail and describe the differences between analyses within this family.
The similarity analysis has come in many varieties and formulations, including the system of spheres approach informally described in §2.1. That formulation is easiest for comparison to strict analyses. But there is a different formulation that is more intuitive and better facilitates comparison among different similarity analyses. This formulation appeals to a (set) selection function f, which takes a world w, a proposition p, and returns the set of p-worlds most similar to w: \(f(w,p)\). [30] \(\phi>\psi\) is then said to be true when the most f-similar \(\phi\)-worlds to w are \(\psi\)-worlds, i.e., every world in \(f(w,<\llbracket>\phi<\rrbracket>^f_v)\) is in \(<\llbracket>\psi<\rrbracket>^f_v\). The basics of this approach can be summed up thus.
As noted, this formulation makes the limit assumption: \(\phi\)-worlds do not get indefinitely more and more similar to w. While D. Lewis (1973b) rejected this assumption, adopting it will serve exposition. It is discussed at length in the supplement Formal Constraints on Similarity. The logic of counterfactuals generated by a similarity analysis will depend on the constraints imposed on f. Different theorists have defended different constraints. Table 1 lists them, where \(p,q\subseteq W\) and \(w\in W\):
| (a) | \(f(w,p)\subseteq p\) | success |
| (b) | \(f(w,p)=\ | strong centering |
| (c) | \(f(w,p)\subseteq q\) & \(f(w,q)\subseteq p \; <\Longrightarrow>\; f(w,p)=f(w,q)\) | uniformity |
| (d) | \(f(w,p)\) contains at most one world | uniqueness |
Table 1: Candidate Constraints on Selection Functions
Modulo the limit assumption, Table 2 provides an overview of which analyses have adopted which constraints.
Table 2: Similarity Analyses, modulo Limit Assumption
simply enforces that \(f(w,p)\) is indeed a set of p-worlds. Recall that \(f(w,p)\) is supposed to be the set of most similar p-worlds to w. The other constraints correspond to certain logical validities, as detailed in the supplement Formal Constraints on Similarity. This means that Pollock (1976) endorses the weakest logic for counterfactuals and Stalnaker (1968) the strongest. It is worth seeing how, independently of constraints (b)–(d), this semantics invalidates an antecedent monotonicity pattern like Antecedent Strengthening.
Consider an instance of Antecedent Strengthening involving \(\mathsf
Table 3: A space of worlds W, and truth-values at each world
Now evaluate \(\mathsf C>\) and \(\mathsf<(A\land B)>C>\) in \(w_<5>\) using a selection function \(f_1\) with the following features:
While Stalnaker (1968) and D. Lewis (1973b) remain the most popular similarity analyses, there are substantial logical issues which separate similarity analyses. These issues, and the constraints underlying them, are detailed in the supplement Formal Constraints on Similarity. Table 4 summarizes which validities go with which constraints.
| Constraint | Validity |
|---|---|
| Strong Centering | Modus Ponens \(\phi>\psi, \phi\vDash \psi\) Conjunction Conditionalization \(\phi\land\psi \vDash \phi>\psi\) |
| Uniformity | Substitution of Subjunctive Equivalents (SSE) \(\phi_1>\phi_2,\phi_2>\phi_1,\phi_1>\psi\vDash \phi_2>\psi\) Limited Transitivity (LT) \(\phi_1>\phi_2,(\phi_1\land\phi_2)>\psi\vDash \phi_1>\psi\) Limited Antecedent Strengthening (LAS) \(\phi_1>\phi_2,\neg(\phi_1>\neg\psi)\vDash(\phi_1\land\phi_2)>\psi\) |
| Uniqueness | Conditional Excluded Middle \(\vDash (\phi>\psi)\lor(\phi>\neg\psi)\) Conditional Negation (CN) \(<\Diamond>\phi\land\neg(\phi>\psi)\,\leftmodels\vDash <\Diamond>\phi\land\phi>\neg\psi\) Consequent Distribution (CD) \(\phi>(\psi_1\lor\psi_2)\vDash(\phi>\psi_1)\lor(\phi>\psi_2)\) |
| Limit Assumption | Infinite Consequent Entailment If \(\Gamma=<\<\phi_2,\phi_3,<\ldots>\>>\), \(\phi_1>\phi_2,\phi_1>\phi_3,<\ldots>\) are true and \(\Gamma\vDash\psi\) then \(\phi_1>\psi\) |
Table 4: Selection Constraints & Associated Validities
A few comments are in order here, though. Strong centering is sufficient but not necessary for Modus Ponens, weak centering would do: \(w\in f(w,p)\) if \(w\in p\). LT and LAS follow from SSE, and allow similarity theorists to say why some instances of Transititivity and Antecedent Strengthening are intuitively compelling.
2.4 Comparing the Logics
The issue of whether a second wave strict analysis (§2.2.1) or a similarity analysis provides a better logic of counterfactuals is very much an open and subtle issue. As sections 2.2.1 and 2.3 detailed, both analyses have their own way of capturing the non-monotonic interpretation of antecedents. Both analyses also have their own way of capturing instances of monotonic inferences that do sound good. Perhaps this issue is destined for a stalemate. [31] But before declaring it such, it is important to investigate two patterns that are potentially more decisive: Simplification of Disjunctive Antecedents, and a pattern not yet discussed called Import-Export.
Both SDA and Import-Export are valid in a strict analyses and invalid on standard similarity analyses. Crucially, the counterexamples to them that have been offered by similarity theorists are significantly less compelling than those offered to patterns like Antecedent Strengthening. Import-Export relates counterfactuals like (33a) and (33b).
(33) a. If Jean-Paul had danced and Simone had drummed, there would have been a groovy party. b. If Jean-Paul had danced, then if Simone had drummed, there would have been a groovy party.
It is hard to imagine one being true without the other. The basic strict analysis agrees: it renders them equivalent.
But it is not valid on a similarity analysis. [32] While Import-Export is generally regarded as a plausible principle, some have challenged it. Kaufmann (2005: 213) presents an example involving indicative conditionals which can be adapted to subjunctives. Consider a case where there is a wet match which will light if tossed in the campfire, but not if it is struck. It has not been lit. Consider now:
(34) a. If this match had been lit, it would have been lit if it had been struck. b. If this match had been struck and it had been lit, it would have been lit.
One might then deny (34a). This match would not have lit if it had been struck, and if it had lit it would have to have been thrown into the campfire. (34b), on the other hand, seems like a straightforward logical truth. However, it is worth noting that this intuition about (34a) is very fragile. The slight variation of (34a) in (35) is easy to hear as true.
(35) If this match had been lit, then if it had been struck it (still) would have been lit.
This subtle issue may be moot, however. Starr (2014) shows that a dynamic semantic implementation of the similarity analysis can validate Import-Export, so it may not be important for settling between strict and similarity analyses.
(29) a. If Spain had fought for the Axis or the Allies, she would have fought for the Axis. b. If Spain had fought for the Allies, she would have fought for the Axis.
Starr (2014: 1049) and Warmbrōd (1981a: 284) observe that (29a) seems to be another way of saying that Spain would never have fought for the Allies. While Warmbrōd (1981a: 284) uses this to pragmatically explain-away this counterexample to his strict analysis, Starr (2014: 1049) makes a further critical point: it sounds inconsistent to say (29a) after asserting that Spain could have fought for the Allies.
(36) Spain didn’t fight for either the Allies or the Axis.
She really could have fought for the Allies.
# But, if she had fought for the Axis or the Allies, she would have fought for the Axis.
Nute (1980b: 33) considers a similar antecedent simplification pattern involving negated conjunctions:
Nute (1980b: 33) presents (37) in favor of SNCA.
(37) a. If Nixon and Agnew had not both resigned, Ford would never have become President.
\(\mathsf<\neg(N\land A)>\neg F>\) b. If Nixon had not resigned, Ford would never have become President.
\(\mathsf<\neg N>\neg F>\) c. If Agnew had not resigned, Ford would never have become President.
\(\mathsf<\neg A>\neg F>\)
Champollion, Ciardelli, and Zhang (2016) consider a light which is on when switches A and B are both up, or both down. Currently, both switches are up, and the light is on. Consider (38a) and (38b) whose antecedents are Boolean equivalents:
(38) a. If Switch A or Switch B were down, the light would be off.
\(\mathsf<(\neg A\lor\neg B)>\neg L>\) b. If Switch A and Switch B were not both up, the light would be off.
\(\mathsf<\neg (A\land B)>\neg L>\)
While (38a) is intuitively true, (38b) is not. [33] This is not a counterexample to SNCA, since the premise of that pattern is false. But such a counterexample is not hard to think up. [34]
Suppose the baker’s apprentice completely failed at baking our cake. It was burnt to a crisp, and the thin, lumpy frosting came out puke green. The baker planned to redecorate it to make it at least look delicious, but did not have time. We may explain our extreme dissatisfaction by asserting (39a). But the baker should not infer (39b) and assume that his redecoration plan would have worked.
(39) a. If the cake had not been burnt to a crisp and ugly, we would have been happy.
\(\mathsf<\neg(B\land U)>H>\) b. If the cake had not been ugly, we would have been happy.
\(\mathsf<\neg U>H>\)
Willer (2017: §4.2) suggests that such a counterexample trades on interpreting \(\mathsf<\neg(B\land U)>H>\) as \(\mathsf<\neg B\land\neg U)>H>\), and provides an independent explanation of this on the basis of how negation and conjunction interact. If this is right, then an analysis which validates SDA and SNCA without rendering \(\neg(\phi_1\land\phi_2)>\psi\) and \(\neg\phi_1\lor\neg\phi_2>\psi\) equivalent is what’s needed. Ciardelli, Zhang, and Champollion (forthcoming) develop just such an analysis. As Ciardelli, Zhang, and Champollion (forthcoming: §6.4) explain, SDA and SNCA turn out to be valid for very different reasons. Champollion, Ciardelli, and Zhang (2016) and Ciardelli, Zhang, and Champollion (forthcoming) also argue that the falsity of (38b) cannot be predicted on a similarity analysis. This example must be added to a long list of examples which have been presented not as counterexamples to the logic of the similarity analysis, but to what it predicts (or fails to predict) about the truth of particular counterfactuals in particular contexts. This will be the topic of §2.5, where it will also be explained why the strict analysis faces similar challenges.
2.5 Truth-Conditions Revisited
In their own ways, Stalnaker (1968, 1984) and D. lewis (1973b) are candid that the similarity analysis is not a complete analysis of counterfactuals. As should be clear from §2.3, the formal constraints they place on similarity are quite minimal and only serve to settle matters of logic. There are, in general, very many possible selection functions—and corresponding conceptions of similarity—for any given counterfactual. To explain how a given counterfactual like (40) expresses a true proposition, a similarity analysis must specify which particular conception of similarity informs it.
(40) If my computer were off, the screen would be blank.
Of course, the strict analysis is in the same position. It cannot predict the truth of (40) without specifying a particular accessibility relation. In turn, the same question arises: on what basis do ordinary speakers determine some worlds to be accessible and others not? This section will overview attempts to answer these questions, and the many counterexamples those attempts have invited. These counterexamples have been a central motivation for pursuing alternative semantic analyses, which will be covered in §3. While this section follows the focus of the literature on the similarity analysis (§2.5.1), §2.5.2 will briefly detail how parallel criticisms apply to strict analyses.
2.5.1 Truth-Conditions and Similarity
What determines which worlds are counted as most similar when evaluating a counterfactual? Stalnaker (1968) explicitly sets this issue aside, but D. Lewis (1973b: 92) makes a clear proposal:
Just as counterfactuals are context-dependent and vague, so is our intuitive notion of overall similarity. In comparing cost of living, New York and San Francisco may count as similar, but not in comparing topography. And yet, Lewis’ (1973b: 92) Proposal has faced a barrage of counterexamples. Lewis and Stalnaker parted ways in their responses to these counterexamples, though both grant that Lewis’ (1973b: 92) Proposal was not viable. Stalnaker (1984: Ch.7) proposes the projection strategy: similarity is determined by the way we “project our epistemic policies onto the world”. D. Lewis 1979) proposes a new system of weights that amounts to a kind of curve-fitting: we must first look to which counterfactuals are intuitively true, and then find ways of weighting respects of similarity—however complex—that support the truth of counterfactuals. Since Lewis’ (1973b: 92) Proposal and Lewis’ ( 1979 ) system of weights are more developed, and have received extensive critical attention, they will be the focus of this section. [35] It will begin with the objections to Lewis’ (1973b: 92) Proposal that motivated Lewis’ ( 1979 ) system of weights, and then some objections to that approach.
Fine (1975: 452) presents the future similarity objection to Lewis’ (1973b: 92) Proposal. (41) is plausibly a true statement about world history.
(41) If Nixon had pressed the button there would have a been a nuclear holocaust
(\(\mathsfH>\))
Suppose, optimistically, that there never will be a nuclear holocaust. Then, for every \(\mathsf
Tichý (1976: 271) offers a similar counterexample. Given (42a)–(42c), (42d) sounds false.
(42) a. Invariably, if it is raining, Jones wears his hat. b. If it is not raining, Jones wears his hat at random. c. Today, it is raining and so Jones is wearing his hat. d. But, even if it had not been raining, Jones would have been wearing his hat.
Lewis’ (1973b: 92) Proposal does not seem to predict the falsity of (42d). After all, Jones is wearing his hat in the actual world, so isn’t a world where it’s not raining and he’s wearing his hat more similar to the actual one than one where it’s not raining and he isn’t wearing his hat?
(1979: 472) responds to these examples by proposing a ranked system of weights that give what he calls the standard resolution of similarity, which may be further modulated in context:
While weight 2 gives high importance to keeping particular facts fixed up to the change required by the counterfactual, weight 4 makes clear that particular facts after that point need not be kept fixed. In the case of (42d) the fact that Jones is wearing his hat need not be kept fixed. It was a post-rain fact, so when one counterfactually supposes that it had not been raining, there is no reason to assume that Jones is still wearing his hat. Similarly, with example (41). A world where Nixon pushes the button, a small miracle occurs to short-circuit the equipment and the nuclear holocaust is prevented will count as less similar than one where there is no small miracle and a nuclear holocaust results. A small-miracle and no-holocaust world is similar to our own only in one insignificant respect (particular matters of fact) and dissimilar in one important respect (the small miracle).
It is clear, however, that Lewis’ (1979) System of Weights is insufficiently general. Particular matters of fact often are held fixed.
(43) [You’re invited to bet heads on a coin-toss. You decline. The coin comes up heads.] See, if you had bet heads you would have won! (Slote 1978: 27 fn33) (44) If we had bought one more artichoke this morning, we would have had one for everyone at dinner tonight. (Sanford 1989: 173)
Recall (38) from §2.4. Champollion, Ciardelli, and Zhang (2016) and Ciardelli, Zhang, and Champollion (forthcoming) argue on the basis of this example that any similarity analysis will make incorrect predictions about the truth-conditions of counterfactuals. In this example a light is on either when Switch A and B are both up, or they are both down. Otherwise the light is off. Suppose both switches are up and the light is on.
(38) a. If Switch A or Switch B were down, the light would be off.
\(\mathsf<(\neg A\lor\neg B)>\neg L>\) b. If Switch A and Switch B were not both up, the light would be off.
\(\mathsf<\neg (A\land B)>\neg L>\)
Intuitively, (38a) is true, as are \(\mathsf<\neg A >\neg L>\) and \(\mathsf<\neg B >\neg L>\), but (38b) is false. Champollion, Ciardelli, and Zhang (2016: 321) argue that a similarity analysis cannot predict \(\mathsf<\neg A >\neg L>\) and \(\mathsf<\neg B >\neg L>\) to be true, while (38b) is false. In order for \(\mathsf<\neg A >\neg L>\) to be true, the particular fact that Switch B is up must count towards similarity. Similarly, for \(\mathsf<\neg B >\neg L>\) to be true, the particular fact that Switch A is up must count towards similarity. But then it follows that (38b) is true on a similarity analysis: the most similar worlds where A and B are not both up have to either be worlds where Switch B is down but Switch A is still up, or Switch A is down and Switch B is still up. In those worlds, the light would be off, so the similarity analysis incorrectly predicts (38b) to be true. Champollion, Ciardelli, and Zhang (2016) instead pursue a semantics in terms of causal models where counterfactually making \(\neg \mathsf<(A\land B)>\) true and making \(\mathsf<\neg A\lor\neg B>\) true come apart.
2.5.2 Truth-Conditions and the Strict Analysis
2.6 Philosophical Objections
Recall Goodman’s Problem from §1.4: the truth-conditions of counterfactuals intuitively depend on background facts and laws, but it is difficult to specify these facts and laws in a way that does not itself appeal to counterfactuals. Strict and similarity analyses make progress on the logic of conditionals without directly confronting this problem. But the discussion of § 2.5 makes salient a related problem. Lewis’ (1979) System of Weights amounts to reverse-engineering a similarity relation to fit the intuitive truth-conditions of counterfactuals. While Lewis’ ( 1979 ) approach avoids characterizing laws and facts in counterfactual terms, Bowie (1979: 496–497) argues that it does not explain why certain counterfactuals are true without appealing to counterfactuals. Suppose one asks why certain counterfactuals are true and the similarity theorist replies with Lewis’ ( 1979 ) recipe for similarity. If one asks why those facts about similarity make counterfactuals true, the similarity theorist cannot reply that they are basic self-evident truths about the similarity of worlds. Instead, they must say that those similarity facts make those counterfactuals true. Bowie’s ( 1979: 496–497 ) criticism is that this is at best uninformative, and at worst circular.
A related concern is voiced by Horwich (1987: 172) who asks “why we should have evolved such a baroque notion of counterfactual dependence”, namely that captured by Lewis’ (1979) System of Weights. The concern has two components: why would humans find it useful, and why would human psychology ground counterfactuals in this concept of similarity rather than our ready-at-hand intuitive concept of overall similarity? These questions are given more weight given the centrality of counterfactuals to human rationality and scientific explanation outlined in §1. Psychological theories of counterfactual reasoning and representation have found tools other than similarity more fruitful (§1.2). Similarly, work on scientific explanation has not assigned any central role for similarity (1.3), and as Hájek (2014: 250) puts it:
Science has no truck with a notion of similarity; nor does Lewis’ ( 1979 ) ordering of what matters to similarity have a basis in science.
[w]e cannot add up similarities or weigh them against differences. Nor can we combine them in any other way… No useful comparisons of overall similarity result. (Morreau 2010: §4)
articulates this argument formally via a reinterpretation of Arrow’s Theorem in social choice theory. Arrow’s Theorem shows that it is not possible to aggregate individuals’ preferences regarding some alternative outcomes into a coherent “collective preference” ordering over those outcomes, given minimal assumptions about their rationality and autonomy. As summarized in §6.3 of Arrow’s theorem, Morreau (2010) argues that the same applies to aggregating respects of similarity and difference: there is no way to add them up into a coherent notion of overall similarity.
2.7 Summary
Strict and similarity analyses of counterfactuals showed that it was possible to address the semantic puzzles described in §1.4 with formally explicit logical models. This dispelled widespread skepticism of counterfactuals and established a major area of interdisciplinary research. Strict analyses have been revealed to provide a stronger, more classical, logic, but must be integrated with a pragmatic explanation of how counterfactual antecedents are interpreted non-monotonically. Similarity analyses provide a much weaker, more non-classical, logic, but capture the non-monotonic interpretation of counterfactual antecedents within their core semantic model. It is now a highly subtle and intensely debated question which analysis provides a better logic for counterfactuals, and which version of each kind of analysis is best. This intense scrutiny and development has also generated a wave of criticism focused on their treatment of truth-conditions, Goodman’s Problem, and integration with thinking about counterfactuals in psychology and the philosophy of science (§2.5, §2.6). None of these criticisms are absolutely conclusive, and these two analyses, particularly the similarity analysis, remain standard in philosophy and linguistics. However, the criticisms are serious enough to merit exploring alternative analyses. These alternative accounts take inspiration from a particular diagnosis of the counterexamples discussed in §2.5: facts depend on each other, so counterfactually assuming p involves not just giving up not-p, but any facts which depended on not-p. The next section will examine analyses of this kind.
3. Semantic Theories of Counterfactual Dependence
Similarity and strict analyses nowhere refer to facts, or propositions, depending on each other. Indeed, 1979 was primarily concerned with explaining which true counterfactuals, given a similarity analysis, manifest a relation of counterfactual dependence. Other analyses have instead started with the idea that facts depend on each other, and then explain how these relations of dependence make counterfactuals true. As will become clear, none of these analyses endorse the naive idea that \(\mathsf B>\) is true only when B counterfactually depends on A. The dependence can be more complex, indirect, or B could just be true and independent of A. Theories in this family differ crucially in how they model counterfactual dependence. In premise semantics (§3.1) dependence is modeled in terms of how facts, which are modeled as parts of worlds, are distributed across a space of worlds that has been constrained by laws, or law-like generalizations. In probabilistic semantics (§3.2), this dependence is modeled as some form of conditional probability. In Bayesian networks, structural equations, and causal models (§3.3), it is modeled in terms of the Bayesian networks discussed at the beginning of §1.2.3. Because theories of these three kinds are very much still in development and often involve even more sophisticated formal models than those covered in §2, this section will have to be more cursory than §2 to ensure breadth and accessibility.
3.1 Premise Semantics
Veltman (1976) and Kratzer (1981b) approached counterfactuals from a perspective closer to Goodman (1947) : counterfactuals involve explicitly adjusting a body of premises, facts or propositions to be consistent with the counterfactual’s antecedent, and checking to see if the consequent follows from the revised premise set—in a sense of “follow” to be articulated carefully. Since facts or premises hang together, changing one requires changing others that depend on it. The function of counterfactuals is to allow us to probe these connections between facts. While D. Lewis (1981) proved that the Kratzer (1981b) analysis was a special case of similarity semantics, subsequent refinements of premise semantics in Kratzer (1989, 1990, 2002, 2012) and Veltman (2005) evidenced important differences. Kratzer (1989: 626) nicely captures the key difference:
[I]t is not that the similarity theory says anything false about [particular] examples… It just doesn’t say enough. It stays vague where our intuitions are relatively sharp. I think we should aim for a theory of counterfactuals that is able to make more concrete predictions with respect to particular examples.
From a logical point of view, premise semantics and similarity semantics do not diverge. They diverge in the concrete predictions made about the truth-conditions of counterfactuals in particular contexts without adding additional constraints to the theory like Lewis’ (1979) System of Weights.
How does premise semantics aim to improve on the predictions of similarity semantics? It re-divides the labor between context and the semantics of counterfactuals to more accurately capture the intuitive truth-conditions of counterfactuals, and intuitive characterizations of how context influences counterfactuals. In premise semantics, context provides facts and law-like relations among them, and the counterfactual semantics exploits this information. By contrast, the similarity analysis assumes that context somehow makes a similarity relation salient, and has to make further stipulations like Lewis’ (1979) System of Weights about how facts and laws enter into the truth-conditions of counterfactuals in particular contexts. This can be illustrated by considering how Tichý’s ( 1976 ) example (42) is analyzed in premise semantics. This illustration will use the Veltman (2005) analysis because it is simpler than Kratzer (1989, 2012) —that is not to say it is preferable. The added complexity in Kratzer (1989, 2012) provides more flexibility and a broader empirical range including quantification and modal expressions other than would-counterfactuals.
Recall Tichý’s ( 1976 ) example, with the intuitively false counterfactual (42d):
(42) a. Invariably, if it is raining, Jones wears his hat. b. If it is not raining, Jones wears his hat at random. c. Today, it is raining and so Jones is wearing his hat. d. But, even if it had not been raining, Jones would have been wearing his hat.
Veltman (2005) models how the sentences leading up to the counterfactual (42d) determine the facts and laws relevant to its interpretation. The law-like generalization in (42a) is treated as a strict conditional which places a hard constraint on the space of worlds relevant to evaluating the counterfactual. [37] The particular facts introduced by (42c) provide a soft constraint on the worlds relevant to interpreting the counterfactual. Figure 9 illustrates this model of the context and its evolution, including a third atomic sentence \(\mathsf
| \(C_0\) | \(\mathsf | \(\mathsf | \(\mathsf |
| \(\boldsymbol | 0 | 0 | 0 |
| \(\boldsymbol | 0 | 0 | 1 |
| \(\boldsymbol | 0 | 1 | 0 |
| \(\boldsymbol | 0 | 1 | 1 |
| \(\boldsymbol | 1 | 0 | 0 |
| \(\boldsymbol | 1 | 0 | 1 |
| \(\boldsymbol | 1 | 1 | 0 |
| \(\boldsymbol | 1 | 1 | 1 |
| \(C_1\) | \(\mathsf | \(\mathsf | \(\mathsf |
| \(\boldsymbol | 0 | 0 | 0 |
| \(\boldsymbol | 0 | 0 | 1 |
| \(\boldsymbol | 0 | 1 | 0 |
| \(\boldsymbol | 0 | 1 | 1 |
| \(\xcancel | \(\xcancel<1>\) | \(\xcancel<0>\) | \(\xcancel<0>\) |
| \(\xcancel | \(\xcancel<1>\) | \(\xcancel<0>\) | \(\xcancel<1>\) |
| \(\boldsymbol | 1 | 1 | 0 |
| \(\boldsymbol | 1 | 1 | 1 |
| \(C_2\) | \(\mathsf | \(\mathsf | \(\mathsf |
| \(w_0\) | 0 | 0 | 0 |
| \(w_1\) | 0 | 0 | 1 |
| \(w_2\) | 0 | 1 | 0 |
| \(w_3\) | 0 | 1 | 1 |
| \(\xcancel | \(\xcancel<1>\) | \(\xcancel<0>\) | \(\xcancel<0>\) |
| \(\xcancel | \(\xcancel<1>\) | \(\xcancel<0>\) | \(\xcancel<1>\) |
| \(\boldsymbol | 1 | 1 | 0 |
| \(\boldsymbol | 1 | 1 | 1 |
Figure 9: Context for (42), Facts in Bold, Laws Crossing out Worlds
On this model a context provides a set of worlds compatible with the facts, in \(C_2\) \(\textit
To evaluate \(\mathsf<\neg R>W>\), one finds the set of worlds from \(\textit
(43) [You’re invited to bet heads on a coin-toss. You decline. The coin comes up heads.] See, if you had bet heads you would have won! (Slote 1978: 27 fn33)
This example relies seamlessly on three pieces of background knowledge about how betting works:
If you bet and it comes up heads, you win: \(\mathsf<\medsquare((B\land H)\supset W)>\)
If you bet and it doesn’t come up heads, you don’t win: \(\mathsf<\medsquare((B\land\neg H)\supset\neg W)>\)
And it specifies facts: \(\mathsf<\neg B\land H>\). The resulting context is detailed in Figure 10:
| C(43) | \(\mathsf\) | \(\mathsf | \(\mathsf |
| \(w_0\) | 0 | 0 | 0 |
| \(\xcancel | \(\xcancel<0>\) | \(\xcancel<0>\) | \(\xcancel<1>\) |
| \(\boldsymbol | 0 | 1 | 0 |
| \(\xcancel | \(\xcancel<0>\) | \(\xcancel<1>\) | \(\xcancel<1>\) |
| \(w_4\) | 1 | 0 | 0 |
| \(\xcancel | \(\xcancel<1>\) | \(\xcancel<0>\) | \(\xcancel<1>\) |
| \(\xcancel | \(\xcancel<1>\) | \(\xcancel<1>\) | \(\xcancel<0>\) |
| \(w_7\) | 1 | 1 | 1 |
Figure 10: Context for (43)
Now, consider the counterfactual \(\mathsfW>\). The first step is to retract the fact that \(\mathsf\) is false from each world in \(\textit
It should now be more clear how premise semantics delivers on its promise to be more predictive than similarity semantics when it comes to counterfactuals in context, and affords a more natural characterization of how a context informs the interpretation of counterfactuals. This analysis was crucially based on the idea that some facts determine other facts, and that the process of retracting a fact is constrained by these relations. However, even premise semantics has encountered counterexamples.
(45) a. If both Switch A and Switch B are up, the light is on.
\(\mathsf<\medsquare((A\land B)\supset L)>\) b. If either Switch A or Switch B is down, the light is off.
\(\mathsf<\medsquare((\neg A\lor\neg B)\supset \neg L)>\) c. Switch A is up, Switch B is down, and the light is off.
\(\mathsf
\(\mathsfL>\)
Figure 11: Context for (45d)
Intuitively, the analysis went wrong in allowing the removal of the fact that Switch A is up when retracting the fact that Switch B is down. Schulz (2007: §5.5) provides a more sophisticated version of this diagnosis: although the fact that Switch A is up and the fact that the light is off together determine that Switch B is down, only the fact that the light is off depends on the fact that Switch B is down. If one could articulate this intuitive concept of dependence, and instead only retract facts that depend on the fact you are retracting (in this case the fact that B is down), then the error could be avoided. It is unclear how to implement this kind of dependence in Veltman’s ( 2005 ) framework. Schulz (2007: §5.5) goes on show that structural equations and causal models provide the necessary concept of dependence—for more on this approach see §3.3 below. After all, it seems plausible that the light being off causally depends on Switch B being down, but Switch A being up does not causally depend on Switch B being down. It remains to be seen whether the more powerful framework developed by Kratzer (1989, 2012) can predict (45).
3.2 Conditional Probability Analyses
While premise semantics has been prominent among linguists, probabilistic theories have been very prominent among philosophers thinking about knowledge and scientific explanation. [40] Adams (1965, 1975) made a seminal proposal in this literature:
However, Adams (1970) was also aware that indicative/subjunctive pairs like (3)/(4) differ in their assertability. To explain this, he proposed the prior probability analysis of counterfactuals (Adams 1976) :
It would seem that this analysis accurately predicts our intuitions in (45) about \(\mathsfL>\). Let \(P_0\) be an agent’s credence before learning that Switch B is down. (45a) requires that \(P_0(\mathsf
Objective probability analyses have been popular among philosophers trying to capture the way that counterfactuals feature in physical explanations, and why they are so useful to agents like us in worlds like ours. Loewer (2007) is a good example of such an account, who grounds the truth of certain counterfactuals regarding our decisions like (46) in statistical mechanical probabilities.
(46) If I were to decide to bet on the coin’s landing heads, then the chance I would win is 0.5
Loewer (2007) proposes that (46) is true just in case (where \(P_<\textit
Loewer (2007) acknowledges that this analysis is limited to counterfactuals like (46). He argues that it can address the philosophical objections to the similarity analysis discussed in §2.6, namely why counterfactuals are useful in scientific explanations, and for agents like us in a world like our own.
3.3 Bayesian Networks, Structural Equations, and Causal Models
Recall from §1.2.3 the basic idea of a Bayesian Network: rather than storing probability values for all possible combinations of some set of variables, a Bayesian Network represents only the conditional probabilities of variables whose values depend on each other. This can be illustrated for (45).
(45) a. If both Switch A and Switch B are up, the light is on.
\(\mathsf<\medsquare((A\land B)\supset L)>\) b. If either Switch A or Switch B is down, the light is off.
\(\mathsf<\medsquare((\neg A\lor\neg B)\supset \neg L)>\) c. Switch A is up, Switch B is down, and the light is off.
\(\mathsf
\(\mathsfL>\)
Sentences (45a)-(45c) can be encoded by the Bayesian Network and structural equations in Figure 12.
Figure 12: Bayesian Network and Structural Equations for (45)
Recall that \(L\dequal A\land B\) means that the value of L equals the value of \(A\land B\), but also asymmetrically depends on the value of \(A\land B\): the value of \(A\land B\) determines the value of L, and not vice-versa. How, given the network in Figure 12, does one evaluate the counterfactual \(\mathsfN>\)? Several different answers have been given to this question.
Pearl (1995, 2000, 2009, 2013: Ch.7) proposes:
On this approach, one simply deletes the assignment \(B=0\), replaces it with \(B=1\), and solves for L using the equation \(L\dequal A\land B\). Since the deletion of \(B=0\) does not effect the assignment \(A=1\), it follows that \(L=1\) and that the counterfactual is true. This simple recipe yields the right result. Pearl nicely sums up the difference between this kind of analysis and a similarity analysis:
In contrast with Lewis’s theory, counterfactuals are not based on an abstract notion of similarity among hypothetical worlds; instead, they rest directly on the mechanisms (or “laws,” to be fancy) that produce those worlds and on the invariant properties of those mechanisms. Lewis’s elusive “miracles” are replaced by principled [interventions] which represent the minimal change (to a model) necessary for establishing the antecedent… Thus, similarities and priorities—if they are ever needed—may be read into the [interventions] as an afterthought… but they are not basic to the analysis. (Pearl 2009: 239–240)
Hiddleston (2005) presents the following example.
(48) a. If the cannon is lit, there is a simultaneous flash and bang. b. The cannon was not lit, there was no flash, and no bang. c. But, if there had been a flash, there would have been a bang.
(48c) is intuitively true in this context. The network for (48) is given in Figure 13.
Figure 13: Bayesian Network and Structural Equations for (48)
Hiddleston (2005) observes that interventionism does not predict \(\mathsf
(49) If the light had been on, then if you had flipped Switch A down, the light would be off.
And, considering a simple match, Fisher (2017b: §1) observes that (50b) is intuitively false.
(50) a. A match is struck but does not light. b. If the match had lit, then (even) if it had not been struck, it would have lit.
In both cases, interventionism is destined to make the wrong prediction. With (49), the intervention in the first antecedent removes the connection between Switch A and the light, so when the antecedent of the consequent is made true by intervention, it does not result in L’s value becoming 0. And so the whole counterfactual comes out false. Similarly with (50b), when the first antecedent is made true by intervention, it stays true even after the second antecedent is evaluated. Hence the whole conditional is predicted to be true. Fisher (2017a) also observes that interventionism also has no way of treating counterlegal counterfactuals like if Switch A had alone controlled the light, the light would be on.
A few final philosophical remarks are in order about the kinds of analyses discussed here. If one follows Woodward (2002) and Hitchcock (2001) in their interpretation of these networks, a structural equation should be viewed as a primitive counterfactual. It follows that this is a non-reductive analysis of counterfactual dependence: it only explains how the truth of arbitrarily complex counterfactual sentences are grounded in basic relations of counterfactual dependence. However, note in the earlier quotation above from Pearl (2009: 239–240) that he interprets structural equations as basic mechanisms or laws, and so arguably counts as an analysis of counterfactuals in terms of laws. These amount to two very different philosophical positions that interact with the philosophical debates surveyed in §1.3.
It is also worth noting that while many working in this framework apply these networks to causal relations, there is no reason to assume that the analysis would not apply to other kinds of dependence relations. For example, constitutional dependence is at the heart of counterfactuals like:
(51) If Socrates hadn’t existed, the set consisting of Socrates wouldn’t have existed.
From a Bayesian Network approach to mental representation (§1.2.3), this makes perfect sense: the networks encode probabilistic dependence which can come from causal or constitutional facts.
Finally, it is worth highlighting that the philosophical objections directed at the similarity analysis in §2.6 are addressed, at least to some degree, by structural equation analyses. Because the central constructs of this analysis—structural equations and Bayesian Networks—are also employed in models of mental representation, causation, and scientific explanation, it grounds counterfactuals in a construct already taken to explain how creatures like us cope with a world like the one we live in.
3.4 Summary
Premise semantics (3.1), conditional probability analyses (§3.2) and structural equation analyses (§3.3) all aim to analyze counterfactuals by focusing on certain relations between facts, rather than similarities between worlds. These accounts make clearer and more accurate predictions about particular counterfactuals in context than similarity analyses. But, ultimately, both premise semantics and conditional probability analyses had to incorporate causal dependence into their theories. Structural equation analyses do this from the start, and improve further on the predictions of premise semantics and conditional probability analyses. Another strength of this analysis is that it integrates elegantly into the broader applications of counterfactuals in theories of rationality, mental representation, causation, and scientific explanation surveyed in §1.1. There is still rapid development of structural equation analyses, though, so it is too early to say where the analysis will stabilize, or how it will fair under thorough critical examination.
4. Conclusion
Philosophers, linguists, and psychologists remain fiercely divided on how to best understand counterfactuals. Rightly so. They are at the center of questions of deep human interest (§1). The renaissance on this topic in the 1970s and 1980s focused on addressing certain semantic puzzles and capturing the logic of counterfactuals (§2). From this seminal literature, similarity analyses (D. Lewis 1973b; Stalnaker 1968) have enjoyed the most widespread popularity in philosophy (§2.3). But the logical debate between similarity and strict analyses is still raging, and strict analyses provide a viable logical alternative (§2.4). Criticisms of these logical analyses have focused recent debates on our intuitions about particular utterances of counterfactuals in particular contexts. Structural equation analyses (§3.3) have emerged as a particularly prominent alternative to similarity and strict analyses, which claims to improve on both in significant respects. These analyses are now being actively developed by philosophers, linguists, psychologists, and computer scientists.
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