A conditional is a sentence, statement, proposition, or thought of the form
If A then C
"A" is called the antecedent of the conditional and "C" the consequent. Philosophers have traditionally divided conditionals into two main groups, indicative, which can be symbolized as [A→C], and subjunctive ([A□→C]). The so-called counterfactual conditionals that have been the subject of so much discussion in analytic philosophy are subjunctive conditionals of the form
If it were to be the case that X then it would be the case that Y (if X were to happen, then Y would happen)
If it had been the case that X, then it would have been the case that Y (if X had happened, then Y would have happened)
Subjunctive conditionals of the form "If she be gone, he is in despair" are not at issue.
It is because the antecedents of such subjunctive conditionals usually state something that is not in fact the case or "contrary-to-fact," or is at least assumed not to be the case by the thinker or utterer of the conditional, that they have come to be known as counterfactuals.
It is not clear that there is any interesting difference between present and future tense indicative and subjunctive conditionals. It is not clear, for example, that there is any important semantic difference between one saying "If it were raining they would not be playing" and "If it's raining, then they're not playing." Nor is it clear that there is any important semantic difference between one saying "If she goes to the party, he will not go" and "If she were to go, he would not go," or between one saying "If salt is mixed with water it dissolves (will dissolve)" and "If salt were to be mixed with water it would dissolve." The idea that there is an important difference here is perhaps an artifact of the empiricist outlook dominant in analytic philosophy in the last century, which endorsed the "regularity theory of causation" and the associated idea that laws of nature could be adequately expressed by the "material conditional" of standard first-order logic.
However that may be, the difference between indicative and subjunctive conditionals seems clearer in the case of past-tense conditionals. Consider
If Georges Agniel and his friends did not discover the Lascaux caves, then someone else did
If Georges Agniel and his friends had not discovered the Lascaux caves, then someone else would have
The difference of meaning is immediately apparent and is sufficiently shown by the fact that although one takes the first to be true, one has no reason to believe the second.
The commonly used labels ("indicative," "subjunctive," and "counterfactual") do not, however, perspicuously mark out the set of conditionals that concern philosophers when they discuss counterfactuals. The indicative/subjunctive distinction is purely syntactical and simply fails to pick out the right set of conditionals. On the one hand, "If the Palestinians declared statehood now, the Israelis would retaliate" is a counterfactual that is not grammatically subjunctive. On the other hand, one can utter a subjunctive conditional of the form "If X had happened, then Y would have happened" without having any intention to assert or imply the falsity of the antecedent. Suppose I am a detective who suspects that a criminal did A although none of my colleagues believe me. I note that the criminal did something peculiar, that is, B, and remark truly that if she had done A, she would have had to have done B in support of my case, without in any way implying that the state of affairs specified in the antecedent is not the case (alternatively, I may say this before dispatching someone to find out whether she did B). Again, I may set you a puzzle, asking you to work out what I have done, and give you clues, pointing out that if I had done X then this would have happened, that if I had done Y then this other thing would have happened, without ever asserting or implying that I did not do X or Y. Again, I may truthfully assert both "If I had come to the party I would have got drunk" and "If I had not come to the party I would have got drunk" without for a moment thinking or implying, inconsistently, that both these antecedents are false.
The purely syntactical criterion is no good, then, and blanket use of the term "counterfactual" to cover all the subjunctive conditionals that concern philosophers is no better. It remains true, nevertheless, that when one asserts a subjunctive conditional one almost invariably suggests that the state of affairs specified in the antecedent is not in fact the case. This entry will therefore use the traditional term "counterfactual" in this discussion, and contrast counterfactuals generally with indicatives in spite of the difficulties just noted.
Theories of Conditionals
Any theory of counterfactuals will be part of a general theory of conditionals, and the question arises as to what form a general theory of conditionals should take. Many favor a truth-conditional approach, that is, one that analyzes conditionals by offering an account of the conditions under which statements of the form "If A then C" are true or false (possible-worlds and metalinguistic accounts of conditionals are examples of truth-conditional approaches). Others seek to analyze conditionals by reference to the conditions under which they can be justifiably asserted or accepted as true (e.g., see Edgington 1986). An attractive alternative is John L. Mackie's (1973) condensed argument/supposition account, according to which conditionals are condensed arguments or suppositions and so not strictly true or false at all.
A central issue for any theory of conditionals is whether indicatives and counterfactuals should receive a uniform treatment, that is, one that uses the same theoretical apparatus across the board. David K. Lewis (1973, 1976) and Frank Jackson (1977, 1979) both reject this idea, offering nonuniform theories that fix the truth-conditions of indicatives and counterfactuals in different ways. Mackie (1973), by contrast, offers a uniform account of all conditionals in terms of the single basic notion of suppositions, and Robert C. Stalnaker (1968), having given an account of all conditionals in terms of possible worlds, accounts for the intuitive difference between indicatives and counterfactuals by appeal to pragmatic considerations.
Central to this debate is the question whether one bases one's account of indicative conditionals on the material conditional of standard first-order logic, often symbolized as "A⊃C," which is true just in case its antecedent is false or its consequent is true (the truth-value of the whole is determined in a purely truth-functional way by the truth-values of the parts). Lewis and Jackson are among those who think that the material-conditional approach can give an adequate account of all indicative conditionals (others think that it can only provide a necessary and not a sufficient condition), but a unified material-conditional account of both indicatives and counterfactuals seems a nonstarter. The material-conditional account, for example, classifies
If the moon had been made of cheese, I would be immortal
as just as surely true as
If this apple had been made of copper, it would have conducted electricity
simply on the ground that the antecedent is false. But one is much more discriminating about the truth-values of counterfactual conditionals than this account allows. That is why Lewis and Jackson, having accepted the material-conditional theory for indicatives, adopt a nonuniform general theory of conditionals, Lewis (1973) offering a possible-worlds account of counterfactuals and Jackson (1977) a causal account.
A further issue concerns whether one can give a uniform account of the logic of indicatives and counterfactuals. The following inference patterns
(I1) If A then C, therefore, if not-C then not-A (contraposition)
(I2) If A then B, if B then C, therefore, if A then C (hypothetical syllogism)
(I3) If A then B, therefore, if A and C then B (strengthening the antecedent)
are valid for the material conditional, but are widely agreed not to hold for counterfactual conditionals (e.g., consider the failure of (I3), in the move from the true claim "If he had walked on the ice, it would have broken" to the false claim "If he had walked on the ice and had been holding a large bunch of helium balloons, the ice would have broken"). While a nonuniform account can allow that these inference patterns hold for indicatives but fail for counterfactuals (see Lewis  and Jackson , who attempts to explain away apparently invalid indicative cases like "if he has made a mistake, then it is not a big mistake, therefore, if he has made a big mistake, he has not made a mistake" in terms of failure of assertibility), a uniform account must hold that if they fail for counterfactuals then they also fail for indicatives (see Stalnaker 1968).
Theories of Counterfactuals
Turning now to counterfactuals, one finds three main approaches. The metalinguistic account initiated by Nelson Goodman in 1947 (see also Chisholm 1955, Mackie 1973, Tichy 1984) analyses counterfactuals in terms of an entailment relationship between the antecedent plus an additional set of statements or propositions, and the consequent. The causal approach offered by Jackson in 1977 (see also Kvart 1986) is closely related but deserves a separate category because it appeals essentially to causal concepts in its analysis of counterfactuals, thereby ruling out the popular strategy of using counterfactuals in an analysis of causation (one of the first to do this was Hume 1748/1975, p. 76; see also Lewis 1986b). Finally, there is the possible-worlds approach initiated by William Todd (1964), Stalnaker (1968), and Lewis (1973), which analyses counterfactuals in terms of similarity relations between worlds. This entry will consider them in turn, after hereby putting aside, as unimportant to the present concerns, all counterfactuals that are true (or false) as a matter of logic or a priori necessity, such as
If Q had been P it would have entailed P (Q)
If this number had been 2 it would have been even (odd)
If this circle had been square it would have had fewer than (more than) seven sides
the metalinguistic approach
According to Goodman's (1947) metalinguistic approach a counterfactual asserts a certain connection or consequential relation between the antecedent and the consequent. Since in the case of the counterfactuals that concern this discussion the antecedent does not entail the consequent as a matter of logic or a priori necessity, certain other statements, including statements of laws and existing particular conditions, must be combined with the antecedent to entail the consequent. These counterfactuals, then, are true, if true at all, only if (and if) the antecedent combined with a set of statements S that meets a certain condition ϕ entails the consequent as a matter of law. The theory is metalinguistic because counterfactuals are treated as equivalent to metalinguistic statements of the relevant entailments.
A notorious difficulty for this theory has been to give an adequate specification of condition ϕ. Consider [A□→C]. Given that the assumption, in the case of a counterfactual, is that A is false, one may reasonably assert ∼A. However, if ∼A were admissible into S, then with A one would get the contradiction [A&∼A], and since it is generally accepted that anything can be inferred from a contradiction, anything could be inferred from the conjunction of A and S, including C. All counterfactuals would therefore turn out to be true (a priori false counterfactuals have been excluded). To prevent this trivialization, the statements that constitute S must be (logically) compatible with A. This excludes ∼A. A further requirement noted by Goodman is that the statements that constitute S must be compatible with ∼C; for if they were not, C would follow from S itself, and A and the laws would play no role in the inference to C.
With this in hand Goodman offers the following analysis: "A counterfactual is true if and only if (iff) there is some set S of true sentences such that S is compatible with C and with ∼C, and such that [A & S] is self-compatible and leads by law to C; while there is no set S′ compatible with C and with ∼C and such that [A & S] is self-compatible and leads by law to ∼C" (Goodman 1947, p. 120; for a discussion of this last condition, see Bennett 2003; Parry 1957). Restricting S with the notion of compatibility does not seem to be enough, however, for counterfactuals that clearly seem false still threaten to turn out true. Consider
(1) If match m had been struck, it would have flared
(2) If match m had been struck, it would not have been dry
Despite the restrictions on S, one gets the unacceptable result that (1) and (2) both turn out true. To see this, assume that it is a law that (L) when oxygen is present, dry matches flare when struck. Start with the situation of the dry match (D), the presence of oxygen (O), and suppose that the match has not been struck (∼S) and has not flared (∼F). O, D, and L are compatible both with S and with ∼F, and with S, they imply F. Thus, (1) is true. Now, however, suppose ∼F: that in fact the match has not flared. ∼F, O, and L are compatible both with S and with D, but with S they imply ∼D. Thus, (2) is true.
To eliminate this unwanted consequence, Goodman (1947) suggests that the relevant conditions in S must be cotenable with the antecedent. A is cotenable with B if it is not the case that B would have been false if A were true. ∼F is thus compatible with S but not cotenable with it, because if the match had been struck (S), it would have flared (F). So (1) is true and (2) is false. However, this solution results in a circular definition or a regress, for counterfactuals are defined in terms of cotenability and cotenability is defined in terms of counterfactuals. Goodman proposed no solution to this problem (for a short discussion, see Bennett 2003, pp. 310–312).
the condensed argument-suppositional approach
Closely related to the metalinguistic account is Mackie's (1973) condensed argument or suppositional account according to which all conditionals, including all counterfactuals, are condensed or abbreviated arguments that leave certain auxiliary premises unstated. Generally, to assert [A□→C] is to assert C within the scope of the supposition A (Mackie replaced the notion of a condensed argument by that of a supposition in an attempt to cover certain atypical conditionals that do not readily expand into arguments, e.g., "If that's a Picasso I'm a Martian").
There are two central ways in which Mackie's (1973, 1974) account differs from Goodman's (1947). First, Mackie abandons any metalinguistic element. In fact, according to Mackie, this feature of Goodman's account is the reason to reject it. Mackie argues that it simply "does not ring true" that when one asserts counterfactuals one is performing a higher-level linguistic act whose subject is a lower-level linguistic act. If-sentences are about the world, not about what is said about the world.
Second, Mackie relaxes the cotenability requirement on A and S. One does not need to provide an exact criterion of cotenability. All that one needs is the idea that the speaker assumes the cotenability of A and S and a notion of cotenability that can, he claims, be elucidated simply in terms of it being reasonable to combine a belief that S with A.
This suggestion is closely in line with what are sometimes called third-parameter views of counterfactuals (see Tichy 1984, who attributes this view to Chisholm 1955; Mill 1868; Ramsey 1931). According to this view, when a speaker asserts a counterfactual, he or she implicitly assumes a set of propositions. The counterfactual is true just in case the antecedent of the counterfactual and the assumed propositions entail the consequent and the implicitly assumed propositions are true. Since the implicitly assumed propositions depend on the attitudes of the speaker, no analysis of these propositions can be given and so the cotenability problem does not arise.
One point strongly in favor of such views is their ability to deal with ambiguous counterfactuals. Consider
If Caesar had been in command in Korea, he would have used the atom bomb
If Caesar had been in command in Korea, he would have used catapults
Although both counterfactuals can plausibly be asserted, they make different predictions about what would have happened. By introducing a third parameter this ambiguity can be located in the set of implicitly assumed propositions. The first counterfactual is asserted by someone who is assuming that Caesar was alive during the actual Korean War, and the second counterfactual is asserted by someone who is assuming that Caesar was involved in a war in Korea during Caesar's actual lifetime.
Jonathon Bennett (2003, pp. 305–308) objects to Chisholm's (1955) version of this solution to the cotenability problem, arguing that it implausibly requires that the asserter of [A□→C] have the assumed propositions in mind, although one can, for example, be sure that the lights would have gone off if one had turned the oven on again without knowing about the faulty electrical wiring in one's kitchen. He further argues that there are no limits to what a speaker could assume in asserting [A□→C], and that this lets in unwanted counterlogical conditionals like "if that piece of cast iron were gold some things would be malleable and not malleable."
the causal approach
Another theory closely related to the metalinguistic approach is Jackson's (1977) causal theory of counterfactuals, so-called because of the central role that causality plays in it. To determine the truth-value of a counterfactual one takes the causal laws at the actual world at a time. These determine the state of the world at later times. One then takes the state of the world at the antecedent time, changes it as little as possible to make the antecedent true, and determines whether the causal laws predict subsequent states that make the consequent true.
More formally, [A□→C] is true at all the A-worlds satisfying the following:
(i) Their causal laws are identical with ours at the time of the antecedent and after
(ii) Their antecedent time-slices are the most similar to ours in particular facts
(iii) They are identical in particular fact to our world prior to the time of the antecedent
Sequential counterfactuals assert that if something had happened at one time, something else would have happened at a later time, and one difficulty for the theory is presented by asequential counterfactuals like:
If I had had a coin in my pocket, it would have been a Euro.
If Flintoff had not taken the winning wicket, Harmison would have (where this is understood as meaning that sooner or later one of them would have taken the winning wicket)
Jackson (1977) proposes to analyze asequential counterfactuals in terms of sequential counterfactuals. For example, one asserts the counterfactual about Flintoff and Harmison when one thinks that if Flintoff had failed to take the final wicket, events would have ensured Harmison's taking it (they were the only bowlers left and Australia was batting so poorly).
Jackson's account appeals to similarities between worlds. Does that mean that he is really giving a possible-worlds account of counterfactuals? Although he no longer objects to being classified as a possible-worlds theorist, in 1977 he drew a sharp division between his causal account and the possible-worlds account. He argued that a causal theorist about counterfactuals could avoid ontological commitment to possible worlds because the relevant similarities were things like the mass of an object or the magnitude of a force, similarities that could be characterized by reference to features of the actual world without any appeal to possible worlds.
the possible worlds approach
In asserting a counterfactual one is of course standardly considering possibilities, how things would or might have been if certain other things had not been as they were, how things would or might be if things were not as they are, and the most influential treatment of counterfactuals has been the possible-worlds approach, which proposes to analyze counterfactuals by giving a rigorous account of their truth conditions and logical behavior using possible-worlds semantics. Stalnaker (1968) and Lewis (1973) are the most influential proponents of this view, and the basic idea is that the counterfactual [A□→C] is true just in case the closest possible A-worlds (worlds where A is true) are C-worlds (worlds where C is true), and the central notions are those of a possible world and the closeness relation. Both Stalnaker and Lewis introduce the idea of a "logical space," which is, roughly, a space of possible worlds. They locate the actual world in a "similarity structure" in such a logical space and make use of this similarity structure to determine the truth-values of counterfactuals.
More formally, for Stalnaker (1968)
[A□→C] is true iff A is impossible or C is true at f (A, w* )
where f is a "selecting" function that takes the antecedent A and the actual world w* as arguments and delivers a unique possible world as a value. The counterfactual is true if C is true at the possible world that f delivers as the value.
How exactly does the selection function select? The informal answer is that the selection is based on an ordering of possible worlds with respect to their similarity or resemblance to the actual world. More formally, for Lewis (1973)
[A□→C] is true iff either there is no A-world or some [A & C] world is more similar to the actual world than any [A&∼C] world
It is convenient to represent Lewis's truth conditions in this way, with direct reference to similarity, although in his original presentation the ordering relation is explicated in terms of a system of spheres of worlds (for any possible world, all other possible worlds can be placed on spheres centered on that world, the sizes of the spheres representing how close those worlds are to that world. All worlds on a given sphere are equally close to the centered world, and inner spheres are closer to the centered world than outer spheres).
Lewis (1973) and Stalnaker (1968) agree that if the antecedent of a counterfactual is impossible than the counterfactual is trivially true. For Lewis, this is because there is no such A-world; for Stalnaker, function f selects the impossible world in which every statement is true. (It is not however clear that all impossible counterfactuals are alike in respect of truth. There is, intuitively, a difference between "If Picasso had been a sonnet, he would have had fourteen lines" and "If Picasso had been sonnet, he would have had compound eyes," and Daniel Nolan  and others argue that impossible worlds, like possible worlds, can be ranked with respect to comparative similarity to the actual world.) Lewis and Stalnaker also agree that inference patterns like contraposition, hypothetical syllogism, and strengthening the antecedent ((I1) to (I3) earlier) are invalid for counterfactuals. However, they disagree about the conditional excluded middle: [[A□→C] ∨ [A□→∼C] for all A and C. Stalnaker accepts it because according to his account there will always be one closest possible world, whereas Lewis accepts ties among closest possible worlds and so the principle is not universally true.
Stalnaker and Lewis also agree in analyzing the "closeness" relation in terms of similarity between worlds. However, what makes one world more similar to the actual world than another world? Kit Fine (1975) and Bennett (1974) object that Lewis's (1973) theory does not provide the correct truth conditions if closeness of worlds is understood in terms of our everyday intuitive notion of similarity. Intuitively, the counterfactual
If Nixon had pushed the button, there would have been a nuclear holocaust
seems true, and yet it is false by the lights of one commonsense notion of similarity, according to which a world in which a nuclear holocaust does not occur although Nixon presses the button is much more similar to our unholocausted world than a world where a nuclear holocaust does occur.
Lewis responds to this objection in "Counterfactual Dependence and Time's Arrow" (1979), claiming that a possible-worlds theory of counterfactuals does not need to appeal to any everyday notion of overall similarity. It is rather up to the theorist to work out a way of weighing factors relevant to overall similarity that will deliver the right truth-values for counterfactuals. Lewis offers the follows systems of weights:
[i] It is of the first importance to avoid big, widespread, diverse violations of law
[ii] It is of the second importance to maximize the spatiotemporal region throughout which perfect match of particular fact prevails
[iii] It is of the third importance to avoid even small, localized, simple violations of law
[iv] It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly (Lewis 1979, p. 473)
According to this system of weights, the Nixon counterfactual turns out true. Consider a world in which Nixon pushes the button and there is no nuclear holocaust; rather, events proceed in such a way as to match those in our world with perfect similarity. The trouble with claiming that this is the most similar world is that Nixon's pressing the button would have numerous effects (including the button's warming slightly, the subsequent state of Nixon's memory, and so on), and only a large miracle could wipe out all these changes. The worlds closest to ours are the ones that agree with our actual world until Nixon presses the button and then continue on in accordance with the laws of the actual world. (However, for a reformulation of the Nixon objection in the light of this reply, see Tooley 2003).
Many philosophers shy away from the apparent metaphysical commitments of the possible-worlds approach. For what is a possible world? Lewis's (1986c) answer that possible worlds are concrete entities, each as real as the actual world, seems to most hopelessly implausible, but there are many other views. Stalnaker's (1968) and Bennett's (2003) possible worlds, for example, are maximally consistent sets of propositions; Saul Kripke's are stipulations; and others hold that possible worlds are combinatorial constructions out of elements of the actual world.
Whatever one's view, and whether or not one wishes to appeal to possible worlds, counterfactual conditionals are the vehicles of two of the most fundamental forms of thought: "What if?" and "If only." They are central to imagination and invention, essential to curiosity and regret, essential, along with conditionals in general, to the fundamental capacities for debating, supposing, speculating, and hypothesizing that constitute the heart of one's intelligence.
See also Bennett, Jonathan; Chisholm, Roderick; Conditionals; Goodman, Nelson; Hume, David; Kripke, Saul; Lewis, David; Mackie, John Leslie; Modality, Philosophy and Metaphysics of; Response-Dependence Theories; Semantics.
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