## Conditionals

## Conditionals

# CONDITIONALS

Conditionals are sentences like the following:

(1) If Oswald did not kill Kennedy, then someone else did

(2) We will not go on the trip if it rains tomorrow

(3) If Oswald had not killed Kennedy, then someone else would have

(4) We would be playing tennis if it were not raining

Conditionals are often believed to be analyzable into a two-place sentence connective and two constituent sentences, the antecedent (the sentence introduced by "if") and the consequent. (Thus, [3] may be analyzed into a binary connective ["If it had been the case that …, then it would have been the case that …"] and the constituent sentences "Oswald did not kill Kennedy" and "someone else did [kill Kennedy].")

Many philosophers believe that there is an important difference between conditionals like (1) and (2) (which are commonly called "indicative conditionals"), and those like (3) and (4) (called "subjunctive" or "counterfactual"). Following Ernest W. Adams (1970), one can motivate this idea by considering (1) and (3). Suppose that you think that Oswald killed Kennedy, acting alone, and that no one else ever thought of committing this crime. You reject (3). But you accept (1): If you are wrong in thinking that Oswald did it, then someone else must be the culprit. Thus, it can be perfectly reasonable to assign different truth-values to the two conditionals. This indicates that an indicative conditional cannot in general have the same meaning as the corresponding counterfactual. Now suppose that this observation is combined with the suggested analysis of conditionals into two constituent sentences and a binary connective. Since (1) and (3) have the same constituent sentences, it is natural to conclude that their difference in meaning must result from a difference in meaning between the conditional connectives contained in the two sentences. The connective occurring in indicative conditionals, it seems, must differ semantically from the one found in counterfactuals.

This line of reasoning can be resisted in a number of ways. In particular, even if (1) and (3) belong to semantically different kinds of conditional, it is not obvious that the line between the two kinds coincides with that between indicative conditionals and counterfactuals. For all the argument shows, some indicative conditionals might have to be classified with (3) or some counterfactuals with (1), and under the influence of Vic H. Dudman (1984) some philosophers argue that indicative conditionals like (2) belong to the same class as (3).

Nonetheless, the standard view has it that conditionals are to be classified into indicatives and counterfactuals, and this entry will focus on theories that rest on this classification. The symbol "→" will be used for the indicative and "□ →" for the counterfactual conditional connective.

## Indicative Conditionals

Two of the main approaches to indicative conditionals will be considered.

### the equivalence thesis

Consider the mode of inference

(5) either

Bor not-A; therefore, ifA, thenB,

which is instantiated by the argument "Either the butler is guilty, or Fred lied about the ice pick. Therefore, if Fred said the truth, then the butler is guilty." This form of inference might appear to be valid. If it is, then an indicative conditional must be true whenever its antecedent is false and whenever its consequent is true. Moreover, it seems plausible that these are the only cases in which the conditional is true. It cannot be true if it has a true antecedent but a false consequent. (If someone says, "If it rains, she won't come," and it rains but she does come, then the utterance is not true.) This suggests that "*A* → *C* " is true if and only if either *A* is false or *C* is true. In other words, "*A* → *C* " has the same truth-conditions as the so-called material conditional, "*A* ⊃ *C*." This claim is sometimes called the "equivalence thesis."

It is well known that the equivalence thesis yields many seemingly absurd consequences. For instance, it makes (6) come out true, since (6) has a false antecedent and a true consequent:

(6) If Kennedy survived Oswald's assassination attempt, then he died in the assassination attempt.

Yet (6) does not seem to be assertable.

One strategy for dealing with such apparent counterexamples originates in work by Paul Grice (1991): According to the equivalence thesis, knowledge that *A* is false or that *C* is true is sufficient for knowing that "*A* → *C* " is true. But if one's belief in the truth of a conditional rests solely on one's knowledge of the truth-values of its constituents (as in the case of [6]), then there is little point in asserting the conditional. For one could convey more information with fewer words by simply uttering the consequent, or the negation of the antecedent (as the case may be). If one utters the conditional anyway, then the audience, trusting the speaker not to do something pointless, will conclude that the speaker has reasons for believing the conditional that go beyond knowledge of the truth-values of its constituents. The utterance of the conditional would therefore be misleading, and the conditional, although true, is unassertable. When confronted with (6), one notes that it would be a mistake to assert it. This accounts for the feeling that there is something wrong with uttering the conditional. This impression can thus be explained without denying that the conditional is true.

The Gricean account has come in for criticism, but even if it is correct and apparent counterexamples to the equivalence thesis can be explained away, one may wonder whether the thesis is sufficiently well motivated. The previous argument for it rests on the assumption that the inference schema (5) is valid. But this premise has been questioned, because of apparent counterexamples to (5), such as "You will meet nobody, or at least not many people. Therefore, if you meet many people, then you will meet nobody."

The equivalence thesis can be supported in other ways, however: It is the simplest of all candidate truth-conditional theories of indicative conditionals. And Frank Jackson (1987, chapter 2) argues that, although the equivalence theorist must concede that an indicative conditional's degree of assertability can differ from its probability of truth, the equivalence thesis can be used to explain the assertability-conditions and can be supported by appeal to this explanatory power.

### the ramsey test

Another approach to the semantics of indicative conditionals originates in a footnote in a paper by Frank P. Ramsey (1990, p. 155, n. 1) and has been developed in detail by Adams (1975). It starts from the idea, which is sometimes called the "Ramsey test," that the degree to which a speaker ought to accept "*A* → *C* " equals the person's subjective conditional probability P(*C* |*A* ) (i.e., P(*A* and *C* ) / P(*A* )), provided that P(*A* ) is not zero so that P(*C* |*A* ) is defined. (On other versions of this account, P[*C* |*A* ] measures the degree to which the speaker should regard the conditional as assertable. The discussion below will focus on the acceptability-conditional version of the thesis.) This hypothesis is strongly supported by its ability to predict pre-theoretical intuitions about individual conditionals. Suppose that I am about to cast a fair die. My probability that I will throw a six given that I will throw an even number is one-third, and this is also the degree to which I accept, "I will throw a six if I throw an even number."

One might be tempted to try to explain why the degree of acceptability of "*A* → *C* " equals P(*C* |*A* ) by the assumption that

(7) a conditional "

A→C" expresses a proposition, and the probability that this proposition is true equals P(C|A) in all probability distributions for which P(C|A) is defined.

However, David K. Lewis shows that (7) is false (he proves this and some stronger results in his 1991a and 1991b). Instead of stating the proof, this entry will point in a nonrigorous and informal way in the direction of the reason why (7) is false (this seems more intuitively helpful than a formal proof):

Let each point of the rectangle in Figure 1 stand for a possible world, and let the rectangle as a whole represent

the totality of possible worlds. Propositions can be represented by the regions containing all and only the points that stand for worlds in which these propositions are true. One can model a belief system by distributing one kilogram of mud over the rectangle: If P(*X* ) equals *p* in the belief system one intends to represent, then one places *p* kilograms of mud on the region representing the proposition *X*. Every possible way of distributing the mud corresponds to some probability distribution. If (7) were true, then there would have to be some region, namely the one representing the proposition expressed by "*A* → *C*," that bears an amount of mud equal to P(*C* |*A* ), that is, to the ratio of the amount on the *A* & *C* region and that on the *A* region, whenever P(*A* ) is not zero. However, it is easy to make it plausible that there is no such region. Assume that P(*C* |*A* ) equals one-half, which is to say that there is the same (nonzero) amount of mud on the *A* & *C* region as on the *A* & *∼C* region. This is information about the relative amounts of mud on the two regions, and as such it tells one next to nothing about the absolute amount on any specific region. In particular, it seems intuitively plausible that, contrary to (7), there is no region that must be loaded with exactly half a kilogram of mud whenever the *A* & *C* region and the *A* & *∼C* region bear the same (nonzero) amount of mud.

These considerations suggest that there is no one region whose amount of mud equals the ratio of the amount of *A* & *C* mud and the amount of *A* mud whenever this ratio is defined. However, it might be that, whenever the mud is distributed in such a way that the ratio is defined, there is some region whose amount of mud equals the ratio, though it is a different region in different cases. (Note the difference in the scopes of the quantifiers.) Hence, as Bas van Fraassen (1976) points out, for all the argument of the last paragraph shows, it could be that an indicative conditional "*A* → *C* " expresses a proposition and that its probability of truth equals P(*C* |*A* ) whenever this conditional probability is defined, but that the proposition expressed by the conditional varies systematically with the speaker's belief system. Philosophers have attempted to extend Lewis's proof so as to rule out this possibility.

As an alternative to finding truth-conditions that fit the Ramsey test, one might give up the idea that indicative conditionals express propositions and make the Ramsey test itself the centerpiece of one's semantic account. Such a theory raises two questions: (1) What account can be given of the meanings of compound sentences that embed indicative conditionals, such as "Either Fred will give you the money if you ask him, or he is more avaricious than Susie"? If indicative conditionals lack truth-values, then one cannot assign a meaning to this sentence using the usual truth-functional construal of the disjunction operator. However, as Allan Gibbard (1981, pp. 234–236) argues, that a nonpropositional account of indicative conditionals does not assign meanings to all compounds of conditionals might be a good thing. For many such compounds are so hard to understand that one may doubt that they have any clear meanings. (Consider "If Fred arrived yesterday if it rains tomorrow, then Susie was in Paris last week.") The thesis that indicative conditionals lack truth-conditions may explain such difficulties of interpretation. (2) The usual criterion for the acceptability of an inference form relates to whether it preserves truth, that is, to whether the conclusion of an instance of it must be true if the premises are true. If indicative conditionals cannot be true or false, then this criterion cannot be applied to inferences involving such conditionals. Adams (1975, chapter 2) tackles this problem by defining a new and independently motivated criterion of acceptability that is more widely applicable. According to this criterion, an inference must preserve probability, in a sense that Adams makes precise as follows: Call 1–P(*A* ) the "uncertainty" of the proposition *A* ; the uncertainty of a conditional "*A* → *C* " equals 1–P(*C* |*A* ). An inference preserves probability just in case there is no probability distribution in which the uncertainty of the conclusion exceeds the sum of the uncertainties of the premises. Classically valid arguments satisfy this condition, as do intuitively acceptable inferences involving indicative conditionals.

## Counterfactuals

Counterfactuals are used to analyze a wide range of philosophically important concepts, such as dispositions, causation, laws of nature, knowledge, practical rationality (counterfactuals are used in decision theory), and freedom of action ("She would have acted differently if she had chosen to do so"). Theories of counterfactuals are of interest in part because they may make it easier to understand and evaluate counterfactual accounts of other notions.

### goodman's account

In the seminal paper "The Problem of the Counterfactual Conditional" (1991) Nelson Goodman proposes an account of roughly the following form for a certain important class of counterfactuals:

(8) "

A□ →C" is true just in caseCfollows fromA, the laws of nature, and suitable true supplementary premises.

This account fits the ordinary-life practice of evaluating counterfactuals well. In determining what would have happened to a certain match if it had been struck on a specific occasion, one needs to draw on knowledge of the particular circumstances, such as the knowledge that (D) the match was dry and (O) oxygen was present, and of the law that (L) dry matches start to burn when struck in the presence of oxygen. These items of knowledge, when combined with the assumption that the match was struck, entail that (B) it burned. This justifies the conclusion that the match would have burned if it had been struck.

Which truths count as "suitable supplementary premises" in the sense of (8)? Clearly, not every truth does. When evaluating the counterfactual "If the match had been struck …," one cannot regard the truth that it was never struck as a suitable ancillary premise. More generally, if the antecedent is both self-consistent and consistent with the laws, then the suitable auxiliary premises must be consistent with the antecedent plus laws. Otherwise, the antecedent combined with the laws and the supplementary premises would entail everything, so that every counterfactual with the relevant antecedent would come out true—an unwelcome result if the antecedent is consistent.

This condition of consistency does not suffice as a criterion for the suitability of a truth as ancillary premise. For there are different sets of truths that meet the consistency constraint, and depending on which of them one regards as the set of suitable auxiliary premises, different counterfactuals come out true. If one uses (D) and (O) as supplementary premises in evaluating the conditional about the match, one can draw on one's knowledge that (L) is a law to conclude that the match would have burned if it had been struck. Availing oneself instead of (O) and (∼B) as auxiliary premises, one can (again using [L]) establish that the match would not have been dry if it had been struck.

The task of stating conditions for a truth's suitability as supplementary premise is central to Goodman's project. After discussing the issue at length, he ends up proposing that a truth *P* is suitable only if *P* is cotenable with the antecedent of the conditional, where this means: It is not the case that *P* would have been false if the antecedent had been true. (For instance, [∼B] is not cotenable with "the match was struck," since if the latter sentence had been true, (∼B) would have been false. But [D] and [O] would still have been true and are therefore cotenable.) Since this criterion is formulated in counterfactual terms, it renders Goodman's theory circular—a problem of which Goodman is vividly aware. As will become clear below, more recent work on counterfactuals promises to deliver a solution.

### the possible-world account

In the late 1960s and early 1970s another account of counterfactual conditionals was developed by Robert C. Stalnaker (1991b) and Lewis (1973). Lewis neatly expresses the core idea: "'*If kangaroos had no tails, they would topple over* ' seems to me to mean something like this: in any possible state of affairs in which kangaroos have no tails, and which resembles our actual state of affairs as much as kangaroos having no tails permits it to, the kangaroos topple over" (1973, p. 1).

More formally, the theory is formulated in terms of possible worlds. A possible world in which the antecedent of a counterfactual is true is called an "antecedent-world." One can state the theory (in a somewhat simplified form) by saying that a counterfactual is true just in case its consequent is true in those antecedent-worlds that are most similar to the actual world.

Stalnaker's and Lewis's accounts differ in a number of ways. First, Stalnaker intends his theory to cover both indicative conditionals and counterfactuals, whereas the scope of Lewis's account is restricted to counterfactuals. Second, according to Stalnaker's truth-conditions, but not according to Lewis's, there is always one most similar possible antecedent-world. In consequence, Stalnaker's theory validates the principle of conditional excluded middle, (*A* □ → *C* ) or (*A* □ → *∼C* ), whereas Lewis's account does not.

It is an advantage of the possible-world account that it can explain some noteworthy logical features of counterfactuals, namely the failure of a number of inference

schemata that are valid for the material and strict conditionals, such as the following:

A □ → B ∴ ∼B □ → ∼A |
(Contraposition) |

C □ → B, B □ → A ∴ C □ → A |
(Hypothetical syllogism) |

B □ → A ∴ (B & C ) □ → A |
(Strengthening the antecedent) |

To see that these modes of inference are invalid, consider the following counterexamples:

- (Even) if Mary had qualified for the tournament, she (still) would not have won it. Therefore, if she had won the tournament, she (still) would not have qualified for it.
- If Hoover had been born a Russian, he would have been a communist. If Hoover had been a communist, he would have been a traitor. Therefore, if Hoover had been born a Russian, he would have been a traitor (Stalnaker 1991b).
- If we had told them about our plan, they would have been delighted. Therefore, if we had told them about our plan and its likely result, they would have been delighted.

The possible-world account explains the failure of these inference rules, as Figure 2 will render clear. Let the dot labeled "@" stand for the actual world, let the other points in the rectangle represent the other possible worlds, and let the smaller and greater spatial distances between the points represent smaller and greater degrees of similarity between the corresponding worlds. As before, propositions can be represented by regions within the rectangle. In the situation depicted, *B* is true in the possible *A* -worlds most similar to the actual world; but *∼A* is not true in the most similar possible *∼B* -worlds. Hence, while "*A* □ → *B* " is true, "*∼B* □ → *∼A* " is false. This shows that contraposition is invalid. Moreover, "*C* □ → *B* " and "*B* □ → *A* " are true while "*C* □ → *A* " is false, and "*B* □ → *A* " is true while "(*B* & *C* ) □ → *A* " is false, so that the diagram also represents counterexamples to hypothetical syllogism and strengthening the antecedent.

If the antecedent of a counterfactual is impossible, then there are no possible antecedent-worlds, so that it is vacuously true that the consequent is true in all the most similar possible antecedent-worlds. The possible-world account therefore entails that all counterfactuals with impossible antecedents are true. But that is implausible: Most philosophers would agree that Willard Van Orman Quine could not have been a hippopotamus, but it does not seem right to say that, if Quine had been a hippopotamus, he would have been a reptile. According to Daniel Nolan (1997) and others, this problem can be remedied if impossible worlds are allowed to figure in the account alongside possible worlds. On this view, impossible worlds are ordered by their comparative similarity to the actual world, just as possible worlds are. A counterfactual "*A* □ → *C* " with impossible antecedent is true just in case *C* is true in the most similar impossible *A* -worlds. Such an account, however, requires an ontology of impossible worlds, which not all philosophers are happy to accept.

### similarity between worlds

The notion of similarity between worlds that is used in the analysis of counterfactuals cannot be the one that governs ordinary offhand judgments about overall similarity. This was shown by Kit Fine (1975) among others. Fine used (9) as his example:

(9) If Nixon had pressed the button, there would have been a nuclear catastrophe.

(9) sounds correct. But offhand it may seem that an antecedent-world devastated by a nuclear explosion is much less similar to the actual world than an antecedent-world in which the signal disappears in the wire after the button-pressing, so that no harm is done. If the notion of offhand similarity were used in analyzing counterfactuals, the account would yield the incorrect verdict that (9) is false and that everything would have been fine if Nixon had pressed the button.

What are the standards of similarity that govern counterfactuals? Many philosophers who address this question assume that different standards are relevant in different contexts of utterance. This assumption is motivated by examples like the following (which is taken from Jackson 1977, p. 9): Frank is in a room on the tenth floor of a building. There is nothing that could break the fall of someone jumping out of the window. It seems safe to say that Frank would get badly hurt if he were to jump. But suppose that Frank says: "I would never jump from a tenth-floor window, unless I had made sure that there was a safety net. So, if I were to jump, a net would be in place, and I would be fine." Frank's reasoning might convince his audience that his counterfactual is true. And yet his conditional seems to be incompatible with the one stated before. The most obvious diagnosis is that the truth-conditions of counterfactuals are context-dependent. In some contexts worlds in which Frank jumps despite the absence of a net count as more similar than those in which he places a net below the window before jumping. In other contexts it is the other way around.

Some of those who take the truth-conditions of counterfactuals to be context-dependent (notably Lewis 1979), believe that there is such a thing as a default assignment of truth-conditions to them, an assignment that hearers choose when interpreting the utterance of a counterfactual unless their presumption in favor of it is removed by distinctive features of the context. That seems plausible enough in the example of the last paragraph: If presented with the case out of the blue and asked for a judgment, one would say that Frank would get badly hurt if he were to jump. It requires some stage-setting (like that provided by Frank's utterance) to create a context in which it seems right to say that he would be fine.

Attempts to describe the default truth-conditions of counterfactuals often start from a special case: counterfactuals whose antecedents are false and describe nomically possible matters of particular local fact. (9) can serve as an example. Pre-theoretical intuitions about this conditional seem to furnish two data points:

(1)

Counterfactual dependence is temporally asymmetrical. If Nixon had pressed the button, then later on things would have been different from what they were actually like; but matters until shortly before the button-pressing would have been just as they actually were. The most similar antecedent-worlds must therefore be just like the actual world until a short time before the button-pressing, but might be different afterward.(2)

Laws support counterfactuals. If Nixon had pressed the button, then events would still have conformed to the actual laws of nature. The most similar antecedent-worlds must therefore be ones that evolve in accordance with the laws of the actual world. In particular, if the missile system is set up in such a way that the actual laws guarantee that button-pressing leads to a nuclear explosion, then there is a nuclear catastrophe in the most similar antecedent-worlds.

Suppose that determinism is true. In that case at least one of the principles (1) and (2) stands in need of some qualification. For determinism entails that every initial segment of the history of the actual world, together with the laws, determines the entire rest of history, and thus determines that Nixon does not press the button. This implies that no antecedent-world can both perfectly conform to the actual laws and be like the actual world throughout some initial segment of its history. Some philosophers (e.g., Lewis 1979) choose to solve this problem by allowing that the most similar antecedent-worlds contain violations of the actual laws, while others allow for backward counterfactual dependence over arbitrarily long periods of time (e.g. Bennett 1984; but see Bennett 2003, §80).

Note that Goodman's problem of specifying which truths are suitable supplementary premises resurfaces on the possible-world theory, in the shape of the question: Which of the actual matters of particular fact must obtain in an antecedent-world for it to count among the most similar? An account of the similarity relation will address this question and, if successful, will at last provide a noncircular solution to Goodman's problem.

** See also ** Entailment, Presupposition, Implicature; Modal Logic; Paraconsistent Logics; Relevance (Relevant) Logics.

## Bibliography

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Bennett, Jonathan. *A Philosophical Guide to Conditionals*. New York: Oxford University Press, 2003.

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Stalnaker, Robert C. "Letter to van Fraassen." In *Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science*, Vol. 1, *Foundations and Philosophy of Epistemic Applications of Probability Theory*, edited by William L. Harper and Clifford A. Hooker. Dordrecht, Netherlands: D. Reidel, 1976.

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*Boris C. A. Kment (2005)*

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