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Indicatives, concessives, and evidential support a
Igor Douven & Sara Verbrugge
b
a
Faculty of Philosophy , University of Groningen , The Netherlands b
Laboratory of Experimental Psychology, University of Leuven , Belgium Published online: 24 Sep 2012.
To cite this article: Igor Douven & Sara Verbrugge (2012) Indicatives, concessives, and evidential support, Thinking & Reasoning, 18:4, 480-499, DOI: 10.1080/13546783.2012.716009 To link to this article: http://dx.doi.org/10.1080/13546783.2012.716009
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THINKING & REASONING, 2012, 18 (4), 480–499
Indicatives, concessives, and evidential support Igor Douven1 and Sara Verbrugge2 1
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2
Faculty of Philosophy, University of Groningen, The Netherlands Laboratory of Experimental Psychology, University of Leuven, Belgium
This paper discusses the issue of categorical acceptability of indicative and concessive conditionals. It presents experimental results in favour of two claims concerning the role of the evidential support relation for acceptability (or otherwise) of conditionals of both types. In particular, the results show that, contrary to fairly standard philosophical theorising, high probability of a conditional’s consequent given its antecedent is necessary but not sufficient for the acceptability of that conditional, and that the antecedent being evidence for the consequent is a further acceptability condition. The results further show that the evidential support relation is crucial in differentiating between the acceptability of an indicative conditional and the acceptability of the corresponding concessive conditional: typically, the use of a concessive conditional signals that the corresponding conditional probability is high in spite of the fact that the antecedent is evidence against the consequent, or in any case is not evidence for the consequent. Keywords: Indicative conditionals; Concessive conditionals; Evidential support; Acceptability; Probability.
Conditionals are sentences of the form ‘‘If A, [then] B’’ or ‘‘B if A’’, with A being called ‘‘the antecedent’’ and B ‘‘the consequent’’. Indicative conditionals (or indicatives, for short) are conditionals whose antecedent is considered to be an open possibility; this distinguishes them from so-called Correspondence should be addressed to Igor Douven, Faculty of Philosophy, University of Groningen, Oude Boteringestraat 52, 9712 GL Groningen, The Netherlands. E-mail:
[email protected] We are greatly indebted to Aidan Feeney, David Over, and an anonymous referee for very detailed and valuable comments on a previous version of this paper. We also thank Walter Schaeken for helpful discussions on the topic of this paper. A version of this paper was presented at the International Conference on Thinking, London 2012. We thank the members of the audience for their stimulating questions and comments. Ó 2012 Psychology Press, an imprint of the Taylor & Francis Group, an Informa business http://www.psypress.com/tar
http://dx.doi.org/10.1080/13546783.2012.716009
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counterfactual conditionals. Much recent research on indicatives, both in psychology and in philosophy, has centred around their truth conditions. Philosophers have long advocated the so-called material conditional account, according to which an indicative has the truth conditions of the corresponding material conditional. In psychology, this is still the dominant semantics for indicatives among mental model theorists. Over the past ten years, however, evidence has piled up for the so-called Equation, which states that the probability of an indicative equals its conditional probability.1 This has put considerable pressure on the material conditional account, for it is easy to show that only in trivial cases does the probability of a material conditional equal the probability of its consequent conditional on its antecedent. In the literature, there has been less discussion of the use conditions of indicatives, that is, the question of under which conditions we are warranted in asserting or accepting an indicative. The question is not only of interest in itself but also in connection with experimental work on the aforementioned Equation: as will be seen below, Over et al. (2007) report empirical findings concerning probabilities of conditionals suggesting that people’s assignments of probabilities to indicatives may be influenced by the use conditions of those conditionals. That, nonetheless, the question of the use conditions of indicatives has been given little attention is simply because this question is widely (even if not universally) taken to have been settled almost half a century ago by Adams (1965), who proposed that the extent to which an indicative ‘‘If A, B’’ is assertable for/acceptable to a given person is equal to PrðB j AÞ where this designates the person’s subjective conditional probability of B given A.2 This thesis, now generally known under the name ‘‘Adams’ Thesis’’ (AT), was thought to be so obviously correct that for a long time no one saw the need to subject it to empirical testing. The present authors were the first to 1
See Hadjichristidis et al. (2001), Evans, Handley, and Over (2003), Oaksford and Chater (2003), Oberauer and Wilhelm (2003), Over and Evans (2003), Evans and Over (2004), Weidenfeld, Oberauer, and Ho¨rnig (2005), Evans, Handley, Neilens, and Over (2007), Evans, Handley, Neilens, Bacon, and Over (2010), Oberauer, Geiger, Fischer, and Weidenfeld (2007), Oberauer, Weidenfeld, and Fischer (2007), Over, Hadjichristidis, Evans, Handley, and Sloman (2007), Douven and Verbrugge (2010, 2012), Pfeifer and Kleiter (2010), and Politzer, Over, and Baratgin (2010). 2 As a caveat, we note that terminological usage in Adams’ writings on conditionals may easily lead to confusion. In the 1965 paper, Adams is explicitly concerned with the assertability of conditionals, though in the formulas occurring in that paper he uses the probability operator to indicate the degree of assertability of a conditional. In later work, such as Adams (1998), he simply speaks of the probability of conditionals without, however, explicitly distancing himself from his earlier view that conditionals possess degrees of assertability rather than probabilities. Also, through Adams’ personal communications with various theorists it has become known that he thought that whatever he had said about the assertability of conditionals also, and even foremost, applied to their acceptability; see Douven & Verbrugge (2010, Sect. 3).
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conduct experiments on AT, finding clear negative results (Douven & Verbrugge, 2010). In the same study, we investigated various theses weaker than AT and found support only for the rather weak thesis that a correlation exists between the degrees to which indicatives are deemed assertable/acceptable and people’s corresponding conditional probability judgements. The previous study was concerned exclusively with quantitative theses about the relation between assertability/acceptability and probability, that is, theses attributing degrees of assertability/acceptability. However, while we often do use these notions in a graded way, we also often use them categorically, as in when we say right out that something is assertable or acceptable. Indeed, assertability and acceptability share this property with a host of other broadly epistemic notions (cf. believing something versus believing something to a certain degree, or deeming something probable versus deeming it probable to a certain degree). A number of authors, including Lewis (1976) and Jackson (1987, p. 31), have proposed a qualitative correlate of AT according to which high probability is both necessary and sufficient for the categorical assertability/ acceptability of an indicative conditional. In Douven (2008), this qualitative thesis has been criticised on largely theoretical grounds. In the same paper, an alternative thesis about the assertability/acceptability of indicative conditionals is proposed, one that assigns a crucial role to the relation of evidential support. In the present paper, we report experimental results that favour the latter thesis at the expense of the Lewis–Jackson proposal. Our experimental work had a second aim, concerning so-called concessive (‘‘even if’’) indicative conditionals. For we conjectured that the evidential support relation is also key to distinguishing between the acceptability of an indicative conditional and the acceptability of the corresponding concessive conditional. Below, we state this conjecture in more detail and present empirical evidence for it. As a preliminary disclaimer, we note that we will be concerned mostly with what one might call ‘‘normal indicatives’’, thus excluding so-called speech act conditionals—‘‘If you’re hungry, there are cookies in the kitchen’’; Dutchman conditionals—‘‘If Sheila marries John, I’m a Dutchman’’; and possibly also conditionals occurring in logical or mathematical contexts. Whether indicatives belonging to any of these special classes have acceptability conditions at all—as opposed to (merely) conditions under which it is appropriate to assert them—and, if so, what these conditions are, are questions we will not seek to answer in this paper. However, we make an exception for so-called non-interference conditionals—such as ‘‘If it snows in July, the government will fall’’—which we will briefly discuss. Moreover, we will limit our attention to so-called simple conditionals— both simple indicatives and simple concessives—that is, conditionals whose
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antecedent and consequent do not embed any conditionals. According to some theorists (e.g., Jackson [1987, pp. 127–137]; Edgington [1995, pp. 382ff]), this is not much of a restriction at all, given that all or almost all compounded conditionals that we can make sense of are reducible to simple conditionals.3
FROM ADAMS’ THESIS TO THE EVIDENTIAL SUPPORT THEORY
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Prima facie, the following thesis, which is Lewis’ (1976) and Jackson’s (1987) qualitative version of AT, strikes one as a natural and plausible answer to the question of when an indicative is categorically assertable/acceptable: QUALITATIVE ADAMS’ THESIS (QAT) An indicative conditional ‘‘If A, B’’ is assertable for/acceptable to a person if and only if the person’s degree of belief in B given A, PrðB j AÞ; is high.4 Arguably, the most fundamental intuition about the assertability/ acceptability of indicatives is that for an indicative to be assertable/ acceptable, there must be some sort of connection between its antecedent and consequent.5 And the requirement of high conditional probability seems to secure precisely such a connection. However, appearances are misleading here. For note that any proposition that is highly probable unconditionally is also highly probable conditional on any other proposition that is probabilistically independent of it, and often the absence of probabilistic relevance signals the absence of relevance in a broader sense. In fact, QAT is completely insensitive to differences in probabilistic relevance: it lumps together all cases in which the probability of the consequent given the antecedent is high, regardless of whether that probability is higher than, lower than, or equal to the unconditional probability of the consequent. To see why this should be cause for concern, consider the following sentences, each of which may be supposed to satisfy the assertability/acceptability condition of QAT:
3 This assessment of the situation regarding compounded conditionals may be overly optimistic, however; see Douven (2011a). 4 In its original version, AT is restricted to simple conditionals. McGee (1989) is an attempt to generalise AT, but see Lance (1991) and Dietz and Douven (2010). As stated in the introduction, we will only be concerned with simple conditionals. So, for present concerns, we can assume QAT to be restricted to simple conditionals as well. 5 At least this is so for normal indicatives. The most characteristic feature of a noninterference conditional seems to be the patent absence of a connection between antecedent and consequent; see below.
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(1) a. If Arsenal ends first in this year’s Premier League, the sun will rise tomorrow. b. If Obama is re-elected, there will be a heads in the first 10 million tosses with this fair coin. c. If sea levels keep rising, the Euro crisis will not be solved overnight. Many will find these sentences intuitively unassertable/unacceptable. Probably, this is because the truth of the antecedents of these sentences is entirely irrelevant to the truth of their consequents: were one to learn that Arsenal ends first in this year’s League, that would have no effect whatsoever on one’s confidence that the sun will rise tomorrow; similarly for (1b) and (1c). Motivated by examples like these, Douven (2008) suggests that an indicative is assertable/acceptable only if the antecedent is evidence (in the Bayesian sense of the word) for the consequent, that is, if it makes the consequent more probable. Specifically, the proposal is this:6 EVIDENTIAL SUPPORT THESIS (EST) An indicative conditional ‘‘If A, B’’ is assertable/acceptable if and only if PrðB j AÞ is not only high but also higher than PrðBÞ. This was prompted by one philosopher’s intuitive responses to a number of sentences such as those given in (1). A methodology that gives pride of place to intuition is widely accepted in analytic philosophy, and we do not find this methodology intrinsically objectionable. Still, the welter of relatively recent empirical results militating against erstwhile popular philosophical theses concerning conditionals—like the material conditional account or AT—should caution against taking intuition as the final arbiter of the correctness of general theses such as QAT and EST. Therefore, we thought it necessary to subject QAT and EST to empirical testing. As stated in the introduction, the aim of our experimental work was somewhat broader still, pertaining also to the role of evidential support in the assertability/acceptability of concessive indicative conditionals. The assertability/acceptability conditions of concessives have received much less attention in the philosophical literature than have the assertability/acceptability conditions of indicatives. Feeney and Handley 6
Actually, the proposal was a bit more complicated in that a condition was added to prevent indicatives like ‘‘If the lottery has only one winner, then your ticket is a loser’’ from qualifying as assertable/acceptable in a situation in which it is known that either the relevant fair and large lottery has a unique winner or half of the tickets will end up being winners, and that the chances of either of these possibilities obtaining are equal. For present purposes, this complication can be sidestepped.
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(2011) report important empirical results on the probabilities of concessives, but in their experiments they did not ask participants to rate the assertability or acceptability of such conditionals; as we know from our work on AT, probabilities do not necessarily provide much information about degrees of assertability/acceptability. Indeed, to the best of our knowledge, so far psychologists have not worked on the assertability/acceptability conditions of concessives at all.7 Yet it is clear that these conditions merit special consideration: the conditions under which a given indicative is assertable/ acceptable are typically different from the conditions under which the corresponding concessive is assertable/acceptable. For example, while (2a) is perfectly acceptable, the corresponding concessive—(2b)—strikes us as positively odd: (2) a. If Obama is re-elected, his supporters will be elated. b. Even if Obama is re-elected, his supporters will be elated. Conversely, of these sentences, only the concessive seems acceptable: (3) a. If Arsenal loses the next match, it is among the best British teams. b. Even if Arsenal loses the next match, it is among the best British teams. While we were considering various pairs like (2) and (3), it occurred to us that the concept of evidential support, as it figures in EST, may be the key to differentiating between the assertability/acceptability conditions of indicatives and the assertability/acceptability conditions of concessives. In particular, we found that, for conditionals whose consequent was highly probable given their antecedents, whenever we judged that the antecedent was evidence for the consequent—as in (2)—we preferred the ‘‘if’’ form over the ‘‘even if’’ form, and whenever the antecedent failed to be evidence for the consequent, or even undermined the consequent—as in (3)—we preferred the ‘‘even if’’ form over the ‘‘if’’ form. This led us to conjecture that a preference for the use of the concessive over the indicative form signals either probabilistic independence of antecedent and consequent, or negative probabilistic dependence between them. In other words, the conjecture is this:
7
More generally still, concessive indicative conditionals seem to have been largely ignored in the psychology literature. (There is some work on concessive counterfactual conditionals by mental models theorists; e.g., McCloy and Byrne [2002] and Santamarı´ a, Espino, and Byrne [2005].)
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CONCESSIVE ABSENCE OF SUPPORT THESIS (CAST) A concessive indicative conditional ‘‘Even if A, B’’ (or ‘‘If A, then still B’’) is assertable/ acceptable if and only if PrðB j AÞ is less than or equal to PrðBÞ but PrðB j AÞ remains high. Conjecturing further still, one could split up the cases considered by CAST into those in which the conditional probability is strictly lower than the unconditional probability, and those in which the two are equal. It is not clear to us whether anything very general can be said about these further classes of conditionals. Some of the cases in which the conditional probability of the consequent given the antecedent is high and equal or roughly equal to the unconditional probability of the consequent may be thought of as belonging to the class of non-interference conditionals mentioned in the introduction. These conditionals are meant to emphasise the inevitability or obviousness of the consequent, regardless of whether the antecedent obtains, expressing, so to speak, that nothing one might learn would change one’s confidence in the consequent. Douven (2008, p. 32) suggests that these conditionals have the characteristic that their assertability/acceptability is either not at all or positively affected if we substitute ‘‘even if’’ for ‘‘if’’ in them or if we insert the word ‘‘still’’ in their consequent clauses. However, in other cases in which the conditional probability of the consequent given the antecedent is high and equal to the unconditional probability of the consequent, the indicative form may simply be unacceptable, for the reason mentioned above—namely, that the lack of probabilistic relevance is due to the absence of a more general kind of relevant connection between antecedent and consequent. In our experiment, we focus on QAT, EST, and CAST, and we leave for future work an investigation of possible differences in the assertability/acceptability conditions of the distinct kinds of cases grouped together by CAST. Before turning to the experimental results, we have a number of comments on the foregoing. First, Douven is noncommittal on whether EST is a brute fact about indicatives, or whether it follows from their truth conditions,8 or from pragmatic principles like the Gricean maxims of good conversation,9 or from something else altogether. We will not try to settle this question here, either. We remain similarly neutral on the question of whether the assertability/acceptability conditions of concessives are to be 8
As Strawson (1986) might have argued; he suggests that it is part of the conventional meaning of ‘‘if’’ that the antecedent and consequent of a conditional are suitably connected, though he does not characterise the putative connection in probabilistic terms. 9 Douven (2008, p. 22ff) does argue that the Gricean maxims are unable to explain the connection that EST postulates. Of course, other pragmatic principles might do a better job on this count.
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regarded as a primitive fact about such conditionals or whether these conditions should rather roll out of the combination of whatever the assertability/acceptability conditions for indicatives are and an analysis of the word ‘‘even’’ (as for instance Bennett [2003, Ch. 17] holds). Second, results reported in Over et al. (2007) provide some indirect evidence for the relevance of the evidential support relation to at least the use conditions of indicatives. These authors investigated the probabilities of so-called real world conditionals10—of both indicatives and counterfactuals, though not of concessives—and found strong support for the Equation (which, recall, is the claim that the probability of a conditional equals the corresponding conditional probability). However, their experiments also showed a modest effect of evidential connection between antecedent and consequent. (Over et al. do not speak in terms of evidence but in terms of the so-called Dp-rule, which measures the difference between the conditional probability of the consequent given the antecedent and the conditional probability of the consequent given the negation of the antecedent. But this has been defended as a measure of evidential support in Christensen [1999].) A plausible thought is that, at least insofar as their results concern indicatives, this effect is due to the fact that the evidential connection between antecedent and consequent is crucially involved in the use conditions of indicatives. For it is natural to suppose that there may be some intrusion of pragmatic factors in the assignment of probabilities.11 To use a well-worn example, according to standard semantic theorising, (4) They married and she got pregnant. and (5) She got pregnant and they married. have the same semantic content (they express the same proposition). That we sense a difference in meaning nonetheless is said to be due to pragmatic factors; according to standard pragmatic theorising, it is due to Grice’s maxim to be orderly, in particular to recount events in the order in which they occurred. Probabilistically, such pragmatic factors are supposed not to matter: probability theory respects logic and disallows the assignment of different probabilities to logically equivalent sentences. But surely that is an idealisation. To the best of our knowledge, no work has been done so far on the connection between people’s probability judgements and pragmatic 10 That is, conditionals pertaining to events that have occurred in reality, or that might still occur, and on which participants could reasonably be presumed to have opinions. 11 As Over et al. (2007, p. 92) also note.
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factors of the kind just alluded to. But it seems reasonable to expect that, placing (4) and (5) in a context in which ‘‘they’’ refers to children of conservative Catholic parents, say, many people will not assign these sentences the same probability. Third, it was just said that Over et al. (2007) checked for the presence of an evidential connection between antecedent A and consequent B by comparing PrðB j AÞ with PrðB j :AÞ rather than with PrðBÞ. This might suggest alternatives to EST and CAST which postulate as a necessary condition for the assertability/acceptability of an indicative ‘‘If A, B’’, respectively concessive ‘‘Even if A, B’’, that PrðB j AÞ > PrðB j :AÞ respectively PrðB j AÞ PrðB j :AÞ. However, a moment’s reflection shows that, supposing all conditional probabilities to be defined, these are not really alternatives at all. The reason is that PrðB j AÞ > PrðBÞ if and only if PrðB j AÞ > PrðB j :AÞ and also PrðB j AÞ PrðBÞ if and only if PrðB j AÞ PrðB j :AÞ. These are simple theorems of probability theory; they are direct consequences of the fact that, by the law of total probability, PrðBÞ is a convex combination of PrðB j AÞ and PrðB j :AÞ. It is still worth noting that while for qualitative theses like EST and CAST it makes no difference whether we compare the conditional probability of the consequent given the antecedent with the unconditional probability of the consequent or with the conditional probability of the consequent given the negation of the antecedent, for quantitative theses that may very well make a difference. For instance, Over et al. (2007) are concerned with determining to what extent the degree to which the antecedent confirms the consequent impacts the judged probability of the conditional. (Again, this is not the vocabulary used by Over et al., but that is what their procedure amounts to.) They use the Dp-rule for this purpose, but they might as well have tried any of the other confirmation measures that have been proposed in the literature, like the difference measure— according to which the degree to which A confirms B is measured by PrðB j AÞ PrðBÞ—or the ratio measure—according to which the degree of confirmation equals PrðB j AÞ= PrðBÞ (see, e.g., Douven [2011b]).12 And, typically, these measures will output different values. Naturally, just as Over et al. studied the effect of evidential connections on probabilities of conditionals, one may study the effect of those connections on the degrees to which indicatives and concessives are deemed acceptable. We leave this as a topic for future research, but note that in the context of that research one may also want to consider various measures of confirmation.
12 These and other measures of confirmation have also surfaced in the literature on causality as possible measures of causal strength (e.g., Fitelson and Hitchcock [2011]), which is what Over et al. are explicitly interested in.
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Measures of confirmation might be relevant to the acceptability of conditionals in a second way. Instead of relating the categorical acceptability of a conditional to whether or not the antecedent supports the consequent, we might relate the probability that a person judges a conditional to be categorically acceptable to the degree to which, according to that person, the antecedent supports the consequent (if the antecedent supports the consequent at all), supposing the conditional probability of the consequent given the antecedent is high enough for acceptance. And for this hypothesis, we might then again want to consider various measures of confirmation to determine degree of support. We briefly look at this hypothesis later on, but in doing so we confine our attention to the difference measure.
EXPERIMENT A general lesson to be learned from the empirical work on AT that we reported in Douven and Verbrugge (2010) is that philosophers’ intuitions concerning the acceptability or unacceptability of conditionals are far from failsafe. The way to gain more definite insight into the status of theses such as QAT, EST, and CAST is to subject them to empirical testing, which is what we have done in the following experiment.
Method Participants. Sixty-two persons participated in the experiment. In return for their cooperation, they were paid a small amount of money. Participants were recruited and paid via the CrowdFlower interface (http://www. crowdflower.com), which directed them to the Qualtrics platform (http://www.qualtrics.com) on which the experiment was run. The participants were from Canada, the United Kingdom, and the United States. Their native language was English. Design. Every participant had to evaluate 18 items, which were presented on-screen. Items were randomised. Materials and Procedure. All materials were in English, the participants’ native language. To each of the 18 items corresponded one of the 18 conditionals that are presented in the Appendix. These conditionals are all real world conditionals in the sense of note 10. The items consisted of three parts. The first part asked for the probability of the consequent of the conditional corresponding to the item. The second part asked for the probability of the consequent of that conditional given the antecedent of the same conditional. And the third part presented both the ‘‘if’’ and the ‘‘even if’’ form of the conditional and asked whether the
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participant accepted only the ‘‘if’’ form, only the ‘‘even if’’ form, both, or neither. The three parts of each item were presented separately on-screen. The following is the first part of one of the items that was used in the experiment:
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How likely is it that the next Bond movie will become a box office hit? ¤ ¤ ¤ ¤ ¤ ¤ ¤
Very unlikely Unlikely Somewhat unlikely As likely as not Somewhat likely Likely Very likely
The second part of the item to which the above example belonged is this: Suppose some second-rate actor plays the role of James Bond in the next Bond movie. Then how likely is it that the movie will become a box office hit? ¤ ¤ ¤ ¤ ¤ ¤ ¤
Very unlikely Unlikely Somewhat unlikely As likely as not Somewhat likely Likely Very likely
While it may still be more common in the psychological literature to determine a conditional probability PrðB j AÞ via the Kolmogorovian ratio definition—first determine PrðA ^ BÞ; then determine PrðAÞ; then divide the former by the latter—in eliciting conditional probabilities in the way we did, we have been explicitly relying on the Ramsey test. According to this test, one determines PrðB j AÞ by hypothetically adding A to one’s stock of beliefs and then determining how probable B is from that (hypothetical) perspective. Our approach should raise no eyebrows. To the contrary, various theorists hold that the Ramsey test actually gives us the best definition of conditional probability.13
13
See Edgington (1995, p. 266; 1997, p. 108), Bennett (2003, p. 53), Oaksford and Chater (2007, p. 109), and Zhao, Shah, and Osherson (2009).
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Note that the participant’s answers to the first and second part of each question allowed us to determine whether the participant regarded the antecedent as being evidence for the consequent, or rather as being evidence against the consequent, or as being evidentially neutral with regard to the consequent. And finally there is the part of the item presenting the conditional in both the ‘‘if’’ and ‘‘even if’’ form:
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Consider the following statements: 1. If some second-rate actor plays the role of James Bond in the next Bond movie, that movie will become a box office hit. 2. Even if some second-rate actor plays the role of James Bond in the next Bond movie, that movie will become a box office hit. Of these statements: ¤ ¤ ¤ ¤
I I I I
accept accept accept accept
only 1. only 2. both. neither.
At the beginning of the experiment, the participants were instructed about the notion of acceptability. It was stressed that acceptability was not meant in the sense of ‘‘grammatically correct’’ or ‘‘socially acceptable’’ (polite, politically correct, etc.), but rather was to be understood as indicating what is credible, given all one’s information. We told the participants that, as a heuristic for answering the questions about whether a given statement is acceptable, they might imagine that they were in a conversation with some friends, and then had to determine whether, given their current knowledge, they would find it reasonable if they or a friend asserted the statement which they were asked to judge. Given the way we were understanding the notion, the statement would qualify as acceptable if the participant thought the assertion was reasonable, and unacceptable otherwise. In this connection, it is also worth mentioning that, in our study of AT, participants were asked to rate the acceptability of conditionals on a 7-point Likert scale. As part of that study, we conducted a control experiment (Douven and Verbrugge [2010, Experiment 2]) to check whether the participants were interpreting the notion of acceptability as we intended it to be understood, namely, as an epistemic notion, and not, for instance, in the sense of being socially acceptable. It turned out that participants’ acceptability judgements closely matched their reasonable-to-believe
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judgements. The feedback they gave on the question of how they thought the notion of acceptability was meant in the items further confirmed that they had understood that notion as an epistemic one.
Results and discussion The 62 participants and 18 items yielded 62 18 ¼ 1; 116 participant–item pairs. These pairs were categorised according to:
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(i)
whether the participant deemed the consequent of the conditional associated with the item to be highly probable given the antecedent of the same conditional; if so, the pair was categorised as HIGH, otherwise as NOT HIGH; (ii) whether the option the participant had chosen in the second part of the item was higher on the scale, the same as, or lower than the option the participant had chosen in the first part of the item, and so whether the participant judged the antecedent of the conditional of the item to be positively relevant to (POS), independent of (IND), or negatively relevant to (NEG) the conditional’s consequent; (ii) whether the participant had chosen the IF, the EVEN IF, the BOTH, or the NEITHER option in the third part of the item. As for (i), we set the threshold for what is to count as highly probable at the midpoint of the scale (following Achinstein [2000, p. 156]). Accordingly, participant–item pairs were classified as HIGH if and only if the participant had chosen ‘‘Somewhat likely’’, ‘‘Likely’’, or ‘‘Very likely’’ in the second part of the item. Table 1 displays all logically possible combinations of falling into and not falling into the aforementioned categories, and for each subcategory gives the number of participant–item pairs that fall into it. Figure 1 represents the percentages of participant–item pairs in the POS, IND, and NEG categories
TABLE 1 Distribution of 1,116 participant–item pairs over the various subcategories
Not high
High
Neg Ind Pos Neg Ind Pos
If
Even if
Both
Neither
8 14 28 2 27 165
55 10 2 48 86 16
8 11 0 26 106 45
125 245 75 1 6 7
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Figure 1. Percentages of POS (black), IND (dark grey), and NEG (light grey); positive percentages are percentages of pairs categorised as HIGH, and negative percentages are percentages of pairs categorised as NOT HIGH.
that fall into the IF, EVEN IF, BOTH, and NEITHER categories; positive percentages indicate proportions of pairs that fall into the HIGH category and negative percentages indicate proportions of pairs that fall into the NOT HIGH category. We performed chi-square tests to see whether there were the associations between probabilistic relevance and choice of the IF, EVEN IF, BOTH, and NEITHER options that are predicted by the hypotheses QAT, EST, and CAST. We found a strong association between HIGH choices and choices of IF, EVEN IF, or BOTH, w2 ð1Þ ¼ 629:49, p < :001; F ¼ :751, p < :001. We found a strong association between IF choices and choices of both HIGH and POS, w2 ð1Þ ¼ 413:06, p < :001; F ¼ :608, p < :001. We found a moderate association between EVEN IF choices and choices of both HIGH and either IND or NEG, w2 ð1Þ ¼ 164:24, p < :001; F ¼ :384 p < :001. And we also found a moderate association between BOTH choices and choices of both HIGH and IND, w2 ð1Þ ¼ 169:95, p < :001; F ¼ :39 p < :001. To obtain further information relevant to the hypotheses, we took a closer look at cell frequencies. As for the question of whether high conditional probability of consequent given antecedent is necessary for the acceptability of the corresponding conditional (whether indicative or concessive), Figure 1 already supports a positive answer. The left graph in Figure 2 makes it easier to see the relevant percentages: 80 per cent of the IF choices, 69 per cent of the EVEN IF choices, and 90 per cent of the BOTH choices were associated with high conditional probability. The right graph
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Figure 2. Left: percentages of choices of HIGH among the IF, EVEN IF, BOTH, and NEITHER participant–item pairs. Right: percentages of choices of IF, EVEN IF, BOTH, and NEITHER among the HIGH participant–item pairs.
in the same figure makes the claim that high conditional probability is also sufficient for the acceptability of a conditional—as Jackson (1987) and Lewis (1976) hold at least for indicatives—appear more doubtful. Only 36 per cent of participant–item pairs that were categorised as HIGH fall into the IF category. Of the HIGH pairs, 28 per cent fall into the EVEN IF category, 33 per cent into the BOTH category, and 3 per cent into the NEITHER category. Of course, participant–item pairs that fall into the BOTH category consist of a participant and an item such that the participant deemed also acceptable the indicative form of the conditional corresponding to the item. So, one might argue that in total 69 per cent of the HIGH participant–item pairs were associated with a judgement to the effect that the conditional corresponding to the item was acceptable in indicative form. On the other hand, we saw that theorists have advanced reasons for believing that conditionals which are deemed acceptable both in indicative and in concessive form are best thought of as constituting a separate category (the category of non-interference conditionals). In any case, the picture looks very different when we distinguish among the POS, IND, and NEG participant–item pairs. As for the necessity direction of EST, we find that 68 per cent of the IF pairs also fall into both the HIGH and the POS category. As for the sufficiency direction, we see that of the pairs that are both HIGH and POS, 71 per cent fall into the IF category. Furthermore, we find that 62 per cent of EVEN IF pairs accord with the necessity direction of CAST by being also either HIGH and IND (40 per cent) or HIGH and NEG (22 per cent) pairs. As for the sufficiency direction of CAST, we find that 44 per cent of the pairs that are either HIGH and NEG or HIGH and IND are also EVEN IF responses. In view of these percentages, it is fair to say that EST is more strongly supported by our evidence than CAST. In Section 1, we mentioned the possibility of relating the probability that a participant judges a conditional to be acceptable to the degree to which, in
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that person’s opinion, the antecedent is evidence for the consequent, assuming that the relevant conditional probability meets the threshold for acceptance. We there also mentioned the difference measure of evidential support as a popular measure for determining degree of evidential support. Figure 3 shows a graph which represents the percentages of choices of IF, and so on, for varying degrees of support as determined via the difference measure (as applied to scores on the 7-point Likert scale that we used). Inspection of the graph suggests that the probability that someone opts for IF is greater the more the relevant conditional probability exceeds the relevant unconditional probability. In summary, while in Section 1 we already encountered some indicatives that appeared to refute QAT, the data from our experiment constitutes a more decisive refutation of QAT. According to QAT, a high value of PrðB j AÞ is sufficient for the acceptability of ‘‘If A, B’’. But we saw that it is not: while people do find acceptable some indicatives with a high corresponding conditional probability, they find a large percentage of indicatives with a high corresponding conditional probability not acceptable. By contrast, we found support for EST and, to a somewhat lesser extent, also for CAST. Together these theses predict that, given that people judge PrðB j AÞ to be high, they will prefer ‘‘If A, B’’ or ‘‘Even if A, B’’, depending on whether or not they judge A to be evidence for B. And what we found was indeed that if A is judged to be evidence for B, then there is a clear tendency to prefer the indicative form, and vice versa; if, on the other hand, A is judged not to be evidence for B, there is a tendency to prefer the concessive form, and vice versa. As for the BOTH responses, they may indicate that some participants interpreted some of the conditionals as noninterference conditionals. As said, however, we leave a deeper investigation
Figure 3. Percentages of choices of IF (black), EVEN IF (dark grey), BOTH (light grey), and NEITHER (white) options, grouped according to degree of confirmation, among the HIGH pairs.
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of the assertability/acceptability conditions of non-interference conditionals for another occasion.
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CONCLUSION We have seen that QAT only identifies one necessary condition for the assertability/acceptability of indicatives, namely, high conditional probability of consequent given antecedent. This is not enough to guarantee any kind of connection between antecedent and consequent, the existence of which seems required by what may well be the ur-intuition about the assertability/acceptability of conditionals. We have tested the hypothesis EST according to which the requisite connection is of an epistemic kind, more in particular, that the antecedent should be evidence for the consequent. Our evidence was seen to corroborate this hypothesis, and thereby to refute QAT. As for our second goal, we conjectured, on the basis of a handful of pairs of indicatives and corresponding concessives, that it is precisely the presence or absence of this evidential connection that determines whether we find an indicative or rather the corresponding concessive acceptable. Our experimental results were seen to corroborate this conjecture, laid down in CAST, as well. To end, we recall an interesting avenue for future research briefly touched upon in Section 1. We were led to our present investigations by our earlier work on AT, which is a quantitative thesis about the acceptability of conditionals. As mentioned at the outset, that earlier work led to the conclusion that degrees of acceptability do not match conditional probabilities, although a correlation between the two could be established. However, in our work on AT we did not consider the evidential support relations between the antecedents and consequents of the conditionals that we used in our materials. If, as was shown in this paper, such relations play a crucial role in determining the qualitative acceptability of indicatives, then it is a plausible conjecture that they are in at least some way involved in the determination of degrees of acceptability as well. Manuscript received 10 February 2012 Revised manuscript received 16 July 2012 First published online 24 September 2012
REFERENCES Achinstein, P. (2001). The book of evidence. Oxford, UK: Oxford University Press. Adams, E. W. (1965). The logic of conditionals. Inquiry, 8, 166–197. Adams, E. W. (1998). A primer of probability logic. Stanford: CSLI Publications.
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Bennett, J. (2003). A philosophical guide to conditionals. Oxford, UK: Clarendon Press. Christensen, D. (1999). Measuring confirmation. Journal of Philosophy, 96, 437–461. Dietz, R., & Douven, I. (2010). Ramsey’s test, Adams’ thesis, and left-nested conditionals. Review of Symbolic Logic, 3, 467–484. Douven, I. (2008). The evidential support theory of conditionals. Synthese, 164, 19–44. Douven, I. (2011a). Indicative conditionals. In L. Horsten & R. Pettigrew (Eds.), The Continuum companion to philosophical logic (pp. 383–405). London, UK: Continuum Press. Douven, I. (2011b). Relativism and confirmation theory. In S. Hales (Ed.), The Blackwell companion to relativism (pp. 242–265). Oxford, UK: Blackwell. Douven, I., & Verbrugge, S. (2010). The Adams family. Cognition, 117, 302–318. Douven, I., & Verbrugge, S. (2012). The probabilities of conditionals revisited. Cognitive Science. Forthcoming. Edgington, D. (1995). On conditionals. Mind, 104, 235–329. Edgington, D. (1997). Commentary. In D. Wiggins (Ed.), Conditionals: M. Woods (pp. 95–137). Oxford: Clarendon Press. Evans, J. St. B. T., Handley, S. J., Neilens, H., Bacon, A. M., & Over, D. E. (2010). The influence of cognitive ability and instructional set on causal conditional inference. Quarterly Journal of Experimental Psychology, 63, 892–909. Evans, J. St. B. T., Handley, S. J., Neilens, H., & Over, D. E. (2007). Thinking about conditionals: A study of individual differences. Memory & Cognition, 35, 1759–1771. Evans, J. St. B. T., Handley, S. J., & Over, D. E. (2003). Conditionals and conditional probability. Journal of Experimental Psychology: Learning, Memory and Cognition, 29, 321–355. Evans, J. St. B. T., & Over, D. E. (2004). If. Oxford, UK: Oxford University Press. Feeney, A., & Handley, S. J. (2011). Suppositions, conditionals, and causal chains. In C. Hoerl, T. McCormack, & S. R. Beck (Eds.), Understanding counterfactuals, understanding causation (pp. 242–262). Oxford, UK: Oxford University Press. Fitelson, B., & Hitchcock, C. (2011). Probabilistic measures of causal strength. In P. Illari, F. Russo, & J. Williamson (Eds.), Causality in the sciences (pp. 600–627). Oxford, UK: Oxford University Press. Hadjichristidis, C., Stevenson, R. J., Over, D. E., Sloman, S. A., Evans, J. St. B. T., & Feeney, A. (2001). On the evaluation of ‘if p then q’ conditionals. In J. D. Moore & K. Stenning (Eds.), Proceedings of the 23rd Annual Meeting of the Cognitive Science Society, Edinburgh, UK, 2001 (pp. 381–386). Mahwah, NJ: Lawrence Erlbaum. Jackson, F. (1979). On assertion and indicative conditionals. Philosophical Review, 88, 565–589. Jackson, F. (1987). Conditionals. Oxford, UK: Blackwell. Lance, M. (1991). Probabilistic dependence among conditionals. Philosophical Review, 100, 269–276. Lewis, D. K. (1976). Probabilities of conditionals and conditional probabilities. Philosophical Review, 85, 297–315. McCloy, R., & Byrne, R. M. J. (2002). Semifactual ‘even if’ thinking. Thinking & Reasoning, 8, 41–67. McGee, V. (1989). Conditional probabilities and compounds of conditionals. Philosophical Review, 98, 485–541 Oaksford, M., & Chater, N. (2003). Conditional probability and the cognitive science of conditional reasoning. Mind & Language, 18, 359–379. Oaksford, M., & Chater, N. (2007). Bayesian rationality. Oxford, UK: Oxford University Press. Oberauer, K., Geiger, S. M., Fischer, K., & Weidenfeld, A. (2007). Two meanings of ‘if’? Individual differences in the interpretation of conditionals. Quarterly Journal of Experimental Psychology, 60, 790–819. Oberauer, K., Weidenfeld, A., & Fischer, K. (2007). What makes us believe a conditional? The roles of covariation and causality. Thinking & Reasoning, 13, 340–369.
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APPENDIX This appendix contains the 18 items that appeared in the materials used in the experiment. We present only the indicatives that were used in the third parts of the items. It will be obvious from the example given in Section 2 what the rest of the items looked like. If some second-rate actor plays the role of James Bond in the next Bond movie, that movie will become a box office hit. If there are no more scandals in the British Royal family, the power of the monarch will gradually decrease in Great Britain. If the British government is unsuccessful in creating more jobs, there will be more riots like the ones we saw recently. If the United States Army prolongs its mission in Afghanistan, that country will be prosperous soon. If Harvard loses money in the current financial crisis, it will be among the world’s richest universities.
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If Germany and Great Britain fail to qualify for the European Championship football to be held next year, Luxembourg will win the Championship. If the energy consumption of the Danish population diminishes, the problem of climate change will remain. If Obama is able to solve the problems that the American economy is facing, the Republican candidate will win the presidential election by a wide margin.
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If Greece and Italy are willing to take very severe austerity measures now, the European debt crisis will be solved within just six months. If the tuition fees for British universities go down again, only children from very rich families will be able to go to college in the UK. If the economic crisis in Ireland deepens, there will be an unprecedented famine in the country. If he stops writing new music, Paul McCartney will earn a considerable amount of money in the coming years. If the Palestinians get their own state, all tensions in the Middle East region will be resolved. If oil prices go up, more countries will turn to alternative sources of energy. If the frequency of traffic jams increases in the coming years, more people will start to consider travelling by public transport. If there is another large-scale terrorist attack, security measures at airports will be upgraded. If Manchester United ends last in this year’s Premier League (the English football league), they will kill their coach. If an international peacekeeping force helps local Iraqi authorities, Iraq will remain a troubled society for the next decade.
