1. Introduction

Consider the following experimental scenario, which was first used in an empirical study conducted by the psychologists Amos ^{Tversky and Daniel Kahneman (1981)}:

Imagine the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume the exact scientific estimates of the consequences are as follows:

In one condition, the options are as follows:

If Program A is adopted, 200 people will be saved.

If Program B is adopted, there is a one-third probability that 600 people will be saved and a two-thirds probability that no people will be saved.

In another condition, the options are:

If Program C is adopted, 400 people will die.

If Program D is adopted, there is a one-third probability that nobody will die and a two-thirds probability that 600 people will die.

In both conditions, participants are then asked:

Which of the two programs would you favor?

Tversky and Kahneman found that most participants in the ‘be saved’ condition favoured Program A -the ‘sure option’ (72% chose this option, compared with just 28% who favoured the ‘risky option’, Program B). In contrast, in the ‘die’ condition, most participants favoured the risky option, Program D (78%, compared with 22% who favoured the sure option, Program C). This is puzzling on the standard interpretation of the two conditions, according to which they describe exactly the same pair of prospects. The mere use of positive or negative language (‘be saved’ vs. ‘die’) seems to produce a reversal in people’s preferences -and in their attitude to risk-taking.

Tversky and Kahneman’s ‘Asian Disease Problem’ (ADP) has inspired a programme of psychological research into the effects of framing information in different words -typically, by using language that is either positively or negatively valenced. In this paper, I will focus solely on ADP-style ‘risky-choice’ framing paradigms, which elicit

choices between outcomes occurring with varying degrees of probability.¹ In particular, I will address the question of why seemingly equivalent options end up being evaluated differently, and whether this can be a rational response on the part of a decision-maker. The effects of framing on choice have standardly been interpreted as evidence of counter-normative (irrational or sub-rational) cognitive processing. As Bermúdez has recently put it: “The consensus view is that susceptibility to framing effects is a paradigm of irrationality” (^{2018, p. 180}). The idea that people are subject to such cognitive biases has proved extremely influential, both within and beyond academia, as part of a wider focus on insights from the behavioural sciences.²Over the last couple of decades, however, various challenges to the prevailing view have begun to gain momentum. Here I will focus on the idea, tested empirically by ^{Mandel (2014)}, that the number expressions in framing problems may be interpreted lower-boundedly, as denoting minimum rather than exact quantities. I will argue that this hypothesis may support a rationalising explanation of framing effects, but only on the basis of a fully worked-out theory of the meanings of number expressions. Crucially, providing that foundation will, in turn, require engagement with a conceptual distinction heavily discussed in the philosophy of language, concerning the difference between semantics and pragmatics. Thus, I aim to show how empirical and philosophical research is importantly intertwined, at least in attempts to explain framing effects.

The structure of the paper is as follows: in section 2, I describe the standard gloss of risky-choice framing effects, which are thought to reveal counter-normative reasoning. In section 3, I introduce the hypothesis that recipients of the ADP interpret number expressions as denoting lower-bounded quantities. I assess the empirical support for this hypothesis and argue that, although it may provide a partial explanation of risky-choice framing effects, it does not yet support a rationalising explanation; that would require showing that the number expressions have lower-bounded meanings in the relevant context. In section 4, I survey various accounts of the semantics and pragmatics of number expressions and describe how these could potentially license lower-bounded interpretations of the number expressions in the ADP. In section 5, I argue that endorsing any one of these accounts will ultimately depend on philosophical arguments about how to draw the boundary between semantics and pragmatics. I conclude that empirical and philosophical research remain inextricably linked in debates about framing, rationality, representation, and meaning.

2. Framing Effects and (Ir)rationality

Framing effects are standardly thought to be counter-normative because they violate the principle of ‘description invariance’, which is taken to be an essential condition for rational decision-making. Roughly speaking, the principle of description invariance requires that, when two expressions obviously describe the same possible outcome, that outcome should be represented and evaluated in the same way under each description.³ It is thought to be obvious that, in each framing condition of the ‘Asian Disease Problem’ (ADP), the same pair of possible outcomes is being described and, therefore, that these should be represented and evaluated by recipients in exactly the same way. As we have seen, though, the empirical data show that people actually evaluate them differently. In Tversky and Kahneman’s experiment, a majority preferred the sure option (Program A) in the ‘be saved’ condition, whereas, in the ‘die’ condition, a majority preferred the risky option (Program D).⁴

Tversky and Kahneman explain the observed framing effect via their theory of decision making under risk, known as ‘Prospect Theory’ (1979). Prospect Theory is put forward as a descriptively accurate account of human decision-making, which diverges in important ways from the normative ideal embodied in classical ‘Expected Utility Theory’. I will focus solely on the following point of divergence here: whereas Expected Utility Theory requires rational agents to have consistent attitudes to risk, Prospect Theory predicts that people are relatively risk-averse when faced with potential gains, and risk-seeking when faced with potential losses. To see the distinction, consider a choice between the following options:

Compare this with the choice between the following options:

According to Expected Utility Theory, a rational agent who is riskaverse will select (1) and (3), while a rational agent who is risk-seeking will select (2) and (4). However, that theory rules out the alternative combinations -(1) and (4), or (2) and (3)- since these would imply a change in risk attitude, based purely on whether one stands to gain or lose the $200.

Prospect Theory predicts, in contrast, the selection of (1) and (4). This is because people are assumed to be risk-averse when facing gains (as in the choice between (1) and (2)) but risk-seeking when facing losses (as in the choice between (3) and (4)). The shift in risk attitude is dubbed a ‘reflection effect’ (since it seems to be caused by ‘reflecting’ the prospects around the zero-utility point).

The core idea is further extended to tasks in which the prospects merely seem like gains or losses. As Kahneman and Tversky put it:

The reference point usually corresponds to the current asset position, in which case gains and losses coincide with the actual amounts that are received or paid. However, the location of the reference point, and the consequent coding of outcomes as gains or losses, can be affected by the formulation of the offered prospects, and by the expectations of the decision maker. (p. 273)

A prominent way of making prospects seem like gains or losses is by describing them with positive or negative wording, as in the two conditions of the ADP. In the first condition, the options are said to sound like gains because of the focus on the people who will ‘be saved’. Therefore, participants tend to be risk-averse in this condition, preferring the sure option (Program A). Conversely, it is held that the options in the second condition sound like losses because of the focus on the people who will ‘die’. And this is argued to explain why participants tend to be risk-seeking in the ‘die’ condition, preferring the risky option (Program D).

Offering an explanation of the underlying psychological mechanism at work, ^{Kahneman and Tversky (1979, pp. 273-274)} posit an initial ‘editing’ phase in decision-making. This involves decisionmakers mentally representing the choice options in particular ways. Of most relevance for current purposes, it is hypothesised that when the options are worded positively they are represented (or ‘coded’) as gains. Conversely, when the options are worded negatively they are coded as losses. The implications of this editing process become evident in the subsequent ‘evaluation’ phase, in which decision-makers assess the choice options before them. Where the alternatives have been coded as gains, decision-makers are predicted to make riskaverse choices. Conversely, where they have been coded as losses, decision-makers are predicted to make risk-seeking choices. As a result, the choices are held to be doubly counter-normative: reasoning errors are committed at both the editing phase (where outcomes are irrationally represented in different ways) and the evaluation phase (where this difference in representation irrationally affects choice). Ultimately, what is distinctive about framing effects (as opposed to mere ‘reflection effects’) is that the positive or negative wording in the choice options cues the audience to represent the outcomes as gains or losses during the editing phase.

It is worth noting that there is limited discussion in the literature of exactly what counts as positive or negative wording.⁵ For my purposes here, though, it will suffice to think of the valence of a linguistic expression as depending straightforwardly on the goodness or badness of what is denoted. Thus, for example, the expression ‘be saved’ is positive because being saved is typically a good thing, whereas ‘die’ is negative because dying is typically bad. Of course, there will not always be consensus about what counts as a good or bad thing. However, I take it that the ADP scenario, which will be my primary focus here, is sufficiently clear-cut.

As noted previously, it is standardly assumed that, in each condition of the ADP, exactly the same two prospects are described. Specifically, both of the sure options are thought to describe a certain prospect of 200 people being saved and 400 dying. Meanwhile, both of the risky options are thought to describe a one-third probability that 600 people will be saved (with none dying), together with a two-thirds probability that 600 people will die (with none being saved). This is taken to be obvious, at least once the conditions are considered side by side.⁶ The only difference, then, is that, in the first condition, the alternatives are framed positively (in terms of the numbers of people who will ‘be saved’), whereas in the second condition they are framed negatively (in terms of the numbers of people who will ‘die’). Since this is seen as a superficial linguistic difference, distinct representation of the prospects (as gains or losses) is taken to violate the normative principle of description invariance. In sum, it is standardly assumed that the pairs of alternative ‘frames’ -(5) and (7) below; and (6) and (8)- have the same meaning, even though they are typically represented differently in each framing condition.

If this is the right gloss of what is happening, framing effects clearly evidence counter-normative behaviour. However, there are various ways of challenging the claim that alternative frames have the same meaning.⁷ Here I will focus on the ‘lower-bounding hypothesis’.

3. Lower-Bounded Representations

For the alternative frames in the ADP to have the same meanings, the number expressions would have to denote complementary quantities. Consider again, for example, frames (5) and (7) reproduced below:

These frames can only describe the same prospect if the quantities of people being saved and dying sum to 600 (the total number of people the disease is expected to kill).⁸ The complementarity of the number expressions in turn requires that they have ‘bilateral’ meanings, whereby ‘200’ means no more and no less than 200, and ‘400’ means no more and no less than 400. In other words, the number expressions must denote exact quantities.⁹

It has been suggested, however, that the number expressions in the ADP are often interpreted unilaterally -specifically, lowerboundedly, as denoting minimum quantities. For example, ‘200’ in (5) may be represented as denoting at least 200, and ‘400’ in (7) as denoting at least 400. Crucially, this ‘lower-bounding hypothesis’ stands to explain the classic risky choice framing effect. To see why, consider what happens if (5) and (7) are interpreted as in (5a) and (7a):

It is consistent with (5a) that more than 200 people will be saved under Program A. In contrast, (7a) ensures that no more than 200 people will be saved under Program C, since at least 400 of the 600 will die. Therefore, Program A has the potential to save more lives than Program C -it is a better prospect. In principle, then, this could explain the observed shift in responses, towards the sure option in the ‘be saved’ framing condition, and away from the sure option in the ‘die’ condition. Importantly, that would be a different explanation than the Prospect Theoretic one which, as we saw, depended on the positive wording (‘be saved’) in the first condition giving the impression of gaining, while the negative wording (‘die’) in the second condition gives the impression of losing.

The lower-bounding hypothesis has been mooted at various times in the framing literature (going back, at least, to ^{Macdonald 1986}). However, it is only recently that it has been the subject of direct empirical investigation.¹⁰

4. The Meanings of Number Expressions

If the lower-bounding hypothesis is to provide a genuinely rationalising explanation of risky-choice framing effects, it must be shown that experimental participants are rational to represent the number expressions as denoting lower-bounded quantities. Specifically, it must be shown that such interpretations are licensed, on the basis of the expressions having genuinely lower-bounded meanings in the relevant contexts. In this section, I briefly survey various possible arguments to that effect, drawing on an ongoing debate about the semantics and pragmatics of number expressions.²⁰ Although I will not seek to provide a resolution, I will indicate how each position could potentially support a rationalising account of framing effects. Note that none of them actually does so as yet. Jumping ahead for a moment, in section 5 I will argue that choosing between the positions, in pursuit of the requisite justification, will ultimately require deeper philosophical engagement.

While it is widely recognised that number expressions obtain lower-bounded readings in certain contexts, some theorists have sought to trace this back to the expressions’ conventional semantics, while others have attributed it to their pragmatic enrichment in particular contexts of use. I begin by discussing the traditional neo-Gricean view, which has been put forward, for example, by ^{Horn (1989)} and ^{Levinson (2000)}. They take number expressions to be lower-bounded as a matter of conventional meaning. On the neoGricean view, then, ‘200’ conventionally denotes at least 200, while ‘400’ conventionally denotes at least 400. If correct, this would render sentences like (5) and (7) inequivalent to one another at a basic semantic level. While the theorists in this camp readily acknowledge that number expressions are sometimes interpreted as denoting bilateral (or upper-bounded) quantities, this is thought to be the result of pragmatic modulation of the conventional meaning, in light of wider contextual considerations.

If number expressions have conventionally lower-bounded meanings, that might already be thought sufficient to license their being represented as lower-bounded. For example, according to some semantic theories, conventional meanings are always psychologically realised during successful linguistic communication.²¹ Of course, showing that the conventional meanings should also be deployed in making evaluative judgements would be a further step.²² Indeed, participants’ choice behaviour might still be thought to violate rational choice principles. This is because it is unclear whether they ought to evaluate the options on the basis of (conventionally-derived) lowerbounded representations, or (pragmatically-derived) bilateral ones. Consider, for example, how pragmatic enrichment of the number expression might be expected to proceed on a standard Gricean analysis:²³ given a statement of (5), if more than 200 people would be saved under Program A, then presumably that would have been stated explicitly -at least if we assume that the number of people who will be saved is known, and that the information being provided is accurate, relevant, and so on. Since (5) does not state that more than 200 people will be saved, it may be legitimate to infer that no more than 200 will be saved -in other words, that exactly 200 people will be saved.²⁴

Given this Gricean rationale, is it still possible to justify a lowerbounded interpretation of the number expression (and an evaluation based on that interpretation), rather than a bilateral one? Perhaps. After all, it is not obvious that the number of people who will be saved is really known; the vignette explicitly states that the figures are estimates. Therefore, the reason for failing to state that more than 200 people will be saved could be a purely epistemic one -it is simply unknown how many people beyond the first 200 will eventually be saved. On that basis, participants might be rational to stick with the conventional lower-bounded meaning of the number expression, leaving open the possibility that more than 200 people will be saved, rather than adopting a pragmatically-enriched, bilateral interpretation.²⁵

On the other hand, participants are explicitly instructed to assume that the options provide exact scientific estimates of the consequences. That might be considered to give them a good reason to adopt bilateral interpretations of the number expressions. Ultimately, then, whether or not the deployment of lower-bounded interpretations can be justified -and the framing effects that result- will depend on precisely how the discourse context functions to constrain meaning. As Mandel indicates in the earlier quote, this is an issue for wider research in linguistic pragmatics to address. For now, then, a rationalising explanation remains possible but not secure.

A distinct approach to the conventional semantics of number expressions takes these expressions to be ambiguous, or polysemous, having multiple lexical entries, or senses. ^{Geurts (2006)} develops an analysis along these lines. He argues that number expressions have both bilateral and lower-bounded senses (although he takes the bilateral senses to be more basic and generally dominant).

Adopting an ambiguity or polysemy view of number expressions, one would then need to show that it is reasonable to adopt lowerbounded interpretations of the number expressions in the ADP. As before, the task of showing that is made somewhat harder by the fact that participants are told that the options provide exact scientific estimates, which might be expected to prime a bilateral interpretation (and, if that interpretation is also the dominant one, as Geurts suggests, then selecting the lower-bounded sense would seem especially odd). Only if a satisfactory explanation can be given here would risky-choice framing effects be consistent with the principle of description invariance. In the end, this will depend on our theory of how alternative senses should be selected in context.

Another approach claims that number expressions have underspecified conventional semantics. In other words, the expressions themselves have neither bilateral nor lower-bounded denotations. This view has been advocated by ^{Kempson and Cormack (1981)} and ^{Carston (1988}, ¹⁹⁹⁸⁾. They argue that, on each occasion of use, the underspecified conventional meaning of a number expression must be pragmatically enriched in one way or other, so that the audience arrives at a lower-bounded or bilateral (or, potentially, upper-bounded) interpretation.²⁶

Given an underspecification view, the lower-bounding hypothesis would provide a rationalising explanation of risky-choice framing effects just if participants could be justified in enriching number expressions lower-boundedly in those contexts. Whether that is so will depend on precisely how such enrichment is supposed to proceed, i.e. which contextual factors may be taken into account, and how they are to be integrated within the interpretation process. That, in turn, will require a rich, psychological theory of interpretation.²⁷

The final account I canvas here has been advocated by ^{Breheny (2008)}, who analyses number expressions as having straightforwardly bilateral conventional meanings. Breheny recognises that such expressions may obtain lower-bounded interpretations in certain contexts, but this is taken to be the result of pragmatic enrichment (specifically, he appeals to the role of background knowledge in generating unilateral meanings). In this sense, Breheny’s position is diametrically opposed to the neo-Gricean approach discussed above.

On a Breheny-style account, the question of whether lower-bounded interpretations are rational would depend, again, on our final pragmatic theory (including how individuals’ background knowledge should be brought to bear during interpretation). So long as participants can be justified in deriving enriched, lower-bounded interpretations of the number expressions in the ADP, they would not be violating the principle of description invariance; there would be an important sense in which the number expressions actually have lower-bounded meanings in that context.

It is clear from the preceding discussion that the ability of the approaches described here to support a fully rationalising explanation of framing effects will depend on our final theory of one or more aspects of linguistic communication. In the meantime, the jury is out on whether participants ultimately behave rationally in deriving and deploying lower-bounded representations of number expressions during risky-choice framing tasks. At most, each approach explains how participants arrive at those interpretations (semantically or pragmatically), but not whether they are the best interpretations to adopt, all things considered. That said, it is equally clear that it would be premature to conclude that framing effects are counter-normative. Thus, the lower-bounding hypothesis at least leaves in play the possibility that framing effects are rational. The importance of this result should not be downplayed, given how widespread and influential the opposite conclusion has become.

The contribution of the discussion in this section has been to outline various ways in which Mandel’s empirical analysis could potentially be deepened, by appeal to a foundational account of the meanings of number expressions. One such account will ultimately be required if the lower-bounding hypothesis is to support a fully rationalising explanation of framing effects. Having established this, the point I want to emphasise in the next section of the paper is that deciding between these foundational positions will require empirically-engaged philosophy.

5. The Semantics-Pragmatics Divide

The question of what number expressions mean, at semantic and pragmatic levels, depends in large part on how we define the concepts of semantics and pragmatics, and where we draw the boundary between the two. This is a project which has exercised philosophers of language for many years. Although we are, of course, free to stipulate how technical terminology is used, presumably an important objective is for the concepts of semantics and pragmatics to have genuine explanatory value, and this should guide their definition and distinction. Even granting this point, though, the semantics-pragmatics boundary is unlikely to fall out automatically from pure empirical inquiry, since we must first agree which real-world phenomena we are trying to explain. For illustration, below I briefly consider three metaphysical distinctions we may want the semantics-pragmatics divide to track (and there are almost certainly others). Depending on how this conceptual issue is resolved, it may prove possible to endorse one of the approaches described in section 4, thus providing the basis for a genuinely rationalising explanation of framing effects.

First, the purpose of semantics could be thought of as tracking certain conventional effects on interpretation, which depend on arbitrary features of language, such as the fact that the word ‘fire’ in English means fire. Meanwhile, pragmatics could track meanings that emerge from non-conventional contextual effects: for example, the sentence ‘There’s a fire’ is used to vastly different effect when it is uttered to other passengers in a crowded underground station, as opposed to one’s friends as they walk into a warm pub. ^{Lepore and Stone (2015)} have recently elaborated a view which treats conventionality as the hallmark of semantics (but, in the process, absorbs all kinds of interpretive effects into the category of semantics, which have not traditionally been considered part of conventional meaning).

Others have proposed a psychological grounding for semantics and pragmatics, seeking to map them to distinct kinds of cognitive process. According to Relevance Theory, for example, semantic processing involves the decoding of linguistic stimuli, whereas pragmatic processing is governed by an entirely different cognitive efficiency principle (^{Carston 2002}; ^{Sperber and Wilson 1995)}. Borg also maps semantics and pragmatics to different (modular vs. non-modular) psychological systems (^{Borg 2004}, ²⁰¹²) -although it should be noted that her view of the contextual effects on meaning diverges importantly from that of Relevance Theorists.

Another way to draw the boundary would be by appeal to normative criteria derived from our everyday communicative practices. The idea here would be that semantics could track those meanings that are hardest for speakers to retract, and to which they remain strictly committed. In contrast, pragmatic meanings may be relatively easy to cancel or deny. ^{Saul (2013)} adopts this kind of approach in mapping the difference between lying and misleading to the difference between what is ‘said’ and what is conveyed less directly. See also ^{Borg 2017} for discussion of how speakers’ liabilities track semantic and pragmatic differences.

If we are lucky, each of these ways of understanding semantic and pragmatic meaning will be perfectly aligned, sorting communicated contents in exactly the same way into one basket or the other. Perhaps more likely, though, we will need to make explicit decisions about which distinction to track. Such decisions are ones to which philosophers of language are particularly well-equipped to contribute.

The question of how to draw the semantics-pragmatics boundary will feed directly into the analysis of the meaning of number expressions. For example, if semantics and pragmatics are held to track distinct cognitive processes, we would expect the semantic and pragmatic meanings of number expressions to have different psychological signatures (perhaps being derived at different speeds, involving different regions of the brain, or being susceptible to different kinds of interference). Alternatively, if we adopt a normative criterion, the semantics and pragmatics of number expressions ought to reflect our ordinary communicative practices. Therefore, in order to evaluate the views surveyed in the previous section, we need to establish which real-world phenomena the posited meanings should be tracking.

The upshot of this (necessarily cursory) discussion, then, is that distinguishing and deciding between the defining criteria of semantic and pragmatic meanings is essential for choosing between analyses of the meanings of number expressions. As we saw, settling on one of these analyses will, in turn, be a necessary step towards rendering the lower-bounding hypothesis a rationalising explanation of riskychoice framing effects. In this way, then, further exploration of an empirical debate in psychological framing research has led us inexorably to a foundational philosophical question concerning the nature of semantic and pragmatic meaning. Looking back in the other direction, it is clear that empirically informed discussions in the philosophy of language have demonstrable application in advancing framing research and its associated real-world impacts.

6. Conclusion

In the case of risky-choice framing effects -and, in particular, the hypothesis that lower-bounded interpretations of number expressions can explain these effects- empirical and philosophical inquiries are importantly intertwined. I have argued that competing explanations of the representation and evaluation of alternative frames depend, ultimately, on foundational issues in the philosophy of language, concerning semantics, pragmatics, and the interface between the two. In doing so, I hope to have indicated how, by reaching across disciplinary boundaries, philosophers can shed light on the conceptual questions pervading empirical work, and can contribute positively to the development of scientific research. Of course, it remains to be seen how representative this case study is of wider epistemic inquiry. However, it would perhaps not be too surprising to discover that science and philosophy are profoundly and pervasively entangled across the piece.²⁸