Semantics and Pragmatics of Vague Probability Expressions

Two experiments assessed the membership functions that German speakers assign to 12 adverb phrases and 17 modal verb forms that express probability assessments. These expressions fall largely into three rather homogeneous classes. The membership functions are used as part of the semantic knowledge base of the natural language dialog system PRACMA, one of whose purposes is to model pragmatic and contextual influences on the use of vague expressions. The system’s normative model accounts for the role, in the selection and interpretation of vague probability expressions, of the listener’s prior expectations, the speaker’s dialog motivation, and the expressions that the speaker could have used but did not. In an old episode of the television series “L. A. Law”, adoption agent Ross tells Ann and Stuart that they may be forced to give up their adopted daughter, because her natural mother is suing to reclaim her. Ann: You never told us she might do that! Ross: I told you it can happen. I said it hardly ever happens. Stuart: No, no! What you said was that the chances were 1 in 100. Now this is the second damned time: Something that never happens is happening to us! This exchange illustrates several results that have emerged from extensive experimental studies of the use of verbal and numerical expressions of probability judgments. People tend to prefer verbal to numeric formulations when they express probability judgments, although this preference is markedly less strong when it comes to receiving the judgments (Wallsten, Budescu, Zwick, & Kemp, 1993). And the same probability (here, 1%) can be expressed verbally in quite different ways by different speakers, which makes the interpretation of such expressions a challenging and error-prone task.1 The example also illustrates two points that have received littleattention in research to date. First, in everyday discourse, probability assessments are often expressed using modal verbs (as in “it can happen”), as well as the adverbs whose meaning has usually been studied. Second, the motivation of the speaker influences his or her choice of a vague probability expression in important and predictable ways; for example, the different paraphrases by Ross and Stuart do not represent random interindividual variation. In experiments, to the extent to which speaker motivation has been identifiable at all, it has generally consisted in the desire to convey as accurate Zwick and Wallsten (1989, p. 72) discuss evidence that communication failures of this sort may have contributed to the Bay of Pigs fiasco and the Challenger space shuttle disaster. as possible an impression of the probability the speaker has in mind. The present paper aims to shed new light on both of these last two issues, as well as on some previously documented empirical phenomena. In the next section, we report on a pair of experiments in which the probability implications of adverb phrases and modal verb forms, respectively, are investigated with German-speaking subjects. These experiments were conducted in a neutral context designed to yield data that could be used to explain differing uses and interpretations of the expressions in different contexts. In the subsequent section, we present an idealized computational model of the selection and interpretation of vague probability expressions that takes speaker motivation (and its perception by the listener) into account. In the final discussion, we consider some empirical results reported by other researchers in the light of the model. Empirical Derivation of Membership Functions Although the most novel aspect of these new experiments is their inclusion of modal verbs, to ensure comparability the first experiment looks at German-language adverb phrases such as those used in previous (mainly English-language) research. These phrases satisfy the condition that they can express probability estimates when they are used alone as well as when they are used in combination with modal verbs.2 The Adverb Experiment Method. Subjects were 24 graduate and undergraduate students, all native speakers of German. The subjects were asked to imagine that they were spectators at a game of chance. In this game, one of eleven wheels of fortune is spinned (cf. Wallsten, Budescu, Rapoport, Zwick, & Forsyth, 1986). The wheels differ widely in the sizes of their black and white portions. A player wins if the arrow to the right of the wheel points into the black sector when the wheel stops. The subjects were asked to imagine that a player was asked by the emcee about the result of the game before the player could see where the wheel had stopped. Given a particular wheel and a particular adverb phrase, the subjects were to indicate how “realistic” they judged this phrase to be as an answer of the player in the given situation; subjects did this by placing a mark on a graphical rating scale (cf. Rapoport, Wallsten, & At the time of this writing, we are analysing the results of a further experiment which looked at combinations of adverbs and modal verbs, to see how the interpretation of such an expression can be predicted on the basis of the meanings of its parts. Cox, 1987). Twelve hypothetical answers of the player were presented to the subjects. Each such statement used a different adverb phrase and had the same form as “Ich habe vermutlich gewonnen” (“I presumably won”). Along with each such statement, 11 wheels were presented, representing the probabilities 5%, 15%, ..., 45%, 50%, 55%, ..., and 95%. The order of the wheels was varied, and a cardboard mask was used to prevent subjects from seeing their ratings for more than one wheel at a given time. Results. For each subject, a curve was obtained for each adverb phrase showing for each probability (corresponding to a wheel) the value assigned by the subject. These curves will be called membership functions (as in fuzzy set theory); they are comparable to the curves resulting from the experiments of Wallsten, Budescu, Rapoport, Zwick, and Forsyth (1986) and Zwick, Budescu, and Wallsten (1988) on English adverbs. The upper part of Figure 1 summarizes the data by giving, for each adverb phrase, a curve showing the mean membership value assigned by the subjects to each probability.3 The curves show that about half of the 12 adverb phrases investigated have monotonically increasing mean membership functions, and that the differences between these functions are rather small. The generally lower membership values for the first three phrases in the figure suggest that they are seen as being truly appropriate only for very high probabilities (above 95%). Three of the phrases in a second group—möglicherweise (possibly), vielleicht (maybe), and eventuell (perhaps)—are mainly judged realistic when associated with medium probabilities. The membership values given for these adverbs are less often close to the extreme values of 0 and 1; and since intermediate membership values are less well-defined, these adverbs show a greater amount of disagreement among subjects (as reflected, e.g., in the average standard deviation of the membership values for a given probability). The membership function for the one low-probabilityexpression included—auf keinen Fall (no way)—is approximately the mirror image of that for the opposite expression, auf jeden Fall (in any case). The Modal Verb Experiment Method. The same experimental arrangement as in the Adverb Experiment was used, but modal verb forms were used instead of adverb phrases. The statements presented had the same form as “Es dürfte der Fall sein, daß ich gewonnen habe” (“It should be the case that I won”). Of the 17 modal verb forms presented, 9 were negated. As before, 24 graduate and undergraduate students, native speakers of German, served as subjects. Results. Here again, for each modal verb a mean membership function was derived (cf. the lower part of Figure 1). For the 8 nonnegated modal verb forms, the results are very similar to those of the Adverb Experiment: Of these forms, 5 show monotonically increasing membership functions, which differ mainly in that two forms—muß (must) and wird (will)—are judged to be generally less realistic than the others for probA mean membership function could conceivably have a shape that was atypical of the shapes of the membership functions for individual subjects;but inspection of the individual functions showed that this was not the case here, in spite of considerable differences among the individual functions. Adverb Phrases auf jeden Fall (in any case) 5 152535455565758595 sicher (surely) 5 152535455565758595 gewiß (doubtless) 5 152535455565758595 bestimmt (certainly) 5 152535455565758595 höchstwahrscheinlich (very probably) 5 152535455565758595 wahrscheinlich (probably) 5 152535455565758595