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Optimal assertions, and what they implicate. A uniform game theoretic approach

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Abstract

To determine what the speaker in a cooperative dialog meant with his assertion, on top of what he explicitly said, it is crucial that we assume that the assertion he gave was optimal. In determining optimal assertions we assume that dialogs are embedded in decision problems (van Rooij 2003) and use backwards induction for calculating them (Benz 2006). In this paper, we show that in terms of our framework we can account for several types of implicatures in a uniform way, suggesting that there is no need for an independent linguistic theory of generalized implicatures. In the final section, we show how we can embed our theory in the framework of signaling games, and how it relates with other game theoretic analyses of implicatures.

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Notes

  1. What we denote by Q 2-implicatures are called R-implicatures by Horn (1984), and I-implicatures by Levinson (2000).

  2. In earlier work, Horn seems to adopt this position as well, but he is less enthusiastic about the view that implicatures are such default inferences in later work (e.g. Horn 2004).

  3. Proponents of the Context-Driven view of Q 1-implicatures include Hirschberg (1985), Carston (1998), and van Rooij and Schulz (2004).

  4. See Chierchia et al. (2001) and Storto and Tannenhaus (2004) for (very) different opinions.

  5. See also Geurts (1998).

  6. Where P(v|C) is short for P({v}|C).

  7. We leave it underdetermined here, but the set F should most naturally be thought of as the set of alternative answers to the question ‘corresponding’ to I’s decision problem.

  8. We assume that there is an \({f\in F:[\![f]\!] = \{w\in\Omega|P_E(w)\neq 0\}}. \)

  9. This model may seem to be somewhat artificial. In a realistic model we have to assume that there are many different places where it might be possible that petrol is available. This means that I has to choose between a larger number of actions. In such a scenario it becomes very natural to assume that only learning G will induce her to do a. But in order to keep the model simple, we consider only a situation where I has to choose between doing nothing and going to that specific garage.

  10. This interpretation rule has, in fact, been proposed already by van Rooij (2004a) in the context of Bidirectional Optimality Theory. Jäger’s (2006) game theoretic analysis of Q 1-implicatures comes down to this rule as well.

  11. As noted in the introduction, the assumption that numerical expressions have an ‘at least’ interpretation is highly controversial, and probably even wrong (see also Clark and Grossman, 2007). For the argument it doesn’t matter much: one could easily think of other examples where the semantic meanings of the alternative expressions form a linear chain with respect to inference. The scale \({\langle\hbox{and, or}\rangle}\) would do as well, just as \({\langle \hbox{all, most, some}\rangle}\), if the quantifiers ‘all’ and ‘most’ give rise to an existential presupposition.

  12. Almost, because if a f contains more worlds, there might in principle be an alternative expression f′ with \({a_{f^{\prime}} \subset a_f}, \) and thus \({P_I(w|a_{f^{\prime}}) > P_I(w|a_f)}. \) We have to assume that there are no such alternatives f′ ∈ F.

  13. In this paper we don’t care what μ(t|f) is in case f is not uttered by s in any type.

  14. Or better, is represented only in case the expert is known to have complete information of the world he is in.

  15. In some of the implicatures treated by Parikh (2001)—e.g. the example where the receiver should conclude from ‘It is 4 p.m.’ that she should go to the talk—he assumes it is crucial to make use of what he calls ‘the value of information’. We would question this assumption, however, and argue that also Parikh himself treats these basically as Q 2-implicatures.

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Acknowledgements

We would like to thank Bart Geurts, Wilfrid Hodges, Kris de Jaegher, and especially Johan van Benthem, Michael Franke and the reviewers for discussion, comments, and suggestions. We also thank Samson de Jager for correcting our English.

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  1. University of Southern Denmark, Kolding, Engstien 1, Kolding, DK 6000, Denmark

    Anton Benz

  2. Institute for Logic, Language and Computation, Universiteit van Amsterdam, Nieuwe Doelenstraat 15, Amsterdam, 1012 CP, The Netherlands

    Robert van Rooij

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  1. Anton Benz
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Correspondence to Robert van Rooij.

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Benz, A., van Rooij, R. Optimal assertions, and what they implicate. A uniform game theoretic approach. Topoi 26, 63–78 (2007). https://doi.org/10.1007/s11245-006-9007-3

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