What is the function of vagueness in language? This paper argues that in contexts in which a cooperative speaker is not perfectly informed about the world, the use of vague expressions can offer an optimal tradeoff between truthfulness (Gricean Quality) and informativeness (Gricean Quantity). Focusing on expressions of approximation such as ``around'', which are semantically vague, it is shown that they allow the speaker to convey indirect probabilistic information, in a way that can give the listener a more accurate representation of the information available to the speaker than any more precise expression would (intervals of the form ``between''). The paper presents a probabilistic treatment of the interpretation of ``around'' which derives various contrasts with non-vague lexical alternatives. It then incorporates it into the Rational Speech Act framework used to model communicative exchanges (Goodman and Stuhlmüller 2013, Lassiter and Goodman 2017). The model is not limited to expressions of approximation, but can be used to rationalize the use of vague expressions more generally.
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