There is a growing body of evidence showing that conversational implicatures are rapidly accessed in incremental utterance interpretation. To date, studies showing incremental access have focussed on implicatures related to linguistic triggers, such as "some" and "or". We discuss three kinds of on-line model that can account for this data. A model built around the notion of linguistic alternatives stored in the lexicon would only account for linguistically triggered implicatures of the kind already studied and not so-called "particularised" implicatures that are not associated with specific linguistic items. A second model built around the idea of focus alternatives could handle both linguistically triggered implicatures and so-called particularised implicatures but would be insensitive to the role that information about the speaker's mental state plays in deriving implicatures. A third more fully "Gricean" model takes account of the speaker's mental state in accessing these implications. In this paper we present a visual world study using a new interactive paradigm where two communicators (one confederate) describe visually-presented events to each other as their eye movements are monitored. In this way, we directly compare the suitability of these three kinds of model. We show hearers can access contextually specific particularised implicatures in on-line comprehension. Moreover, we show that in doing so, hearers are sensitive to the relevant mental states of the speaker. We conclude with a discussion of how a more "Gricean" model may be developed and of how our findings inform a long-standing debate on the immediacy of on-line perspective taking in language comprehension. (Contains 7 tables and 3 figures.)