This paper proposes a model-theoretic approach to incremental interpretation where all sentence prefixes have semantic values. The proposed semantics is based on Discourse Representation Theory (DRT), where semantic representations (called DRSs) are interpreted as assignment updates. In our semantics, a partial DRS of a sentence prefix is interpreted as two sets which stipulate the assignment updates. One denotes possible updates and the other denotes necessary updates. With the proposed semantics, we can assign truth values to sentence prefixes.
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