It has been often argued that parentheticals, discourse adverbials and certain parts of speech like interjections do not contribute to the truth conditional content of the assertions of which they are part. In this paper I argue that many of these constructions do contribute a truth conditional content, and I propose a semantics for parentheticals and discourse adverbials that treats these constructions similarly to SDRT's treatment of presuppositions. I also point out differences between standard presupposition triggers on the one hand and parentheticals or discourse adverbials on the other.
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