Many linguists and philosophers of language may have heard of informational independence, but most, not to say virtually all, of them consider it as a marginal feature of the semantics of natural languages. Yet in reality it is a widespread phenomenon in languages like English. In this paper, we shall develop an explicit unified formal treatment of all the different varieties of informational independence in linguistic semantics. This treatment amounts to a new type of logic, which is thereby opened for investigation. We shall also call attention to several actual linguistic phenomena which instantiate informational independence and provide evidence of its ubiquity. Last but not least, we shall show that the phenomenon of informational independence prompts several highly interesting methodological problems and suggestions.