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Knowing what follows from a sentence is an indispensable ingredient of understanding it. Correspondingly, finding out what can be inferred from the formula constructed for a sentence is a very important task in computational semantics. Here are some of the reasons why this is so:
Often, we can only fully understand a sentence by inferring from it (together with our background knowledge). For example if we ask someone whether he has already listened to the latest record of Carla Bley, he may answer ``Oh, I hate Jazz!''. To understand this as an answer to our question, we have to infer that he in fact has not listened to the record (maybe due to his musical half-heartedness).
Inference from a sentence may be necessary to react properly to it, e.g. to answer a question.
Already in the process of meaning construction itself, inference may help us reduce the number of readings that can be constructed. This may greatly reduce the load for subsequent processing stages.
A limited amount of inference may even be essential to any language understanding: Given a complex sentence (resp. a complex formula for it), we have to infer which basic atomic facts have to hold for the sentence to be true. This sort of inference is known as model generation and we will come back to it at some length in Chapter 9.
Up to now, we've only seen definitions that capture the notion of logical consequence (see Section 1.2.6) semantically. In this section, we will develop a method to get a grip on this notion operationally: We will see how we can use syntactic calculi to actually compute what follows from a formula. But let us proceed step by step.
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