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Up to now we have seen various ways to construct logical formulae as meaning representations for sentences. But we don't yet know what to do further with such formulae. We will now learn how to do useful work with such meaning representations.
If we utter a sentence, we transport information. One way of exploiting this information is to find out what follows from the sentence. The parallel task on the level of meaning representations is that of inference from the formula for that sentence. 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.
In this chapter, we will develop a method to get a grip on the notion of logical consequence operationally: We will see how we can use syntactic calculi to compute what follows from a formula. But in order to understand this, we first have to repeat some of the basic semantic concepts of first-order logic.
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