7.1.1 What we already know about Logics

A recapitulation of formal logics.

We have already learned a lot about the syntax and semantics of first-order logic. But logical systems usually have a third component - a calculus. Before we look in detail at this third component, we briefly recap the key logical concepts that we have seen so far.

Why is logic useful for us?

Logic studies formal languages and their relation to the world. This task is closely related to our task in computational semantics, i.e. computing the meaning of natural language utterances. Two of the advantages of using logic for this task are that logics are are mathematically precise and relatively simple.

Consider the following points where we have already gained a lot from this precision and simplification:

Formulae

of formal languages (such as the language defined by the syntax of first-order logic) simplify sentences of natural languages: Problems of grammaticality no longer arise. Furthermore, well-formedness can in general be decided by a simple recursive procedure.

Models

are what we use as a simplification of real-world situations. Models simplify the real world by concentrating on mathematically well-understood structures, namely sets and relations. They allow us to make predictions about truth conditions of natural language sentences. Moreover they make it possible to precisely define semantic notions, such as logical consequence.