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From the Syntax to the Semantics of Formal Languages

Up to now, any string of a language, even if it follows a strict syntax, for instance, a string representing a formula like $1 < 2$, is nothing else but a meaningless concatenation of letters. Semantics governs the way we can interpret a given formula of a formal language. Only with semantics, it is possible to distinguish between true and false formulae.

In this chapter, we will stepwise approach the concept of semantics of formal languages. In order to do so, we will first introduce the truth values and further concepts, including interpretation, axioms, rules of interference, derivability, and model. It will turn out that semantics is s model relation we can use to decide if a formula is true or false.