Formal languages and systems

Formal languages and systems are concerned with symbolic structures considered under the aspect of generation by formal (syntactic) rules, that is, irrespective of their or their components’ meaning(s). In the most general sense, a formal language is a set of expressions. The most important way of describing this set is by means of grammars. Formal systems are formal languages equipped with a consequence operation yielding a deductive system. If one further specifies the means by which expressions are built up (connectives, quantifiers) and the rules from which inferences are generated, one obtains logical calculi of various sorts, especially Frege–Hilbert-style and Gentzen-style systems.