There are two kinds of languages, formal and informal. Informal languages rely on both semantics and syntax to construct intelligible thoughts. Formal languages have a special property which allows us to use either syntax or semantics to get a sentence. But that’s a long way from a thought expressed in a natural language. Let’s take a little trip through mathematical logic, the linguistics of mathematics.
A formal language is defined by three things:
- A lexicon which unambiguously defines the set of ‘words’ in the language. We call these words symbols. A formal lexicon completely defines the symbols allowed in the formal language.
- We also need a notion of sentences derived by syntax rules. A sentence of a formal language constructed properly according the the grammar rules of the syntax is called a formula.
- Finally, we require a semantics which correlates symbols and their meanings.
Mathematical logic considers two fundamental questions about such formal languages.