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.