Given some alphabet $\Sigma$ we can define some language $L$ as a subset of the set of all possible words $\Sigma^*$ that use the symbols found in the alphabet. It seems that there are two ways of doing this.
- Define a grammar that describes just those words that should be included in the language (well-formed-formulas).
- Exhaustively list the words that should be included in the language.
Am I correct that a formal language, by definition, does NOT require grammar even if it is usually more practical to define a language this way?
By exhaustively listing the words that should be included in the language you are giving a grammar. A grammar consists of two parts: a) The alphabet (i.e., the symbols that make up grammatical expressions of your language), b) A set of rules (usually given in the form of a recursive definition) that elucidates how you may combine symbols from the alphabet to produce "grammatical" sentences. In your second example, the alphabet is your list of desired "words." The set of rules is this: 1. Every symbol from the alphabet is a grammatical expression, 2. No other expression is a grammatical expression. And that's it. Given that your language has zero "combinatorial potential", it will probably be a very boring language.