Formal language Glossary
A formal language is a set of strings of symbols from an alphabet, satisfying some rules. The strings generated by the rules are the "words" or well-formed formulae of the language.
In the case of first-order logic, the alphabet consists of upper and lower case letters (of which we (take it to have an infinite supply, generating them by adding the "prime" symbol as necessary) and symbols for the connectives and quantifiers. If formulae are seen as strings, we need parentheses in the alphabet as well; if they are seen more abstractly as trees, we don't. The formation rules defining the grammar are given in the definition of a formula.
The formal language is a completely artificial construct, unlike the natural languages in which real reasoning is carried out. This enables us to give precise semantics and inference rules, making formal logic possible. It does mean, however, that the formal system is not an exact analysis of the natural one, so creativity and interpretation are needed at the boundary between the natural and formal languages.
The formal language of logic is extensible. We can add more connectives or other expressions to it, to give an account of more features of arguments. For example, free logic extends it with an existence predicate 'EST'. We could also extend it with connectives marking future and past tenses, so that reasoning could take account of time, or we could add another connective indexed with a name, to relativise assertions to the agents who believe them, or to computer programs whose execution makes the propositions true. These notes do not cover such extensions of the language and logic, but clearly they could if we wished.
- With an alphabet consisting of the usual 26 letters, one formal language might consist of the set of strings in which every letter occurs an even number of times. This would be fairly useless, but is certainly a formal language according to the definition.
- The set of formulae of first order logic is a formal language (indeed, it is the formal language which interests us in these notes), but for instance the set of logical formulae in which no existential quantifier occurs inside the scope of a universal one is also a formal language, in which we may or may not be interested.
- The set of theorems of first order logic is a formal language. We may regard the natural deduction calculus as a way of encoding a grammar for this "language".