Interpretation (logic) - First-order Logic

First-order Logic

Unlike propositional logic, where every language is the same apart from a choice of a different set of propositional variables, there are many different first-order languages. Each first-order language is defined by a signature. The signature consists of a set of non-logical symbols and an identification of each of these symbols as a constant symbol, a function symbol, or a predicate symbol. In the case of function and predicate symbols, a natural number arity is also assigned. The alphabet for the formal language consists of logical constants, the equality relation symbol =, all the symbols from the signature, and an additional infinite set of symbols known as variables.

For example, in the language of rings, there are constant symbols 0 and 1, two binary function symbols + and ·, and no binary relation symbols. (Here the equality relation is taken as a logical constant.)

Again, we might define a first-order language L, as consisting of individual symbols a, b, and c; predicate symbols F,G, H, I and J; variables x,y,z; no function letters; no sentential symbols.