# What is the formal definition of interpretation of a formal language?

Is there a formal version of "An interpretation is an assignment of meaning to the symbols of a formal language?"

I'm looking for something akin to the formal definition of "is an element of" relation, which would be "subset of U x P(U)," i.e. a subset of the cartesian product of the domain set and its power set.

I'm aware an interpretation is an activity outside the logical system, but was wondering if there is a formal definition in metalogic or something.

• Yes these are called structure/model in most logics such as in the classic FOL: The most common way of specifying an interpretation (especially in mathematics) is to specify a structure (also called a model; see below). The structure consists of a domain of discourse D and an interpretation function I mapping non-logical symbols to predicates, functions, and constants... Aug 21, 2022 at 21:22
• Yes, it is described in every math logic textbook. Aug 22, 2022 at 6:08
• See Interpretation (logic) as well as Classical Logic: Semantics. Aug 22, 2022 at 8:37
• And see e.g. this post Aug 22, 2022 at 11:17
• Just read any proper logic text (i.e. written by a logician) such as mentioned here. Neither wikipedia nor SEP are suitable for learning the basics. Aug 22, 2022 at 11:22