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.

  • 1
    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.
    – user21820
    Aug 22, 2022 at 11:22

1 Answer 1


Yes, the typical model theoretic approach is as follows. Fix a first order language L and signature S. Fix also some set A. A S-structure is a tuple (A, I), where I is the interpretation function such that, given any n-ary relation symbol R in S, assigns some subset of A^n, denoted as I(R). Since functions are (in set theory) just specialized relations, this generalizes to functions as well.

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