Given a simple formal system (we shall call this one ) whose alphabet α consists only of three symbols and whose formation rule for formulas is: 'Any string of symbols of which is at least 6 symbols long, and which is not infinitely long, is a formula of . Nothing else is a formula of .' The single axiom schema of is: " " (where " " is a metasyntactic variable standing for a finite string of " "s ) A formal proof can be constructed as follows: In this example the theorem produced " " can be interpreted as meaning "One plus three equals four." A different interpretation would be to read it backwards as "Four minus three equals one." Other concepts of interpretation There are other uses of the term "interpretation" that are commonly used, which do not refer to the assignment of meanings to formal languages. In model theory, a structure A is said to interpret a structure B if there is a definable subset D of A, and definable relations and functions on D, such that B is isomorphic to the structure with domain D and these functions and relations. In some settings, it is not the domain D that is used, but rather D modulo an equivalence relation definable in A. For additional information, see Interpretation (model theory). A theory T is said to interpret another theory S if there is a finite extension by definitions T′ of T such that S is contained in T′. See also Conceptual model Free variables and Name binding Herbrand interpretation Interpretation (model theory) Logical system Löwenheim–Skolem theorem Modal logic Model theory Satisfiable Truth References External links Stanford Enc. Phil: Classical Logic, 4. Semantics mathworld.wolfram.com: FormalLanguage mathworld.wolfram.com: Connective mathworld.wolfram.com: Interpretation mathworld.wolfram.com: Propositional Calculus mathworld.wolfram.com: First Order Logic Formal languages Interpretation (philosophy) Model theory Philosophy of language Philosophy of mind Semantics