 Important differences among these terms cannot be mapped out here. However, in order to better understand correspondence rules, two of their functions will be considered: (i) theory interpretation (Carnap, Hempel) and (ii) theory reduction (Nagel, Schaffner). The dominant perspective on correspondence rules is that they interpret theoretical terms. Unlike “mathematical theories,” the axiomatic system of physics “cannot have… a splendid isolation from the world” (Carnap 1966, 237). Instead, scientific theories require observational interpretation through correspondence rules. Even so, surplus meaning always remains in the theoretical structure (Hempel 1958, 87; Carnap 1966). Second, correspondence rules are seen as necessary for inter-theoretic reduction (van Riel and Van Gulick 2014). For instance, they connect observation terms such as “temperature” in phenomenological thermodynamics (the reduced theory) to theoretical concepts such as “mean kinetic energy” in statistical mechanics (the reducing theory). Correspondence rules unleash the reducing theory’s epistemic power. Notably, Nagel (1961, Chapter 11; 1979) and Schaffner (1969, 1976, 1993) allow for multiple kinds of correspondence rules, between terms of either vocabulary, in the reducing and the reduced theory (cf. Callender 1999; Winther 2009; Dizadji-Bahmani, Frigg, and Hartmann 2010). Correspondence rules are a core part of the structure of scientific theories and serve as glue between theory and observation.