Descriptive interpretation


According to Rudolf Carnap, in logic, an interpretation is a descriptive interpretation if at least one of the undefined symbols of its formal system becomes, in the interpretation, a descriptive sign. In his Introduction to Semantics he makes a distinction between formal interpretations which are logical interpretations and descriptive interpretations: a formal interpretation is a descriptive interpretation if it is not a logical interpretation.
Attempts to axiomatize the empirical sciences, Carnap said, use a descriptive interpretation to model reality.: the aim of these attempts is to construct a formal system for which reality is the only interpretation. - the world is an interpretation of these sciences, only insofar as these sciences are true.
Any non-empty set may be chosen as the domain of a descriptive interpretation, and all n-ary relations among the elements of the domain are candidates for assignment to any predicate of degree n.

Examples

A sentence is either true or false under an interpretation which assigns values to the logical variables. We might for example make the following assignments:
Individual constants
Predicates:
  • Fα: α is sleeping
  • Gαβ: α hates β
  • Hαβγ: α made β hit γ
Sentential variables:p "It is raining."
Under this interpretation the sentences discussed above would represent the following English statements:p: "It is raining."F: "Socrates is sleeping."H: "Plato made Socrates hit Aristotle."x: "Everybody is sleeping."z: "Socrates hates somebody."x'y'z: "Somebody made everybody hit somebody."x'z'G): Everybody is sleeping and Socrates hates somebody.x'y'z ''H''): Either Socrates hates somebody or somebody made everybody hit somebody.