According to Rudolf Carnap, in logic, an interpretation is a descriptive interpretation (also called a factual interpretation) if at least one of the undefined symbols of its formal system becomes, in the interpretation, a descriptive sign (i.e., the name of single objects, or observable properties).Carnap, Rudolf, Introduction to Symbolic Logic and its Applications In his Introduction to Semantics (Harvard Uni. Press, 1942) he makes a distinction between formal interpretations which are logical interpretations (also called mathematical interpretation or logico-mathematical interpretation) 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 Concept and the Role of the Model in Mathematics and Natural and Social Sciences - the world is an interpretation (or model) 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 a: Socrates b: Plato c: Aristotle 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(a): "Socrates is sleeping." H(b,a,c): "Plato made Socrates hit Aristotle." x(F(x)): "Everybody is sleeping." z(G(a,z)): "Socrates hates somebody." xyz(H(x,y,z)): "Somebody made everybody hit somebody." xz(F(x)G(a,z)): Everybody is sleeping and Socrates hates somebody. xyz (G(a,z)H(x,y,z)): Either Socrates hates somebody or somebody made everybody hit somebody.
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