Descriptive interpretation

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).^[1] 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.^[1]

Attempts to axiomatize the empirical sciences, Carnap said, use a descriptive interpretation to model reality.:^[1] the aim of these attempts is to construct a formal system for which reality is the only interpretation.^[2] - the world is an interpretation (or model) of these sciences, only insofar as these sciences are true.^[2]

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.^[3]

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."
• $\forall$x(F(x)): "Everybody is sleeping."
• $\exists$z(G(a,z)): "Socrates hates somebody."
• $\exists$x$\forall$y$\exists$z(H(x,y,z)): "Somebody made everybody hit somebody."
• $\forall$x$\exists$z(F(x)$\wedge$G(a,z)): Everybody is sleeping and Socrates hates somebody.
• $\exists$x$\forall$y$\exists$z (G(a,z)$\lor$H(x,y,z)): Either Socrates hates somebody or somebody made everybody hit somebody.

Sources

1. Carnap, Rudolf, Introduction to Symbolic Logic and its Applications
2. The Concept and the Role of the Model in Mathematics and Natural and Social Sciences
3. Mates, Benson (1972). Elementary Logic, Second Edition. New York: Oxford University Press. p. 56. ISBN 0-19-501491-X.