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]

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

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