Descriptive interpretation

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See also: Formal interpretation, Logical interpretation

According to Rudolf Carnap, in logic, a formal 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 an descriptive interpretation, and all n-ary relations among the elements of the domain are candidates for assignment to any predicate of degree n. ^[3]

[edit] 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)&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.

[edit] Sources

^ ^a ^b ^c Carnap, Rudolf, Introduction to Symbolic Logic and its Applications
^ ^a ^b The Concept and the Role of the Model in Mathematics and Natural and Social Sciences
^ Mates, Benson (1972). Elementary Logic, Second Edition. New York: Oxford University Press, p. 56. ISBN 019501491X.