Descriptive interpretation
# 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). 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 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."
* $\forall$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 2 3 Carnap, Rudolf, Introduction to Symbolic Logic and its Applications
* 1 2 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. pp. 56. ISBN 0-19-501491-X.
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