title: Descriptive interpretation
text: # 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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