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).[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."
  • \forallx(F(x)): "Everybody is sleeping."
  • \existsz(G(a,z)): "Socrates hates somebody."
  • \existsx\forally\existsz(H(x,y,z)): "Somebody made everybody hit somebody."
  • \forallx\existsz(F(x)&G(a,z)): Everybody is sleeping and Socrates hates somebody.
  • \existsx\forally\existsz (G(a,z)\lorH(x,y,z)): Either Socrates hates somebody or somebody made everybody hit somebody.

Sources

  1. ^ a b c Carnap, Rudolf, Introduction to Symbolic Logic and its Applications
  2. ^ a b 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. pp. 56. ISBN 019501491X. 

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