 Descriptive interpretation

See also: Interpretation (logic)
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 logicomathematical 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 nonempty set may be chosen as the domain of a descriptive interpretation, and all nary 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."
 x(F(x)): "Everybody is sleeping."
 z(G(a,z)): "Socrates hates somebody."
 xyz(H(x,y,z)): "Somebody made everybody hit somebody."
 xz(F(x)&G(a,z)): Everybody is sleeping and Socrates hates somebody.
 xyz (G(a,z)H(x,y,z)): Either Socrates hates somebody or somebody made everybody hit somebody.
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. pp. 56. ISBN 019501491X.
Categories: Semantics
 Formal languages
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