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."
- [math]\displaystyle{ \forall }[/math]x(F(x)): "Everybody is sleeping."
- [math]\displaystyle{ \exists }[/math]z(G(a,z)): "Socrates hates somebody."
- [math]\displaystyle{ \exists }[/math]x[math]\displaystyle{ \forall }[/math]y[math]\displaystyle{ \exists }[/math]z(H(x,y,z)): "Somebody made everybody hit somebody."
- [math]\displaystyle{ \forall }[/math]x[math]\displaystyle{ \exists }[/math]z(F(x)[math]\displaystyle{ \wedge }[/math]G(a,z)): Everybody is sleeping and Socrates hates somebody.
- [math]\displaystyle{ \exists }[/math]x[math]\displaystyle{ \forall }[/math]y[math]\displaystyle{ \exists }[/math]z (G(a,z)[math]\displaystyle{ \lor }[/math]H(x,y,z)): Either Socrates hates somebody or somebody made everybody hit somebody.
Sources
- ↑ 1.0 1.1 1.2 Carnap, Rudolf, Introduction to Symbolic Logic and its Applications
- ↑ 2.0 2.1 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. https://archive.org/details/elementarylogic00mate/page/56.
 |  |
