Descriptive interpretation

See also: Formal interpretation, Logical interpretation

A formal 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).

Rudolf Carnap, in his Introduction to Semantics 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 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.^[1]

Examples

NB Any non-empty set may be chosen as the domain of an interpretation, and all n-ary relations among the elements of the domain are candidates for assignment to any predicate of degree n. ^[2]

The following example is in first-order logic with propositional variables. The underlying language has the following non-logical symbols: Predicate symbols F (unary), G (binary) and H (ternary), and a propositional variable p. The domain of discourse is D = Persons (past living and future)

Example of an informal interpretation

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:
Domain: Persons (past living and future)
Individual Constants

a: Socrates
b: Plato
c: Aristotle

Predicates:

Fα: α is sleeping (strictly: F is the set of those who are sleeping)
Gαβ: α hates β (strictly: G is the binary relation that holds between between α and β when α hates β)
Hαβγ: α made β hit γ (strictly: H is the tertiary relation that holds between between α and β and γ when α made β hit γ)

Sentential variables:

p: It is raining. (strictly: p is assigned the truth-value of the judgment that it is raining)

Under this interpretation the sentences on the left below would be true just in those cases (if any) that the English statements on the right below were true:

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."

In other words p is true under the intepretation just in case it is raining, F(a) is true under the interpreatation just in case Socrates is sleeping, &c.

This is not a formal interpretation (structure) as defined above. One could argue that it becomes a structure by fixing an exact location and point in time, but even then it is not clear that a statement such as "It is raining" can always be assigned a truth-value ("Is this rain or snow?"); and given the fact that Socrates died before Aristotle was born, ascribing a truth-value to "Socrates hates Aristotle" raises other issues as well. On the other hand if "It is raining" cannot always be assigned a truth-value then it is not, by definition, a statement. Similarly statements like "Humans feared dinosaurs" or "Socrates like LSD" would nornmally be considered unproblematically false. To avoid the later issue we could of course assign Moore, Russell and Witgenstein t a, b and c respectively.

Example of a formal interpretation

F is true for Socrates, but not for Plato or Aristotle.
G is false for all pairs of elements of the domain.
H is true for the triple (Socrates, Plato, Aristotle) but false for everything else. I.e. Hαβγ holds if and only if α = Socrates, β = Plato and γ = Aristotle.
p is false.

Under this interpretation, the above sentences have truth-values:

p: false.
F(a): true.
H(b,a,c): false.
$\forall$ x(F(x)): false.
$\exists$ z(G(a,z)): false.
$\exists$ x $\forall$ y $\exists$ z(H(x,y,z)): false.
$\forall$ x $\exists$ z(F(x) $\wedge$ G(a,z)): false.
$\exists$ x $\forall$ y $\exists$ z (G(a,z) $\lor$ H(x,y,z)): false.

Sources

^ 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. p. 56. ISBN 019501491X. {{cite book}}: |pages= has extra text (help)

Carnap, Rudolf, Introduction to Symbolic Logic and its Applications
Carnap, Rudolf, Introduction to Semantics

[1] The Concept and the Role of the Model in Mathematics and Natural and Social Sciences

[2] Mates, Benson (1972). Elementary Logic, Second Edition. New York: Oxford University Press. pp. p. 56. ISBN 019501491X. {{cite book}}: |pages= has extra text (help)

[1]

[2]