View source for 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).<ref name="itslaia">[[Rudolf Carnap|Carnap, Rudolf]], ''Introduction to Symbolic Logic and its Applications''</ref> In his ''Introduction to Semantics'' (Harvard Uni. Press, 1942) he makes a distinction between formal interpretations which are ''[[logical interpretation]]s'' (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''.<ref name="itslaia"/>

Attempts to [[Axiomatic system|axiomatize]] the [[empirical science]]s, Carnap said, use a ''descriptive interpretation'' to model reality.:<ref name="itslaia"/>  the aim of these attempts is to construct a [[formal system]] for which [[reality]] is the only interpretation.<ref name="tcarotmim">The Concept and the Role of the Model in Mathematics and Natural and Social Sciences</ref> - the world is an interpretation (or model) of these sciences, only insofar as these sciences are true.<ref name="tcarotmim"/>

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.<ref>{{cite book
  | last = Mates
  | first = Benson
  | title = Elementary Logic, Second Edition
  | publisher = [[Oxford University Press]]
  | year = 1972
  | location = New York
  | pages = [https://archive.org/details/elementarylogic00mate/page/56 56]
  | isbn = 0-19-501491-X
  | url-access = registration
  | url = https://archive.org/details/elementarylogic00mate/page/56
  }}</ref>

==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."
* {{all}}''x''(''F''(''x'')): "Everybody is sleeping."
* {{exist}}''z''(''G''(''a'',''z'')):  "Socrates hates somebody."
* {{exist}}''x''{{all}}''y''{{exist}}''z''(''H''(''x'',''y'',''z'')):  "Somebody made everybody hit somebody."
* {{all}}''x''{{exist}}''z''(''F''(''x''){{and}}''G''(''a'',''z'')): Everybody is sleeping and Socrates hates somebody.
* {{exist}}''x''{{all}}''y''{{exist}}''z'' (''G''(''a'',''z''){{or-}}''H''(''x'',''y'',''z'')): Either Socrates hates somebody or somebody made everybody hit somebody.

== Sources ==
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{{DEFAULTSORT:Descriptive Interpretation}}
[[Category:Semantics]]
[[Category:Formal languages]]
[[Category:Interpretation (philosophy)]]


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