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<p><span class="autocomment">Sources</span></p> <p><b>New page</b></p><div>{{See also|Interpretation (logic)}}<br /> <br /> According to [[Rudolf Carnap]], in [[logic]], an interpretation is a &#039;&#039;&#039;descriptive interpretation&#039;&#039;&#039; (also called a &#039;&#039;&#039;factual interpretation&#039;&#039;&#039;) if at least one of the undefined symbols of its [[formal system]] becomes, in the interpretation, a &#039;&#039;[[descriptive sign]]&#039;&#039; (i.e., the name of single objects, or observable properties).&lt;ref name=&quot;itslaia&quot;&gt;[[Rudolf Carnap|Carnap, Rudolf]], &#039;&#039;Introduction to Symbolic Logic and its Applications&#039;&#039;&lt;/ref&gt; In his &#039;&#039;Introduction to Semantics&#039;&#039; (Harvard Uni. Press, 1942) he makes a distinction between formal interpretations which are &#039;&#039;[[logical interpretation]]s&#039;&#039; (also called &#039;&#039;mathematical interpretation&#039;&#039; or &#039;&#039;logico-mathematical interpretation&#039;&#039;) and &#039;&#039;descriptive interpretations&#039;&#039;: a formal interpretation is a &#039;&#039;descriptive interpretation&#039;&#039; if it is not a &#039;&#039;logical interpretation&#039;&#039;.&lt;ref name=&quot;itslaia&quot;/&gt;<br /> <br /> Attempts to [[Axiomatic system|axiomatize]] the [[empirical science]]s, Carnap said, use a &#039;&#039;descriptive interpretation&#039;&#039; to model reality.:&lt;ref name=&quot;itslaia&quot;/&gt; the aim of these attempts is to construct a [[formal system]] for which [[reality]] is the only interpretation.&lt;ref name=&quot;tcarotmim&quot;&gt;The Concept and the Role of the Model in Mathematics and Natural and Social Sciences&lt;/ref&gt; - the world is an interpretation (or model) of these sciences, only insofar as these sciences are true.&lt;ref name=&quot;tcarotmim&quot;/&gt;<br /> <br /> 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.&lt;ref&gt;{{cite book<br /> | last = Mates<br /> | first = Benson<br /> | title = Elementary Logic, Second Edition<br /> | publisher = [[Oxford University Press]]<br /> | year = 1972<br /> | location = New York<br /> | pages = [https://archive.org/details/elementarylogic00mate/page/56 56]<br /> | isbn = 0-19-501491-X<br /> | url-access = registration<br /> | url = https://archive.org/details/elementarylogic00mate/page/56<br /> }}&lt;/ref&gt;<br /> <br /> ==Examples==<br /> <br /> A sentence is either &#039;&#039;true&#039;&#039; or &#039;&#039;false&#039;&#039; under an &#039;&#039;interpretation&#039;&#039; which assigns values to the logical variables. We might for example make the following assignments:<br /> <br /> &#039;&#039;&#039;Individual constants&#039;&#039;&#039;<br /> * a: Socrates<br /> * b: Plato<br /> * c: Aristotle<br /> &#039;&#039;&#039;Predicates&#039;&#039;&#039;:<br /> * Fα: α is sleeping<br /> * Gαβ: α hates β<br /> * Hαβγ: α made β hit γ<br /> <br /> &#039;&#039;&#039;Sentential variables:&#039;&#039;&#039;<br /> * &#039;&#039;p&#039;&#039; &quot;It is raining.&quot;<br /> <br /> Under this interpretation the sentences discussed above would represent the following English statements:<br /> * &#039;&#039;p&#039;&#039;: &quot;It is raining.&quot;<br /> * &#039;&#039;F&#039;&#039;(&#039;&#039;a&#039;&#039;): &quot;Socrates is sleeping.&quot;<br /> * &#039;&#039;H&#039;&#039;(&#039;&#039;b&#039;&#039;,&#039;&#039;a&#039;&#039;,&#039;&#039;c&#039;&#039;): &quot;Plato made Socrates hit Aristotle.&quot;<br /> * {{all}}&#039;&#039;x&#039;&#039;(&#039;&#039;F&#039;&#039;(&#039;&#039;x&#039;&#039;)): &quot;Everybody is sleeping.&quot;<br /> * {{exist}}&#039;&#039;z&#039;&#039;(&#039;&#039;G&#039;&#039;(&#039;&#039;a&#039;&#039;,&#039;&#039;z&#039;&#039;)): &quot;Socrates hates somebody.&quot;<br /> * {{exist}}&#039;&#039;x&#039;&#039;{{all}}&#039;&#039;y&#039;&#039;{{exist}}&#039;&#039;z&#039;&#039;(&#039;&#039;H&#039;&#039;(&#039;&#039;x&#039;&#039;,&#039;&#039;y&#039;&#039;,&#039;&#039;z&#039;&#039;)): &quot;Somebody made everybody hit somebody.&quot;<br /> * {{all}}&#039;&#039;x&#039;&#039;{{exist}}&#039;&#039;z&#039;&#039;(&#039;&#039;F&#039;&#039;(&#039;&#039;x&#039;&#039;){{and}}&#039;&#039;G&#039;&#039;(&#039;&#039;a&#039;&#039;,&#039;&#039;z&#039;&#039;)): Everybody is sleeping and Socrates hates somebody.<br /> * {{exist}}&#039;&#039;x&#039;&#039;{{all}}&#039;&#039;y&#039;&#039;{{exist}}&#039;&#039;z&#039;&#039; (&#039;&#039;G&#039;&#039;(&#039;&#039;a&#039;&#039;,&#039;&#039;z&#039;&#039;){{or-}}&#039;&#039;H&#039;&#039;(&#039;&#039;x&#039;&#039;,&#039;&#039;y&#039;&#039;,&#039;&#039;z&#039;&#039;)): Either Socrates hates somebody or somebody made everybody hit somebody.<br /> <br /> == Sources ==<br /> {{reflist}}<br /> <br /> {{DEFAULTSORT:Descriptive Interpretation}}<br /> [[Category:Semantics]]<br /> [[Category:Formal languages]]<br /> [[Category:Interpretation (philosophy)]]<br /> <br /> <br /> {{semantics-stub}}</div>

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