Intended interpretation

:"See also:Formal interpretation

One who constructs a syntactical system usually has in mind from the outset some interpretation of this system. While this intended interpretation can have no explicit indication in the syntactical rules --since these rules must be strictly formal --the author's intention respecting interpretation naturally affects his choice of the formation and transformation rules of the syntactical system. For example, he chooses primitive signs in such a way that certain concepts can be expressed: He chooses sentential formulas in such a way that their counterparts in the "intended interpretation" can appear as meaningful declarative sentences; his choice of primitive sentences must meet the requirement that these primitive sentences come out as true sentences in the interpretation; his rules of inference must be such that if by one of these rules the sentence $mathcal\{I\}$_j is directly derivable from a sentence $mathcal\{I\}$_i, then "$mathcal\{I\}$_i imp $mathcal\{I\}$_j" turns out to be a true sentence (under the customary interpretation of 'imp'). These requirements ensure that all provable sentences also come out to be true. [Rudolf Carnap, "Introduction to Symbolic Logic and its Applications"]

Most formal systems have many more models than they were intended to have (the existence of non-standard models is an example). When we speak about 'models' in empirical sciences, we mean, if we want reality to be a model of our science, to speak about an "intended model". A model in the empirical sciences is an "intended factually-true descriptive interpretation" (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same domain of discourse as the intended one, but other assignments for non-logical constants. [The Concept and the Role of the Model in Mathematics and Natural and Social Sciences]

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