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 $\to$ ${\mathcal {I}}$_j turns out to be a true sentence (under the customary interpretation of '$\to$'). These requirements ensure that all provable sentences also come out to be true.

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.