# Intended interpretation

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One who constructs a syntactical system usually has in mind from the outset some interpretation of this system. While this **intended interpretation** (called **standard model** in mathematical logic—a term introduced by Abraham Robinson in 1960)^{[1]} 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 is directly derivable from a sentence , then turns out to be a true sentence (under the customary interpretation of → as meaning implication). These requirements ensure that all provable sentences also come out to be true.^{[2]}

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. ^{[3]}

## Example

Given a simple formal system (we shall call this one ) whose alphabet α consists only of three symbols { , , } and whose formation rule for formulas is:

- 'Any string of symbols of which is at least 6 symbols long, and which is not infinitely long, is a formula of . Nothing else is a formula of .'

The single axiom schema of is:

- "
**********" (where "*****" is a metasyntactic variable standing for a finite string of " "s )

A formal proof can be constructed as follows:

In this example the theorem produced " " can be interpreted as meaning "One plus three equals four." A different interpretation would be to read it backwards as "Four minus three equals one."^{[4]}

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Rudolf Carnap,
*Introduction to Symbolic Logic and its Applications* - ↑ The Concept and the Role of the Model in Mathematics and Natural and Social Sciences
- ↑ Geoffrey Hunter,
*Metalogic*

nl:Bedoelde interpretatie pt:Interpretação pretendida zh:预期释义