Intended interpretation

{{ safesubst:#invoke:Unsubst||$N=Merge to |date=__DATE__ |$B= Template:MboxTemplate:DMCTemplate:Merge partner }} {{ safesubst:#invoke:Unsubst||$N=Context |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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) 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 → as meaning implication). 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.

Example

Given a simple formal system (we shall call this one ${\mathcal {FS'}}$) whose alphabet α consists only of three symbols { $\blacksquare$, $\bigstar$, $\blacklozenge$ } and whose formation rule for formulas is:

'Any string of symbols of ${\mathcal {FS'}}$ which is at least 6 symbols long, and which is not infinitely long, is a formula of ${\mathcal {FS'}}$. Nothing else is a formula of ${\mathcal {FS'}}$.'

The single axiom schema of ${\mathcal {FS'}}$ is:

" $\blacksquare$ $\bigstar$ * $\blacklozenge$ $\blacksquare$ * " (where " * " is a metasyntactic variable standing for a finite string of " $\blacksquare$ "s )

A formal proof can be constructed as follows:

(1) $\blacksquare$ $\bigstar$ $\blacksquare$ $\blacklozenge$ $\blacksquare$ $\blacksquare$

(2) $\blacksquare$ $\bigstar$ $\blacksquare$ $\blacksquare$ $\blacklozenge$ $\blacksquare$ $\blacksquare$ $\blacksquare$

(3) $\blacksquare$ $\bigstar$ $\blacksquare$ $\blacksquare$ $\blacksquare$ $\blacklozenge$ $\blacksquare$ $\blacksquare$ $\blacksquare$ $\blacksquare$

In this example the theorem produced " $\blacksquare$ $\bigstar$ $\blacksquare$ $\blacksquare$ $\blacksquare$ $\blacklozenge$ $\blacksquare$ $\blacksquare$ $\blacksquare$ $\blacksquare$ " 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."