Now, we are ready to explain, what it means to logically *intepret* a word of a formal language. We are able to provide a first strict definition of the concept of logical *interpreation*. Later, when we will be discussing special types of logic, including the propositional logic, the first-order and higher-order predicate logics, we will need some customized definitions of *interpretations*, more precisely serving specific purposes of each case.

For the time being, it is sufficient to understand that an *interpretation* inside a logical system is a rule assigning truth values to the strings of the underlying formal language.

Let \(L\subseteq (\Sigma^*,\cdot) \) be a formal language in a given domain of discourse $U$. The **interpretation** $I$ is an appropriate partial function $I: L\subset \to \mathbb B$, $s\to I(s)$, depending on $L$ and $U$.

In other words, given such an appropriate $I$, for any string $s\in L$, the value of the function $I(s)$ can take one of three values:

$$I(s):=\cases{1,&\text{if }s\text{ is interpreted as being “true”,}\\

0,&\text{if }s\text{ is interpreted as being “false”,}\\

undefined,&\text{if }s\text{ neither can be interpreted as being “true” or “false”.}}$$

- Please note that in general, not all strings $s\in L$ can be interpreted, even if they are syntactically correct. In this case, they either have no meaning or do have a meaning but no truth value can be assigned to them. We will learn examples of such strings later on.
- Note also that the
*truth*of strings is always a matter of the choice of the specific function $I$. It means that logical systems do not allow any kind of “absolute, objective interpretation”. For instance, the same string $s$ can be true in one interpretation, false in another or no interpretation might be possible in yet another interpretation.

| | | | | created: 2018-02-03 08:27:58 | modified: 2020-05-04 19:29:19 | by: *bookofproofs*

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