Now, we are ready to explain, what it means to logically interpret a word of a formal language. We are able to provide a first strict definition of the concept of logical interpretation. 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.

Definition: Interpretation of Strings of a Formal Language and Their Truth Function

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:

$$I(s):=\cases{1,&\text{if }s\text{ is interpreted as being “true”,}\\ 0,&\text{if }s\text{ is interpreted as being “false”,}\\ undefined,&\text{if }s\text{ neither can be interpreted as being “true” or “false”.}}$$

Insights worth mentioning

• Please note that, in general, not all strings $s\in L$ can be interpreted, even though 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 support 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.

Chapters: 1 2
Definitions: 3 4 5 6 7 8 9 10 11 12
Examples: 13 14

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