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

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

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