Definition of linguistic variable.

Question

A linguistic variable is defined as a quintuple $(X,T(X),U,G,M)$ where: $$ X\text{ is the name of the variable.} $$ $$ T(X)\text{ is the collection of linguistic values of $X$.} $$ $$ U \text{ is a universe of discourse.} $$ $$ G\text{ is a syntactic rule which generates the terms in $T(X)$.} $$ $$ M \text{ is a semantic rule which associates with each linguistic value X its meaning.} $$

Let us take $X=$ Age, and $T(X)=\{$ young, very young, old, very old, middle-aged, $\ldots\}$. What could we have as $U,G,M$? I think I do not really understand the definition of linguistic variable, so I cannot imagine a possible choice for $U,G,M$.

Answer 1

See Linguistic variables and truth-values in fuzzy logic :

A linguistic variable $\mathcal X$ is a nonfuzzy variable which ranges over a collection $T (\mathcal X)$ of structured fuzzy variables $X_1, X_2, \ldots$ with each fuzzy variable in $T (\mathcal X)$ carrying a linguistic label $X_i$, which characterizes the fuzzy restriction which is associated with $X_i$.

As an illustration, $Age$ is a linguistic variable if its values are assumed to be the fuzzy variables labeled young, not young, very young,... rather than the numbers $0,1,2, \ldots$ [...] Thus, if the base variable for $Age$ (i.e. numerical age) is assumed to range over the universe $U= \{ 0, 1, \ldots, 100 \}$, then the linguistic values of $Age$ may be interpreted as the labels of fuzzy subsets of $U$.

More generally, a linguistic variable is characterized by a quintuple $( \mathcal X, T(\mathcal X), U, G, M )$ where $\mathcal X$ is the name of the variable, e.g. $Age$; $T(\mathcal X)$ is the term-set of $\mathcal X$, that is, the collection of its linguistic values, e.g. $T(\mathcal X) = \{$ young, not young, very young, $\ldots \}$; $U$ is a universe of discourse, e.g., in the case of $Age$, the set $\{ 0, 1, \ldots, 100 \}$; $G$ is a syntactic rule which generates the terms in $T(\mathcal X)$; and $M$ is a semantic rule which associates with each term $X_i$ in $T(\mathcal X)$ its meaning $M(X_i)$, where $M(X_i)$ is a fuzzy subset of $U$ which serves as a fuzzy restriction on the values of the fuzzy fariable $X_i$.