 To capture global compositionality is more complicated. Here is an attempt. Let us say that the expressions e and \(e'\) are local equivalents just in case they are the results of applying the same syntactic operation to lists of expressions such that corresponding members of the lists are synonymous. (More formally: for some natural number k there is a k-ary F in E, and there are some expressions \(e_1, \ldots, e_k\), \(e_1 ', \ldots, e_k '\) in E, such that \(e = F(e_1 ,\ldots ,e_k)\), \(e' = F(e_1 ',\ldots ,e_k ')\), and for every \(1\le i \le k\), \(m(e_i) = m(e_i ')\).) It is clear that m is locally compositional just in case locally equivalent pairs of expressions are all synonyms. Let us say that the expressions e and \(e'\) are global equivalents just in case they are the results of applying the same syntactic operation to lists of expressions such that corresponding members of the lists are either (i) simple and synonymous or (ii) complex and globally equivalent. (Here is the recursive definition more formally. Let us say that the expressions e and \(e'\) are 1-global equivalents just in case they are synonymous simple expressions. Let us say that the expressions e and \(e'\) are n-global equivalents just in case for some natural number k there is a k-ary F in E, and there are some expressions \(e_1 ,\ldots ,e_k, e_1 ',\ldots ,e_k '\) in E, such that \(e = F(e_1 ,\ldots ,e_k), e' = F(e_1 ',\ldots ,e_k ')\), and for every \(1 \le i \le k\) there is a \(1 \le j \lt n\) such that \(e_i\) and \(e_i '\) are j-global equivalents. Finally, let us say that the expressions e and \(e'\) are global equivalents just in case for some natural number n they are n-global  equivalents.)[15]  I suggest that m is globally compositional just in case globally equivalent pairs of expressions are all synonyms.