For the specification of formal systems for a grammar formalism, conventional mathematical wisdom dictates that we start with primitives (basic primitive structures or building blocks) as simple as possible and then introduce various operations for constructing more complex structures. Alternatively, we can start with complex (more complicated) primitives that directly capture crucial linguistic properties and then introduce some general operations (language independent operations) for composing them. This latter approach has led to the so-called strongly lexicalized grammars, providing some new insights into syntactic description, semantic composition, discourse structure, language generation, psycholinguistic and statistical processing, all with computational implications. In this paper, we will illustrate some of these insights in the context of the lexicalized tree-adjoining grammar (LTAG).