The core capacity of human language is described as the faculty to combine words into hierarchical structures. This review aims to isolate the fundamental computation behind the language faculty together with its neural implementation. First, we present our central hypothesis by confronting recent linguistic theory with evolutionary arguments: linguistic humaniqueness is reflected in the labeling of word combinations forming asymmetric hierarchical structures. Second, we review the neurolinguistic literature, especially focusing on dual-stream connectivity models. We put forward that the dorsal pathway, especially the arcuate fascicle, is responsible for the rule-based combinatorial system, implementing labeling and giving rise to hierarchical structures. Conversely, the ventral stream is rather responsible for semantic associative operations. We further present evolutionary neuroanatomical evidence grounding our hypothesis. We conclude by suggesting further avenues of research as well as open questions to be addressed. With the aim to expand our knowledge on the neurobiology of language, we hope to provide a testable hypothesis for the origin of language syntax bringing together evidence from different fields.