In recent years, dependency grammars have established themselves as valuable tools in theoretical and computational linguistics. To many linguists, dependency grammars and the more standard constituency-based formalisms are notational variants. We argue that, beyond considerations of formal equivalence, cognition may also serve as a background for a genuine comparison between these different views of syntax. In this paper, we review and evaluate some of the most common arguments and evidence employed to advocate for the cognitive or neural reality of dependency grammars in linguistics, psycholinguistics, or neurolinguistics. We then raise the possibility that the abilities to represent and track, alternatively or in parallel, constituency and dependency structures co-exist in human cognition and are constitutive of syntactic competence. In Sect. 2 , we briefly introduce dependency grammar and we contrast some of its core features with those of phrase structure grammars. In Sect. 3 , we describe the ‘received position’, the concept of ‘notational variance’, and some formal theory issues. In Sect. 4 , we assess whether linguistic arguments fare better than formal language theory in deciding between dependency grammar and phrase structure. In Sect. 5 , we consider further theoretical reasons for favoring dependency analyses, and we conclude that they are not any more decisive than the cross-linguistic data. In Sect. 3 The Received Position The relationship between dependency grammar and phrase structure grammars has been a long-standing problem in linguistics. Footnote 4 Two early formal results aimed to settle the matter: those of Gaifman ( 1961 ) and Hays (1961, 1964 ). To appreciate these results we need to briefly state two important criteria for comparing grammar formalisms in formal language theory, namely weak and strong generative capacity. The standard definitions of these terms have been held relatively stable since their introduction in Chomsky ( 1963 ). Relatedly, Stabler ( 2019 ) has argued that different mathematical foundations for syntax, present in phrase structure grammars, in dependency grammars, and in optimality theoretic formalisms, have common roots in formal tree languages. Stabler’s formal arguments for the insufficiency of each individual framework, together with their joint relevance for modeling linguistic competence, dovetail with the conclusion we reach in Sect. 6 on more cognitive grounds. The situation we are left with is one in which the received position largely has its roots in early research in formal language theory and in mathematical comparisons between the different grammars. Not only does it deal effectively with wh -constructions, but also with anaphora, which may involve long-distance dependencies and agreement, such as of number and gender. This is even clearer in inflected languages with freer word order. Dependency grammar captures a wide range of syntactic constructions without the need for additional formal mechanisms such as movement, thus extending its cross-linguistic coverage to languages in which such constructions are ubiquitous. Yet, projectivity may come at a coverage cost. The condition itself formally amounts to a ban on crossing edges. But it is surprising that his formal, deductive method for testing the effects of alternative syntactic representations on cohesion data has not been more widely adopted in psycholinguistics. To date, this is arguably the clearest demonstration of the empirical superiority of dependency grammar over traditional PSGs vis-á-vis syntactic judgments. The fourth argument is based on on-line behavioral evidence that human language processing tracks syntactic dependencies between words. For example, in a recent eye tracking study, Lopopolo et al. Essentially, LFG rejects the idea that the phrase structure and dependency graph are notational variants. In fact, it motivates their union via interface principles. Much like dependency grammar, LFG is a lexicalized formalism, because “every partial tree licensed by a grammar rule contains a lexical element” (Müller, 2018: 392). Further similarities include Computational complexity, tractability, learnability etc., also act as constraints on theory (van Rooij & Baggio, 2021 ). See Ferrer-I-Cancho ( 2016b ) for a formal framework for word order variation and alternation. From a theoretical perspective, dependency grammar does not assume a universal syntactic level of representation that explains syntactic structure in all languages (e.g., UG in generative syntax), but does produce a uniform method of analysis applicable to many languages. Chen, X., Gómez-Rodríguez, C., & Ferrer-i-Cancho, R. (2018). A dependency look at the reality of constituency. Glottometrics, 40 , 104–106. 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