Syntactic Structures and Recursive Devices: a Legacy of Imprecision Author(S): Marcus Tomalin Source: Journal of Logic, Language, and Information, Vol

Syntactic Structures and Recursive Devices: A Legacy of Imprecision Author(s): Marcus Tomalin Source: Journal of Logic, Language, and Information, Vol. 20, No. 3, MATHEMATICS OF LANGUAGE (Summer 2011), pp. 297-315 Published by: Springer Stable URL: https://www.jstor.org/stable/41488480 Accessed: 23-04-2020 06:16 UTC

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms J Log Lang Inf (201 1) 20:297-315 DOI 10.1007/S10849-01 1-9141-1

Syntactic Structures and Recursive Devices: A Legacy of Imprecision

Marcus Tomalin

Published online: 17 April 201 1 © Springer Science+Business Media B.V. 201 1

Abstract Taking Chomsky's Syntactic Structures as a starting point, this paper explores the use of recursive techniques in contemporary linguistic theory. Specif- ically, it is shown that there were profound ambiguities surrounding the notion of recursion in the 1950s, and that this was partly due to the fact that influential texts such as Syntactic Structures neglected to define what exactly constituted a recursive device. As a result, uncertainties concerning the role of recursion in linguistic theory have pre- vailed until the present day, and some of the most common misunderstandings that have appeared in recent discussions are examined at some length. This article shows that debates about such topics are frequently undermined by fundamental misunderstandings concerning core terminology, and the full extent of the prevailing haziness is revealed. An attempt is made, for instance, to distinguish between such things as iterative constructional devices and self-similar syntactic embedding, despite the fact that these are usually both unhelpfully classified as examples of recursion. Consequently, this article effectively constitutes a plea for much greater accuracy and clarity when such important issues are addressed from a linguistic perspective.

Keywords Syntax • Recursion • Minimalism

1 Introduction

Although recursive devices have been used in linguistic theorising implicitly since the fourth century ВС (at least), and explicitly since the 1950s, during the past few years

This article is an extensively revised version of a paper that was delivered at MoLlO in 2007. The original paper was delivered as a 'key-note' address which celebrated the 50th anniversary of the publication of Syntactic Structures (1957).

M. Tomalin (E3) Downing College, University of Cambridge, Cambridge CB2 1DQ, UK e-mail: [email protected]

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms 298 M. Tomalin such techniques have been subjected to unprecedented discussion and debate. This is partly due to the strong claims that Chomsky (in particular) has made since 1995 concerning the role of recursion in the Minimalist Program. In essence, a strongly minimalist view of language can be expressed by the simple equation: 'interfaces + recursion = language', and this suggests that recursion is the single most important component of the faculty of language (FL).1 While some linguists have enthusias- tically championed this view, others have claimed that it is far too parsimonious. For Sigrid Beck, '[g]rammer is not limited to recursive structure building' (Beck 2007,278). More provocatively still, others have recently challenged the minimalist assumption that recursion is a necessary component of FL by presenting analyses of languages which appear not to utilise recursive structures. Dan Everett, in particular, has claimed that the Amazonian language Pirahã does not make use of nested clauses.2 If such analyses prove to be correct, then recursion would seem not to be universal, and this would in turn undermine its privileged status in modern syntactic theory. Given the numerous arguments made for and against the claim that recursion is fundamental to FL, it is appropriate that the subject should have become prominent in recent discussions of the evolution of natural language. For example, Simon Kirby has explored computational models for the acquisition of 'recursive syntax' (Kirby 2002). In addition, other linguists have recently argued that the traditional view that recursion enables an infinite number of linguistic structures to be generated from a finite set of discrete units is a lazy stance that requires drastic reconsideration. In particular, Geoff Pullum and Barabara Scholtz have proposed that 'the use of a rule-application analog of recursion in grammars does not entail infinitude for human languages, and infinitude does not offer independent evidence that a human language must have a recursive generative grammar' (Pullum and Scholtz 2010,8) Predictably, such considerations have attracted the attention of non-linguists, and the alleged centrality of recursion in human cognition (in general) has become an increasingly popular research topic in cognitive science more broadly. Michael Corballis, for example, has argued that recursion is a distinctive (possibly unique) aspect of human cognitive function (Corballis 2007). In the light of these ongoing debates, this article will seek to accomplish several tasks, but the main purpose is to attempt to clarify the way in which the term recursion is used in contemporary linguistic theorising. In order to do this, an historiographical perspective will be adopted initially, and the deployment of recursive techniques in Chomsky's Syntactic Structures (SS) will provide a convenient starting point. As will be shown in detail, the notion of 'recursion' was fundamentally ambiguous when it began to be used by linguists in the 1950s, and (as the subtitle of this article implies), these ambiguities have persisted to the present day. This unfortunate (and needless) perpetuation of imprecision has had a deleterious impact upon recent discussions of the role of recursion in linguistic theory.

1 For a discussion of this formulation, see Sauerland and Gätner (2007). When used in this context, 'interfaces' usually denotes the conceptual-intentional (CI) and sensorimotor (SM) systems. These are the language-external systems with which FL interacts. For other perspectives, see van der Hülst (2010). 2 The ongoing debate about these matters can be traced in publications such as Everett (2005, 2009), Nevins et al. (2009a, b).

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2 Recursive Syntactic Structures

Before focusing on the particular recursive techniques mentioned in SS (and some of Chomsky's other works from the 1950s), it is helpful to reflect upon how the word recursion is commonly used in general, non-technical discourse. The OED, for instance, offers the following definition:

The application or use of a recursive procedure or definition; primitive recursion, definition of a function of natural numbers by induction on a single argument or (equivalently) by simple recursion formulae; recursion formula, an equation relating the value of a function for a given value of its argument (or arguments) to its values for other values of the argument(s).

This definition is hardly lucid, but it certainly attempts to identify certain kinds of 'procedure' and 'definition' that have distinctive properties of self-reference. By contrast, that other source of all Truth, Wikipedia, begins its entry with these words:

Recursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition. The term is also used more generally to describe a process of repeating objects in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion.3

One again, the main emphasis here is upon particular types of 'process' and 'definition', and recursive qualities are not associated explicitly with inherent structural configurations. As these informal and rather fragmentary definitions/descriptions indicate, words and phrases such as 'procedure', 'process', 'function', 'definition', 'self-reference', 'circularity', and 'efficiency' appear frequently when recursion is discussed. A more playful definition runs as follows:

Recursion : If you still don't know, See: "Recursion".

At the time of writing (7/1 1/09), the Google search engine essentially implements this formulation: a search for the word 'recursion' returns around three million results, along with the helpful suggestion 'Did you mean: recursion!' . So, informally, there appear to be several inherent properties that are associated with recursive techniques, and self-reference (of some kind) is usually viewed as being central. However, as I have shown in some detail elsewhere, when the word recursion began to be used explicitly in the context of linguistic theory in the 1950s, there were considerable ambiguities as to what the term actually meant.4 Although there is no need to rehearse the same arguments here in detail, a brief overview of the main developments might be useful. In 1 889, the Italian mathematician Giuseppe Peano made extensive use of inductive definitions in his axioms for the natural numbers, and, influenced in part by Peano's

3 http://en.wikipedia.org/wiki/Recursion, accessed on 7/1 1/09. 4 For a detailed discussion, see Tomalin (2007).

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms 300 M. Tomalin work, Kurt Godei developed his theory of recursive functions in the late 1920s and early 1930s.5 He defined primitive recursive functions as follows (Godei 1986[1931]):

0(0, *2, . . . , Xn ) = '¡/(X2 , . . . , xn) ф(к + 19хг,...,хп) = ß(k, ф(к, X2, . . . , xn), X2, . . . , Xn) (1)

In (1) the number-theoretic function ф(х', X2, . . . , x„) is recursively defined in terms of the previously defined number-theoretic functions '¡r{x i, *2, . . . jc„_i) and ß(x' , *2, . . . , xn+' ), assuming that these hold for x' , . . . , Jtw+i, and k. Godei introduced these functions in order to provide a method for encoding syntax, and (at this time) he was not primarily concerned with the task of understanding recursion or computability. However, he knew that there were certain functions which could not be classified as primitive recursive, but which were nonetheless recursive, therefore he introduced the notion of 'general recursive functions' in order to elucidate effectively calculable (i.e., intuitively computable) functions (Godei 1986[1934], 368). During the 1930s and 1940s, several developments complicated the situation:

- À-Definability was proposed in 1936 by Alonzo Church. This introduced the notion of 'effectively calculable', and Church attempted to demonstrate that 4 [e] very [general] recursive function of positive integers is À-definable' and '[e]very À -definable function of positive integers is [general] recursive' (Church 1936,349) - Computability theory was proposed by Alan Turing in 1936. This introduced the notion of (Turing-)computable functions, and Turing proved that 'all effectively calculable (À-definable) sequences are computable', and vice versa (Turing 1936,263)

Consequently, by the end of the 1 930s, three distinct mathematical frameworks - (general) recursive functions, À-definable functions, and Turing-computable functions - were shown to be equivalent. Crucially, Godei strongly rejected Church's approach to computability, even though the formalism made use of Gödel's own recursive functions. To complicate matters further, in unpublished research from the 1920s, Emil Post had largely anticipated the results of Godei, Church, and Turing, and, during the 1940s, he elaborated the notion of recursively enumerable sets. Post used the term 'recursive' to refer both to specific kinds of mathematics objects (e.g., sets), and to certain devices/techniques (e.g., functions) that were used to generate such objects. The results obtained by Peano, Godei, Church, Turing, Post, and others were influ- entially summarised in Stephen Kleene's textbook Introduction to Metamathematics , and this is a text with which many mathematically-minded linguistics in the 1950s were familiar.6 The aforementioned developments were all introduced in the domains of mathematics and logic, but during the first half of the twentieth century, the connections between linguistics and mathematics became stronger and linguists sought to make

Although Peano's work was influential, the development of modern recursion theory is often traced back to Dedekind (1888). 6 For a summary of some of these developments, see Soare (1996, 1999).

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms Syntactic Structures and Recursive Devices: A Legacy of Imprecision 301 the study of language more scientific and rigorous.7 Bar-Hillel was one of the first linguists to argue explicitly (in 1952) that recursive definitions should be used in the study of natural language. Using English as a 'metalanguage' and French as 'an object language', he suggested that sentences could be defined recursively as follows (Bar-Hillel 1953,163):

Définition 1: Sentence (Recursive) 1. X is a sentence ' (a simple sentence ) =df * is a sequence of a nominal and a (intransitive) verbal or a sequence of a nominal, a (transitive) verbal, and a nominal, or ... 2. X is a sentence^ i (a compound sentence of the n - h 1th order) =df x is a sequence of a sentence^,, the word "et", and a sentence^, where either p or m (or both) are equal to n and none is greater than n, or . . .

For Bar-Hillel, Definition 1 takes the form of 'a pair of simultaneous recursive definitions' (Bar-Hillel 1953,164). In a subsequent paper, 'Logical Syntax and Semantics' (1954), he suggested that recursive definitions could be used specifically in order to clarify the nature of the relationship between the various postulated levels of linguistic analysis (e.g., phonological, morphological), and he developed his ideas in response to the work of Kenneth Pike (and others). Specifically, Pike had argued that these levels should not be entirely separate, and he outlined a nine-step procedure that would enable a linguist to produce a phonetic transcription of a spoken utterance (Pike 1952). Bar- Hillel stated that some linguists wanted to keep the levels separate so that the whole system would not be 'in constant danger of succumbing to an infestation of meaning' (Bar-Hillel 1954,234). He also acknowledged anxieties concerning the circularity of defining phonemes in terms of morphemes while simultaneously defining the latter in terms of the former. However, he maintained that it was possible to use seemingly circular 'concept formations' which nonetheless avoided infinite regress:

[...] concept formations of these kinds are in regular use in mathematics, and especially in mathematical logic, where they are known as a specificai case of RECURSIVE DEFINITIONS. It seems rather likely (though a detailed proof would require many man-hours of work) that Pike's nine-step procedure can be formalized and adequately represented by a set of such definitions. (Bar-Hillel 1954,234)

Chomsky was well-aware of Bar-Hillel's work in the mid 1950s and although he responded positively to this particular suggestion in 1955, by the time he wrote 55, he had changed his mind:

Bar-Hillel has suggested in "Logical syntax and semantics", Language 30.230-7 (1954) that Pike's proposals can be formalized without the circularity that many sense in them by the use of recursive definitions. He does not pursue this suggestion in any detail, and my own feeling is that success along these lines is unlikely. (Chomsky 1957,57-58)

7 For a detailed discussion of this general development, see Tomalin (2006).

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But this is only one of the references to recursion that appear in 55, and in order to appreciate Chomsky's view of this problematical subject, it is necessary to explore the text in more detail.

3 Loops and Devices

55 was certainly published by Mouton Publishers (now Mouton de Gruyter) in 1957, as part of the Janua Linguarum series edited by Cornelis van Schooneveld, but the particular month in which it appeared is uncertain. The Preface to the first edition is dated August 1st 1956, and it seems most likely that the volume was in the public domain by February 1957.8 It was conventional at the time for reviews of linguistics monographs (or books that discussed any technical academic subject) to appear a year or two after the text itself was published. Curiously, this was not the case for 55 since Robert Lees published an influential review of 55 in the July-September volume of the journal Language . The rapidity of the reviewing process certainly suggests expressly co-ordinated organisation. Given its mythological status as one of the seminal texts of twentieth-century linguistics, it is worthwhile recalling that 55 is a strange publication which has provoked contrasting responses. In the text itself, Chomsky describes the purpose of the work:

This study deals with syntactic structure both in the broad sense (as opposed to semantics) and the narrow sense (as opposed to phonemics and morphology). It forms part of an attempt to construct a formalized general theory of linguistic structure, and to explore the foundations of such a theory. (Chomsky 1957,5)

This description emphasises the incomplete nature of the project: 55 is merely 'part of' a much more elaborate undertaking which involves the construction of 'a formalized general theory of linguistic structure'. In the above extract, Chomsky's brief summary is strikingly self-deprecating, almost self-effacing. Nonetheless, some of the earliest reviewers identified radical tendencies in 55. Although Robert Lees' review is usually the only one that is still discussed today, there were many other contemporaneous assessments. For instance, William Haas reviewed the text for Archivům Linguisticum in 1958, and he noted in particular that 'Mr Chomsky does look on grammar itself as some kind of mechanical device', an approach which (Haas acknowledges) had 'attracted some valuable new friends to grammatical studies'. Although he criticised Chomsky for making 'extravagant claims' (e.g., concerning the relationship between syntax and semantics), he declared that 55 was 'a real advance' over previous studies in linguistics (Haas 1958,50-54). By contrast, in a review that appeared in the Inter- national Journal of American Linguistics , Charles Voegelin provided an amusing and insightful assessment of 55. He noted that Chomsky's discussion was 'broadly ori- ented', and he expressed doubts as to the originality of the approach adopted: at one point he refers to 'the new theory (if it is new), the revolutionary theory (if it is revolutionary)' (Voegelin 1958,229). Having compared the task of constructing a linguistic theory to the task of constructing a mousetrap, Voegelin concludes his review with

8 For a summary of the various uncertainties, see Noordegraaf (2001).

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms Syntactic Structures and Recursive Devices: A Legacy of Imprecision 303 four questions, all of which convey his doubts about the kind of project that Chomsky had delineated: (i) Are transforms in syntax new? (ii) Will they start a Copernican revolution within linguistics? (iii) Is transform grammar incompatible with linguistic typology? (iv) Is transform grammar elicitable? (Voegelin 1958,229-231). By contrast with entertaining and acute contemporaneous reviews such as these, Robert Lees' assessment is far more sycophantic:

[...] Chomsky's book on syntactic structures is one of the first serious attempts on the part of a linguist to construct within the tradition of scientific theory- construction a comprehensive theory of language which may be understood in the same sense that a chemical, biological theory is ordinarily understood by experts in those fields. It is not a mere reorganisation of the data into a new kind of library catalogue, nor another speculative philosophy about the nature of Man and Language, but rather a rigorous explication of our intuitions about our language in terms of an overt axiom system; and it may well provide an opportunity for the application of explicit measures of simplicity to decide preference of one form over another form of grammar. (Lees 1957, 377-378)

In many respects, Lees' description initiated the critical response which viewed SS as marking a distinct break with prevailing linguistic traditions. The claim that Chomsky's work was both profoundly innovative and unremittingly scientific encouraged the view that SS had triggered a revolutionary break (of some kind) with the post-Bloomfieldian linguistics that had dominated since the 1940s. As Frederick Newmeyer expressed it in 1980: '[SS] completely shattered the prevailing structuralist conceptions of linguistic theory' (Newmeyer 1980, 35). In Newmeyer's account, SS inaugurated a brand-new approach to linguistic theorising which became known as generative grammar (GG). For many years, this remained the conventional historiographical stance, but, since the early 1990s, numerous linguistics have questioned this 'revolutionary' interpretation. In his monograph Grammatical Theory in the United States from Bloomfield to Chomsky 1993, Peter Matthews identified many points of contact which related GG to the work of the post-Bloomfieldians. Viewed in this way, Chomsky's work appeared to emerge naturally from the linguistic culture of the 1940s. As Konrad Koerner put it in 2002 'it appears that the more closely we look into the discussion going on in American linguistics during the late 1940s and early 1950s, the more obvious it becomes that what many people today want to call a 'revolution', namely the movement said to have been initiated by the publication of Chomsky's Syntactic Structures , was at most an evolution of then current work' (Koerner 2002,180). Also, in my own book Linguistics and the Formal Sciences (2006), I argued at length that GG is more accurately viewed as an astonishing act of synthesis which brought together techniques and ideas from such diverse fields as mathematics, formal logic, analytical philosophy, and (of course) post-Bloomfieldian linguistics. Curiously, Chomsky's own subsequent assessment of the importance of SS is far less extreme than that of his more devoted admirers. Indeed, his own evaluation is largely in keeping with the revisionist accounts that have appeared from the early 1990s onwards. In particular, he has consistently acknowledged that SS was (in effect) merely an informal summary of techniques and theoretical approaches that had been elaborated in much greater detail in then unpublished (but widely circulated) texts such a Morphophonemics of

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Modern Hebrew (1951) and The Logical Structure of Linguistic Theory (1975 [1955]), and in published articles such as 'Systems of Syntactic Analysis' (1953), 'Logical Syntax and Semantics: their linguistic relevance' (1955), 'Semantic Considerations in Grammar' (1955), and 'Three Models of the Description of Language' (1956). This is how Chomsky described SS in a 1997 interview:

At the time Mouton was publishing just about anything, so they decided they'd publish it along with a thousand other worthless things that were coming out. That's the story of Syntactic Structures : course notes for undergraduate science students published by accident in Europe. (Dillinger & Palácio 1997,162-163)

The suggestion that SS is best classified as a collection of pedagogical notes, aimed at undergraduates, that were published by chance is certainly plausible. For instance, although Lees (and others) claimed that SS provided 'a rigorous explication' of the ideas it presented, in the text itself Chomsky frequently apologises for the non-rigorous and sketchy nature of the discussion: '[...] this fact may be obscured by the informal- ity of the presentation' (Chomsky 1957,5); '[s]eemy 'Three models for the description of language' [...] for proofs about this and related theorems about relative power of grammars' (Chomsky 1957,30); '[...] this is a complex matter that requires a much more detailed development of transformational theory that we can give it here' (Chom- sky 1957,77). This consistently and conspicuously informal tendency manifests itself elsewhere too. Since SS is a fairly short academic monograph, it contains a surpris- ingly large number of typos and small errors, all of which suggest poor and/or rushed proof-reading and copy-editing. Here are some of the slips that appeared in the first edition:

- '[...] from the viewpoint of transformational nalysis' (Chomsky 1957,68) - 'the boy studying in the ljbrary' (Chomsky 1957,81) - 'SYNTACS AND SEMANTICS' (Chomsky 1957,93, 99; running chapter heading) - '7^' (Chomsky 1957,81; this should be Ts°ubb)

Seemingly, then, SS was an informal, error-strewn, high-level summary of existing research, which was suitably dumbed-down for an undergraduate audience. Nonethe- less, despite this, it was considered by certain professional linguists to be so rigorous and so revolutionary that it shattered prevailing notions in linguistic theory. This is an extraordinary state of affairs, and in order to appreciate how it could have been possible, it may be useful to reflect upon the way in which Chomsky presented recursion in SS, since this provides a convenient case-study of his more general methodology. Recursion is first mentioned overtly in chapter 3 of SS, during a discussion of finite state machines (FSMs). Chomsky notes that he had encountered FSMs in Claude Shannon and Warren Weaver's The Mathematical Theory of Communication 1949, although recursion is not mentioned in this particular text. He states that an FSM is a 'familiar communication theoretic model for language' (Chomsky 1957,18), which suggests that mathematically-inclined linguistics in the 1950s knew something of contemporaneous Information Theory. He goes on to describe an FSM as having 'a finite number of different internal states', including 'an initial state ' and 'a final state9, and probabilities are assigned to each transition from state to state (Chomsky 1 957, 1 8-20). Any language that can be generated by an FSM of this kind is

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Fig. 1 FSG with a closed loop

referred to as 'finite state language ' and the FSM itself constitutes a Finite State Gram- mar (FSG)(Chomsky 1957,19). Crucially, FSGs can contain 'closed loops' which enable the grammar 'to produce a infinite number of sentences' (Chomsky 1957,19). The following example of a simple FSG with a closed loop is presented (Chomsky 1957,19): This FSG permits the generation of an infinite set of sentences: 4he old man comes' , 4he old old man comes', 'the old . . .old man comes', 'the old men come', 'the old old men come', 'the old . . .old men come', and so on. Revealingly, Chomsky later claims that

[...] the assumption that languages are infinite is made in order to simplify the description of these languages. If a grammar does not have recursive devices (closed loops, as in Fig. 1, in the finite state grammar) it will be prohibitively complex. If it does have recursive devices of some sort, it will produce infinitely many sentences (Chomsky 1957,23-24).

This short passage is troublesome for a number of reasons. For a start, it is odd to claim that the potentially infinite generative capacity of natural languages must be assumed merely to simplify the way in which such languages are analysed by linguists. This suggests that the assumption is prompted merely by practical convenience. However, more fundamentally, Chomsky offers no definitions either of the noun 'recursion' or of the phrase 'recursive devices', and the closed loop example is presented as a single instance of such a device within the FSG framework. In addition, in the light of subsequent developments within GG, it is of particular interest that 'recursive devices' are here discussed as being merely useful , not essential , components of linguistic theory: they enable an infinite number of sentences to be generated without the grammar becoming prohibitively complex, but there is no suggestion that they are unavoidable components. So, in Chomsky's theoretical framework, anything that permits the generation of an infinite set of grammatical sentences can be referred to as being 'recursive'. Although he first discusses recursive devices in the context of FSGs, he returns to this topic at the start of chapter 4 when (in a long footnote) he briefly considers a 'system of

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms 306 M. Tomalin word class analysis' that had been proposed by F. Harwood in 1955. He notes that Harwood's system is 'similar' to a phrase structure grammar, but he rejects Harwood's approach, partly because 'it could not generate an infinite language with a finite grammar' (Chomsky 1957,27). Since 'recursive devices' enable 'infinitely many sentences' to be produced, seemingly Harwood's system is inadequate since it does not explicitly contain any such devices. Apparently, then, recursion plays a crucial role in the grammars that are developed in 55, but what does 'recursion' actually denote? 55 offers no rigorous definitions, and the most precise presentation is the closed loop in a FSG (discussed above). Since 55 is essentially a rough summary of earlier work, then Chomsky's pre-55 publications can be of some use. For instance, in his 1956 paper 'Three Models for the Description of Language' ('TMDL') contains the following passage:

In general, the assumption that languages are infinite is made for the purpose of simplifying the description. If a grammar has no recursive steps (closed loops, in the model discussed above) it will be prohibitively complex - it will, in fact, turn out to be little better than a list of strings or of morpheme class sequences in the case of natural language. If it does have recursive devices, it will produce infinitely many sentences. (Chomsky 1956,115-116)

Certainly, 'TMDL' offers a more detailed technical/mathematical account of finite state, phrase structure, and transformational grammars, and since Chomsky lists 55 as 'to appear' in the references, it is difficult to establish precisely which text was completed first. It is most likely, though, that the more detailed account (i.e., 'TMDL') was the primary and initial analysis and that this more comprehensive research was simply summarised informally in 55. Crucially, though, there is no explicit discussion of closed loops in 'TMDL', and no further discussion of recursion. Throughout 55, Chomsky states repeatedly (often in footnotes) that many of the ideas he is considering were presented in greater detail in his mimeographed The logical structure of linguistic theory (1975[1955]; published 1975; from henceforth LSLT). In LSLT recursive devices are used in a number of distinct ways. For example, having specified a set of n conversions or 're-write' rules (e.g., X -» Y) in the phrase structure component of the grammar, Chomsky notes that certain rules can be recursive:

[w]hen we turn to the level of phrase structure, we find that certain rules may have a recursive character. Thus noun phrase (NP) might be analyzed in such a way that one of its components may be a NP as in such sentences as "the man who made the discovery is my brother" [...] (Chomsky 1975[1955], 171-172)

Here 'recursive' is associated explicitly with self-similar syntactic embedding - that is, a phrase (NP2) is embedded within a phrase of the same type (NPi) creating structures such as [npj the man who made [np2 the discovery]], and conversions that take the form NP' -> . . .NPi. . . create structures of this kind. However, Chomsky later claims in LSLT that conversions themselves (whether recursive or not) can be applied recursively too:

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[...] we can understand the linear grammar to be the sequence of conversions

Si, . . . , S„, Si, . . . , S„, Si, . . . , Sn, sively produced by the linear grammar Si, . . . , Sn. We define a proper linear grammar as a linear grammar which is so constructed that it is impossible to run through it over and over again vacuously. (Chomsky 1975[1955], 194-195)

This terminological conflation is regrettable, since the repeated (or iterated) application of a set of rules is not necessarily associated with the notion of self-similar syntactic embedding. Towards the end of LSLT , Chomsky considered whether 'recursive parts of the grammar' should be permitted in the phrase structure component, or whether they should only appear in the transformational component (Chomsky 1975[1955], 5 1 6-5 1 8), but although LSLT offers various examples of recursion in linguistic theory, no definitions are presented, and it is assumed throughout that the term in unprob- lematical. It should be noted, though, that, in Chomsky's work from the 1950s, the word recursion (and its cognate forms) is generally used to refer to particular kinds of constructional procedures rather than to particular kinds of structures, so, to this extent, he avoids at least this confusion. In my 2007 article, I suggested that when the term 'recursion' began to be used by linguists in the 1950s it was already riddled with ambiguities, and I identified five possible interpretations. On reflection, this initial assessment was too modest. Seemingly, when SS appeared, there were as many as nine distinct (though sometimes related) meanings:

- II: recursion = inductive definition (à la Peano 1889) - 12: recursion = primitive recursion (à la Godei 1986[1931], Bar-Hillel 1953) - 13: recursion = general recursion (à la Godei 1986[1931]) - 14: recursion = À-definability (à la Church 1936) - 15: recursion = computability (à la Turing 1936) - 16: recursion = the recursive enumeration of set membership (à la Post 1944) - 17: recursion = self-similar syntactic embedding (à la Chomsky 1975[1955]) - 18: recursion = iterated rule application (à la Chomsky 1975[1955]) - 19: recursion = closed loops in FSGs (à la Chomsky 1957)

Since 'recursion' was clearly a problematical term, it is intriguing that Chomsky did not attempt to clarify its usage in SS. He suggests that recursive devices are essential to linguistic theory, but he does not offer any definitions, and therefore his discussion retains the ambiguities that he had inherited from the mathematical and logical literature. This is just one example of the way in which SS draws upon the literature and terminology of the formal sciences, but does not succeed in presenting a stable 'formalized' analytical linguistic theory: SS passively inherits the contemporaneous confusion concerning the term 'recursion', and the obfuscation is unquestioningly imported from one academic domain into another. In a response to my 2007 article, David Lobina and José García- Albea have suggested that the analysis presented there is too restricted since it only focuses upon the linguistics literature, and does not offer a survey of the range of connotations of 'recursion' that were encountered in such fields as computer science, psychol- ogy, artificial intelligence, and so on (Lobina and Garcia-Albea 2009). This is a

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms 308 M. Tomalin reasonable point, but (to my knowledge) no one has yet undertaken an exhaus- tive analysis of the way in which 'recursion' was deployed in these different academic fields during the period 1930-1960. Such an exploration would involve blowing the dust off of many books and journal articles that were published in a wide range of different fields, and an historiographical assessment of this sort would certainly provide considerable insights into the emergence of cognitive science as an identifiable nexus formed from inter-related disciplines. For instance, as computer science flourished during the 1950s, numerous textbooks were published which presented the rudiments of computer architecture, programming languages, and the like, and frequently these discussions introduced recursive techniques. In his influential text Digital Computer Programming (1957), Daniel McCracken focused on the use of a 'recursion formula' while discussing the internal operation of an interpretative routine, and he presented the following example (McCracken 1957,185-186):

term 0 = 1 X term n = -(term n - 1) (2) n

In (2), the function 'term' explicitly calls itself in the second step, and these equations recall Godei 's primitive recursion equations (given as (1) above). Since computer science emerged out of the logical and mathematical literature of the 1930s, it was perhaps inevitable that Gödel 's formulation of functions that call themselves should have become the dominant framework for recursion. Crucially, though, the clear distinction between iteration and recursion (which is still standardly taught in undergraduate computer science courses) had become canonical by the late 1950s. In An Introduction to Digital Computing (1963), Bruce Arden presents a range of definitions which he takes to be entirely well-established and uncontroversial; he defines recursive functions as '[functions whose definitions include direct or indirect reference to the function being defined', and he offers the following examples (Arden 1963,300-301): Iteration: n /(и) = П<, n > 0 ¿=1 (3) /(0) = 1

Recursion: /(") = nf(n - 1), n > 0

/(0) = 1

As these examples make clear, when SS appeared in 1 957, the use of the term 'recursion' was already well-established in other disciplines that formed part of the cluster from out of which cognitive science emerged. However, most of the linguists who discussed recursion did not offer clear and concise definitions, and this perpetuated the prevailing ambiguities. Unfortunately, the confusions and vagaries that characterised

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms Syntactic Structures and Recursive Devices: A Legacy of Imprecision 309 discussions of recursion in the 1950s have remained prevalent ever since, and, given the recent revival of interest in recursive techniques, they have (if anything) become more problematical during the past decade.

4 Vagaries and Cross-Purposes

As mentioned earlier, recursion has recently become a popular topic (once again) in the linguistics literature. In 2002, Hauser, Chomsky, and Fitch defined the faculty of language in the narrow sense (FLN) as being an abstract linguistic computational system (independent of the other systems with which it interacts and interfaces) which 'takes a finite set of elements and yields a potentially infinite arrangement of discrete expressions' (Hauser et al. 2002,1571; henceforth HCF). Given this assumption, HCF claimed that

FLN - the computational mechanism of recursion - is recently evolved and unique to our species [...] we propose in this hypothesis that FLN comprises only the core computational mechanisms of recursion as they appear in narrow syntax and the mappings to the interfaces. (Hauser et al. 2002,1573)

This implies that FLN (a unique species-specific property) is primarily (and perhaps exclusively) a recursive device of some kind. However, in a manner that is eerily reminiscent of the vague references to recursive elements in early texts such as SS, the informal discussion in HCF offers no detailed definitions, so it is impossible to determine what exactly 'recursion' means in this context. Marc Hauser has recently attempted to clarify things a little by distinguishing between recursive and combinatorial processes:

Recursion is an iterative operation, in which a rule is called up repeatedly to create new expressions, be they embedded phrases within a sentence, new musical scores with repeating themes, or tools within tools (for example, a Swiss army knife). Each expression has a unique interpretation or function depending on the arrangement of the elements. By contrast, combinatorial operations allow discrete elements to be unified and ordered, thus creating new ideas, which could be expressed as novel words (Walkman from walk and man) or novel musical forms. (Hauser 2009,192)

In the light of the foregoing discussion, it should be clear that hazy informal descriptions of this kind are of very little use. For instance (as shown above), in the logical, mathematical, and computer science literature recursion and iteration are generally considered to be entirely different things (as contrasted in equation pairs (3) and (4)). It is not surprising, therefore, that since HCF appeared in 2002, many linguists have criticised the discussion of recursion presented there. Steven Pinker and Ray Jackend- off tried to highlight the confusion when, claiming that phonological structures are not recursive, they observed that:

[...] the segmental/syllabic aspect of phonological structure, though discretely infinite and hierarchically structured, is not technically recursive. (As mentioned, HCF use "recursion" in a loose sense of concatenation within hierarchically

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embedded structures). Recursion consists of embedding a constituent in a constituent of the same type, for example a relative clause inside a relative clause (i a book that was written by the novelist you met last night). This does not exist in phonological structure: a syllable, for instance, cannot be embedded in another syllable. (Pinker and Jackendoff 2005,10)

For Pinker and Jackendoff, then, 'recursion' simply denotes self-similar syntactic embedding, but they do not acknowledge that this is only one of many possible interpretations of the word 'recursion* . As mentioned earlier, Dan Everett is another person who has repeatedly rejected the HCF claim that 'recursion* is fundamental to natural language. Specifically, he has argued that the Amazonian language Pirahã is characterised by 'the absence of embedding' (Everett 2005,621), where 'embedding' means that one constituent contains a constituent of the same kind:

English expresses the content of verbs such as "to say", "to think", and "to want" as clausal complements [here the subscript s labels the embedded clauses as theory-neutral]: "I said that b John will be here]", "I want [s you to come]", "I think [s it's important]". In Pirahã the contents of such verbs, to the degree that equivalent verbs exist at all, are expressed without embedding. (Everett 2005,628)

When he turns to another example, grammatical possession, he comments as follows:

Neither the declarative nor the interrogative form of recursive possession is acceptable. No more than one possessor per noun phrase is ever allowed. [. . .] A cultural observation here is, I believe, important for understanding this restric- tion. Every Pirahã knows every other Pirahã, and they add the knowledge of newborns very quickly. Therefore one level of possessor is all that is ever needed. (Everett 2005,630)

So, nested possessive structures (e.g., Kó'oi's son* s daughter) are described as being 'recursive', thus implying (once again) that recursion denotes self-similar syntactic embedding (i.e., a constituent of one kind is embedded within a constituent of the same kind). Everett claims that Pirahã does not contain structures like this, although he assumes that the level of embedding must be greater than 1 since he does not clas- sify structures such as Kó'oi's daughter as being recursive despite the fact that they can be analysed as nested DPs. Pinker, Jackendoff, and Everett (to select just these three) are largely in agree- ment that recursion (in the context of linguistic theory) denotes self-similar syntactic embedding. The problem is that many other linguists (including Chomsky) adopt a very different view. Although 'recursion' remains undefined in HCF, Chomsky provided a definition (of some sort) in his 1995 monograph The Minimalist Program . Having introduced the computational component of natural language (C hl' he comments that 'the operations of Chl recursively construct syntactic objects' (Chomsky 1995,226). Specifically, syntactic objects (SOs) are defined as follows (Chomsky 1995,243):

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Definition 2: Syntactic Objects 1. Lexical items 2. К = {y {a, ß }}, where a, ß are syntactic objects and у is the label of К

All lexical items (Lis) are SOs, and further SOs can be created by combining existing SOs in a principled manner. Chomsky explicitly notes that it is clause 2 which provides the 'recursive step' (Chomsky 1995,243), and (presumably) this is the 'recursive' procedure to which HCF refers. If this is the case, then a simple declarative clause such as 'the man laughs' is created by means of recursion, even though it does not contain self-similar syntactic embedding: Merge has combined the determiner and the noun, and then has combined this SO with the verb. Given this more general definition of recursion, Everett's counter-claims become fallacious, since Pirahã certainly contains declarative clauses which have been constructed by merging SOs with other SOs. However, it should be more than clear by now that Pinker, Jackendoff, and Everett (on the one hand) and Hauser, Chomsky, and Fitch (on the other) are simply talking about different things, despite the fact that they all use the word 'recursion' : Definition 2 does not exclusively create embedded structures, and therefore, from a minimalist perspective at least, recursion does not necessarily denote self-similar syntactic embedding. Given these differences, it is not surprising that many recent discussions of such matters have been frequently confused and confusing. In a 2006 Language Log posting about Everett's work (for example), Mark Liberman commented as follows:

By "recursion", HCF mean "computational mechanisms . . .providing the capacity to generate an infinite range of expressions from a finite set of elements". "Recursion" in this sense goes beyond the simple combinations of modifiers and heads ("red" + "cow" -> "red cow"), or subjects and verbs ("Joan" -f "disagree" -> "Joan disagrees"), or any other construction that doesn't involve embedding a complex element repeatedly inside another element of the same type. Non-recursive constructions (like modifier+head) are very useful, and such embeddings multiply the set of messages that you can make out of a finite set of elements, but they don't "generate an infinite range of expressions" unless they operate recursively. (Liberman 2006)

The problem with this is that the kind of 'recursive' process presented in Definition 2 does indeed generate 'the simple combinations of modifiers and heads'. In fact, given Definition 2, modifier/head SOs (e.g., red cow ) and SOs that contain embedded structures are generated in exactly the same way: two SOs are Merged, and the features associated with the lexical items and functional categories determine the structure created. Seemingly, several linguists have become frustrated by these laughable (and entirely avoidable) confusions. Derek Bickerton, in particular, has recently criticised the HCF stance on recursion, suggesting that HCF avoided the term 'iteration' because it could not be claimed that this kind of constructive procedure had distinctive recently-evolved species-specific properties.9 However, even Bickerton's discussion, which seeks to tidy up the terminological mess, introduces further inaccuracies. For example, he

9 Bickerton (2009); especially the discussion on pp. 536-537.

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This content downloaded from 194.94.133.193 on Thu, 23 Apr 2020 06:16:45 UTC All use subject to https://about.jstor.org/terms 312 M. Tomalin states (incorrectly) that the use of recursion in syntactic theory began with SS , and he states (incorrectly) that The Minimalist Program (1995) eliminated the need for it:

The irony is that Chomsky is the sole person responsible both for the appearance and disappearance of recursion. His 1957 analysis, created the notion that syntax required recursion. His 1995 analysis removed the necessity for assuming recursion. So how is it that Chomsky in HCF is still proposing recursion as the central, perhaps sole content of FLN? (Bickerton 2009,536)

This bizarre question indicates clearly the way in which the careless use of terminology such as 'recursion* can impede profitable academic debate. Given the fact that Chomsky explicitly refers to the second step in Definition 2 as being 'recursive', it is strange indeed to assert that the discussion in The Minimalist Program signals the demise of recursion in syntactic theory. Such a claim can only be made if 'recursive' is understood to refer to something other than the kind of constructional process that Chomsky defines.

5 Conclusion

This article has attempted to show that recent reflections upon the role of recursion in linguistic theory have been dogged and impeded by a pervasive but ultimately unneces- sary terminological befuddlement. This confusion was already well-established before SS was published in 1957, and Chomsky's influential work did nothing to clarify things, but the continued predominance of such imprecision in recent publications is profoundly frustrating since it could easily be avoided if only the analytical terminology were used with greater care. It is important, for instance, to distinguish carefully between 'processes' and 'structures' whenever recursion is considered. While some linguists use 'recursion' to refer to the kind of constructional procedure captured in Definition 2 above, others use it to refer exclusively to certain kinds of structural configurations (e.g., nested clauses) - but these two usages are not equivalent. In the domain of computer science, although the pair of simultaneous equations given in (3) would usually be classified as being 'recursive' (in accordance with interpretation II), no one would ever claim that the factorial 4! was a 'recursive number'. However, in modern linguists such distinctions are rarely observed with any consistency: sometimes recursion refers to processes, sometimes to structures, occasionally to both, and this sloppiness only deepens the foggy terminological haze. Crucially, the creation of self-similar syntactic embedding does not (of necessity) require self-referential constructional processes: the clausal complement in a structure such as 'I think that he will come' could be inserted iteratively. Given the wide-spread bewilderment concerning such matters, it would be helpful if some of the most important questions concerning the role of recursion in syntactic theory were re-expressed more precisely. Three questions that have been asked in many different ways in recent years are:

- Ql. Does language = interfaces + recursion? - Q2. Are recursive structures manifest in all languages? - Q3. How did recursion evolve (in the context of language use)?

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Even if the domain of enquiry is simplified drastically by reducing the number of possible interpretations of the word 'recursion' from 9 possibilities (11-19) to only two options (specifically, 17 and 18), then the number of questions proliferates:

- Q4. Does language = interfaces + self-similar syntactic embedding? - Q5. Does language = interfaces + iterative construction? - Q6. Is self-similar syntactic embedding manifest in all languages? - Q7. Is iterative construction manifest in all languages? - Q8. How did self-similar syntactic embedding evolve (in the context of language use)? - Q9. How did iterative construction evolve (in the context of language use)?

In total, of course, if all 9 interpretations were used, there would be 27 distinct questions, and it should be clear by now that many of the linguists who claim to disagree about the role of recursion are actually attempting to answer different questions. Ever- ett and Chomsky would (presumably) both answer 'No' to Q4 (for different reasons). Meanwhile, Qs 8 and 9 are currently unanswered questions which focus on distinct, though related, aspects of cognitive function. Although they still lack clarity, at least formulations Q4-9 are more specific than the inherently ambiguous questions Ql-3. Over 50 years have now passed since SS first appeared, and it continues to be a frustrating text. It makes many bold and intriguing claims about linguistic structure, yet it rarely substantiates them; it presents several fascinating theoretical approaches to the analysis of language, yet many important ideas and approaches are left undeveloped and undefined. In particular, as this article has shown, recursion is dealt with in such a scratchy way in SS that no certain conclusions can be reached: it is never defined, and though various recursive devices are mentioned, they are only ever discussed informally. In summary, then, when SS appeared in 1957, 'recursion' was a much used, much discussed, fundamentally ambiguous term that caused extensive but needless misunderstandings. Unfortunately, the popularity of the semi-formal expository style adopted both in SS, and in later publications such as Aspects of the Theory of Syntax (1965), ensured that, in 2010, the term 'recursion' continues to be a much used, much discussed, but fundamentally ambiguous term that causes extensive but needless misunderstandings. This is a deplorable state of affairs, it is hard to understand why such sloppiness should be allowed to masquerade as serious linguistic research. While it was perhaps inevitable that such indeterminacies should have bedevilled a text such as SS (which, in Chomsky's own assessment, was simply an informal overview aimed at undergraduates), it is absurd that similar vagaries should continue to be disseminated in serious academic publications over half a century later. If linguists were not so easily contented with opaque, slip-shod, and chattily informal presentations, then such misunderstandings could be avoided. Over 20 years ago now, in a Carrolian mood, Geoff Pullum mischievously anticipated the imminent demise of formal linguistics. He predicted that it would soon meet the Boojum and would 'softly and suddenly vanish away' (Pullum 1991,48). Pullum may well have been teasing us, but, when one reads (with gritted teeth) the plethora of recent publications by prominent linguists which purport to reassess the role of recursion in syntactic theory, one certainly experiences a profound sense of grief and loss that is akin to mourning.

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