Language Sciences 45 (2014) 5670Contents lists available at ScienceDirectLanguage Sciences
journal homepage: www.elsevier .com/locate/ langsciWhat linguists are talking about when talking about.
David J. LobinaFaculty of Philosophy, University of Oxford, Radcliffe Humanities, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG,England, United Kingdoma r t i c l e i n f o
Article history:Received 2 March 2014Received in revised form 18 May 2014Accepted 28 May 2014Available online
Keywords:Recursive proceduresRecursive rewriting rulesEmbedding operationsRecursive structuresFour conflations among the aforementionedconstructsE-mail address: firstname.lastname@example.org
http://dx.doi.org/10.1016/j.langsci.2014.05.0060388-0001/ 2014 Elsevier Ltd. All rights reserved.a b s t r a c t
A historical look at the manner in which recursion was introduced into linguistics,including how it was used thereafter, shows that Chomsky, the scholar who popularisedthe use of recursive techniques in linguistics, has always understood this notion to be acentral feature of generative procedures, much as it was treated in mathematical logic inthe 193050s. Recursion is the self-reference property that underlies all types of recursivefunctions; recursive definitions (or definitions by induction), in addition, justify everystage of the computations effected by computational procedures such as Post productionsystems or set-operators like merge, making recursion the central feature of a generativegrammar. The contemporary literature, however, has confused this recursive property of agrammar with other constructs, such as self-embedded sentences, self-embedding oper-ations, or certain rewriting rules, thereby obscuring the role of recursion in the theory oflanguage. It is here shown that this is the result of the literature implicitly endorsing anumber of unwarranted conflations, four of them analysed here. It is concluded that mostof the discussion on the centrality and uniqueness of recursion in human language and/orgeneral cognition has been confusing and confused for very fundamental reasons; a storyof conflations, in a nutshell.
2014 Elsevier Ltd. All rights reserved.1. In the way of a justification
The world possibly doesnt need another article on the role of recursion in language. Ever since Hauser et al. (2002)hypothesised that recursion may be the only sui generis feature of human language, a barrage of publications has made itsway into print, many of these engaging very different aspects of the hypothesis. It has been claimed by some, supposingrecursion to refer to self-embedded sentences such as the cat [the dog [the mouse bit] chased] ran away, that recursion couldntpossibly be the central property of language on account of the apparent fact that not all languages exhibit such structures(among others, Parker, 2006; Everett, 2012). Others, equating this recursive property with an embedding or self-embeddingoperation, have pointed out that such an operation appears to be part of the grammar of all languages (e.g., Hauser, 2009 and,to some extent, Nevins et al., 2007). Many more issues have been discussed in the literature, of course; my intention here,however, is not to catalogue them all but to organise our knowledge of all things recursive; that is, I offer a conceptualanalysis, and high time for such an analysis it is.
Granted, similar analyses have been offered before, the most prominent perhaps being those of Tomalin (2007) and Fitch(2010); as I will show below, however, there are some problems with these two publications (in terms of both the quality andthe quantity of what they say), not least the fact that they are not entirely compatible with each otherdindeed, they takek.
D.J. Lobina / Language Sciences 45 (2014) 5670 57recursion to be different thingsdand that is an issue that must be addressed. In any case, neither of these two papers, or anyother from the literature for that matter, seem to have hadmuch of a positive effect; much confusion remains, as evidenced ina recent paper that attempts to clarify what recursion is by focussing on mathematical logicdnamely, Watumull et al.(2014)dbut which commits multiple, and very obvious, mistakes (surprisingly, neither Tomalin, 2007 nor Fitch, 2010 arereferenced).
By quoting a text of Gdels (1931) inwhich the noted mathematician offers a definition of what he therein calls recursivefunctions (these are now known as the primitive recursive class; Davis, 1965, p. 4), Watumull et al. (2014), WEA in thisparagraph and the next, identify three criterial properties of what they call the primitive notion of recursion (p. 2): a) arecursive function must specify a finite sequence (Turing computability, WEA claim); b) this function must be defined interms of preceding functions, that is, it is defined by induction, which WEA associate with strong generativity (i.e., thegeneration of ever more complex structure); and c) this function may in fact just reduce to the successor function (that is,mathematical induction, which WEA associate with the unboundedness of a generative procedure). However, Gdels textdoesnt say that at all, as the full quote demonstrably shows (the italicised words below mark the material WEA purposelyomit from their citation of Gdel (1931); Imquoting fromDavis (1965), which offers a different translation from the oneWEAuse, but this doesnt affect my point in the slightest):1 Thainducti
2 In whopefuA number theoretic function f is said to be recursive if there exists a finite sequence of number-theoretic functionsf1,f2,.,fn which ends with f and has the property that each function fk of the sequence either is defined recursivelyfrom two of the preceding functions, or results [footnote not included] from one of the preceding functions by substi-tution, or, finally, is a constant or the successor function x 1 (pp. 145; underline in the original, my italics).In this quote, Gdel is simply identifying which functions from a finite list are to be regarded as primitive recursive (cf. thedefinition of this class in Kleene, 1952, pp. 2203, which is very similar indeed)1; what the text is clearly not offering is acombination of properties subsuming the recursion conceptdand in any case, by omitting the material in italics, WEA end upwith three criterial properties by design (whatever happened to substitution or a constant?). More importantly, there are nogrounds for identifying Gdels finite sequence of functions with Turing computability (a notion that was unavailable to Gdelat the time in any case), and it is certainly a mistake to equate a definition by induction with strong generativity, or thesuccessor function with mathematical induction and/or unbounded computationsdall these are related but clearly inde-pendent concepts. Having said that, WEAs focus on mathematical logic is to be welcomed, but the clear deficiencies in thatpaper must also be addressed.2
Given the current state of affairs, then, no-one could be faulted for wondering what it all means; or what it could all mean,in any event. Despite the documentary density, I advance that there is some novelty to be had, especially for the philosopher,and of some worth to the engorged literature to boot. To that end, I start this essay with a brief description of how recursionhas been understood within the formal sciences, especially in the fields of mathematical logic and computer science. I thenchronicle both the introduction of recursion into linguistic studies and how its role within generative grammar has evolvedover the ensuing decades; to this end, I will focus on Noam Chomskys writings, the scholar responsible for introducingrecursive techniques into linguistics. Rather surprisingly, given the prominence of this notion in the literature, such an ex-ercise is yet to be done, let alone offered for reflection. This type of analysis, however, brings to light a number of importanthistorical facts: a) recursionwas first employed in the 1950s and 60s in the samemanner as it was used inmathematical logic,a field that exerted a great influence on linguists at the time; and b) its application has not translated much over the years, atleast as far as Chomskys individual writings are concerned. Building upon that, I then demonstrate that the confusion sur-rounding this concept is not solely a matter of imprecision, as claimed by Tomalin (2011), but a story of conflations: betweenrecursive mechanisms and recursive structures; namely, between the self-reference so typical of recursive functions and self-embedded sentences; between the recursive applications of specific rules of Post production systems and self-embeddingoperations; and, lastly, between what an operation does and how it applies. As a result, I will conclude, most of the dis-cussion in the literature as to the centrality and uniqueness of recursion in natural language has centred on issues (such aswhether all languages exhibit self-embedded sentences) that have little to do with the introduction of recursive tools intolinguistics, let alone the reasons for introducing such techniques in the first place. As such, then, some of the strongest claimsto be found in the literature are either fallacious or quite simply misplaced.2. What is recursion, then?
As Brainerd and Landweber (1974) put it, it is useful to define functions using some form of induction scheme., a generalscheme.which we call recursion (p. 54). This is in fact consonant with the original interpretation of recursion withinmathematics as being part of a definition by induction, as chronicled by Soare (1996). Also known as a recursive definition, itconsists in defining a function by specifying each of its values in terms of previously defined values (Cutland, 1980, p. 32); at is, what Gdel is saying is that if we have a list of functions, any one function from this list will be defined as (primitive) recursive if it is defined byon from previous functions, OR is substituted by some of them, OR is the constant function, OR is the successor function.hat follows, I will not discuss the finer details of WEAs mistakes, as this would take us far outfield; instead, I will offer my own narrative, which
lly will provide a much better grounding for the issues at stake here.
D.J. Lobina / Language Sciences 45 (2014) 567058self-referential characteristic (Tomalin, 2006, p. 61). As an example, take the factorial class (fact(n) n n 1 n 2.1),which can be recursively defined in the two-equation system so common of such definitions, as follows: if n 1, thenfact(n) 1 (base case); if n > 1, then fact(n) n fact(n 1) (recursive step). Note, then, that the recursive step involvesanother invocation of the factorial function. Thus, in order to calculate the factorial of, say, 4 (i.e., 4 3!), the function mustreturn the result of the factorial of 3, and so on until it reaches the factorial of 1, the base case, effectively terminating therecursion.
There are a number of different types of recursive functions (the primitive, the general, the partial), and these are allrecursive for the very same reason: they are all underlain by the induction scheme, even if they encompass different inputoutput pairs (hence, the different classes).3 As is well-known, these functions proved to be very important in the 1930s andbeyond as a means to formalise the class of computable functions. Church (1936), in particular, identified the generalrecursive functions with the computable class, but this proposal was not entirely correct (see Soare, 1996, pp. 28991; Sieg,1997 for details). Kleene (1938) replaced the general class of recursive functions with the partial class for the purposes Churchhad in mind, and this type of recursive functions was further polished byMcCarthy (1963), first, and recently byMoschovakis(2001). From a completely different perspective, that of a generative system that lists a set of integers rather than computing afunction, Post (1943) introduced his canonical systems, a construct that directly stems from the generalisation bypostulation method of an earlier work (namely, Post, 1921). According to Post (1943), canonical, or production, systems canbe reduced to what he calls a normal form, which can be described in terms of the mapping from gP to Pg0, where g standsfor a finite sequence of letters (the enunciations of logic) and P represents the operational variables manipulating theseenunciations (p.199). Crucially, thewhole approach naturally lends itself to the generating of sets by themethod of definitionby induction (p. 201), a fact of production systems qua formalisation of a computational system that has moved Sieg (1997) toclaim that it is most natural to consider the generative procedures underlying algorithms to be finitary inductive definitions(p. 166).4 Furthermore, Post (1944) calls the objects his production system generates either recursively enumerable sets or(general) recursive sets, pointing to the fact that every set so constructed is a recursively-defined object (that is, everyoperation of a production system is a definition by induction), thereby placing recursion at the very heart of these generativesystems.
In fact, Soare (1996) describes the spirit of the times of the 1930s and 40s as one in which recursion is taken to be at thevery centre of what a computation is, to the point that systems of recursive equations, or recursion itself, are employed asbeing almost synonymous with computability and/or computable, in detriment of Turings (1936) model, which did not makeuse of recursion at all.5 This state of affairs, what Soare calls the Recursion Convention, involves the following practices: a) usethe terms of the general recursive formalism to describe results of the subject, even if the proofs are based on the formalism ofTuring computability; b) use the term Churchs Thesis (in a narrow sense: the identification between general recursivefunctions (or partial recursive functions) and the computable functions) to denote various theses, including the Turing Thesis(viz., all intuitively computable functions are Turing Machine-computable.); and c) name the subject using the language ofrecursion (e.g., recursion function theory). This issue will be of some importance later on, but I nowmove on to a descriptionof the manner in which computer scientists have used recursive techniques.
Naturally, the way in which computer scientists employ recursion is similar to that of mathematicians, but some differ-ences are worth pointing out. In a programming language such as LISP, for instance, the step-by-step list of instructions forcompleting a taskdthe proceduredcan contain steps that are defined in terms of...