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On the Metatheory of Linguistics Christian Wurm On the Metatheory of Linguistics Christian Wurm Doktorarbeit zur Promotion an der Universiat¨ Bielefeld Gutachter: Marcus Kracht, Jens Michaelis, Gregory Kobele December 18, 2013 2 Contents 1 Introduction 7 1.1 What is Metalinguistics? . 8 1.2 A Note on Syntax and Semantics . 13 1.3 The Goal of This Work . 14 1.4 The Philosophical Context... 15 1.5 ...and the Context of Learning Theory . 16 2 Fundamentals and Problems of Linguistic Metatheory 19 2.1 The Creative Commitment . 20 2.1.1 What and Where is Language? . 20 2.1.2 The Mathematics of the Creative Commitment . 21 2.2 The Epistemic Foundations of Linguistics . 22 2.2.1 The Epistemic Burden of Linguistics as Psychology . 23 2.2.2 The Epistemic Burden of Linguistics as a Formal Science 25 2.2.3 The Epistemic Burden of Linguistic Judgments . 26 2.3 Some Fundamental Concepts of Metalinguistics . 27 2.4 The Projection Problem . 29 2.5 A Sketch of the History of the Problem... 30 2.6 ...and Why the Classical Solution does not Work . 31 2.7 Questions Around the Projection Problem . 32 2.7.1 Language is Not Designed for Usage . 32 2.7.2 Insights by Descriptive Elegance . 33 2.7.3 On Recursion . 33 2.7.4 Patterns and Dependencies . 34 2.7.5 Weak and Strong Generative Capacity . 35 2.7.6 Chunking . 38 2.7.7 pro-drop, Syntactic Complexity and Trivialization . 39 2.8 Ontologies of Linguistics and their Construction . 41 2.8.1 On the Semantics of Linguistic Theories . 41 2.8.2 The Classical Ontology and Its Problems . 42 2.8.3 The Intensional Ontology and its Motivation . 44 2.8.4 The Finitist Conception of \Language" . 47 2.8.5 Finitism in a Broader Sense . 50 3 The Ontology of Metalinguistics 53 3.1 Preliminaries . 54 3.2 Linguistic Judgments . 55 3.3 Partial Languages . 56 3 4 CONTENTS 4 The Classical Metatheory of Language 59 4.1 The Classical Metatheory . 60 4.2 Introducing Pre-Theories . 61 4.3 Substitutional Pre-Theories . 69 4.4 Structural Inference . 74 4.5 Properties of Pre-Theories I . 77 4.5.1 Problems for Infinite Languages . 77 4.5.2 On Regular Projection . 79 4.5.3 On Similarity . 81 4.6 Properties of Pre-Theories II . 82 4.6.1 Characteristic and Downward Normal Pre-Theories . 82 4.6.2 Upward Normality . 88 4.6.3 Normalizing Maps . 92 4.6.4 Normality and a Normal Pre-Theory . 96 4.6.5 Monotonicity . 96 4.6.6 A Weaker Form of Monotonicity . 99 4.6.7 Fixed-point Properties . 100 4.6.8 Closure under Morphisms . 101 4.7 Methodological Universals . 103 4.7.1 Which Languages Do We (Not) Obtain? . 103 4.7.2 Unreasonable Restrictions of the String Case . 105 4.7.3 Linguistic Reason . 106 4.8 Extension I: Pre-Theories on Powersets . 107 4.8.1 Syntactic Concepts . 107 4.8.2 Syntactic Concepts: Definitions . 108 4.8.3 Monoid Structure and Residuation . 110 4.9 Analogies and Inferences with Powersets . 111 4.9.1 Upward Normality and (Weak) Monotonicity . 114 4.9.2 Reducing Lattices to Languages . 116 4.10 Context-freeness and Beyond: SCLn . 119 4.11 Transformational Pre-Theories . 124 4.11.1 Ontological Questions . 124 4.11.2 Detour: an Alternative Scheme . 128 4.11.3 Legitimate Functions . 129 4.11.4 Opaque Functions, and Why They Will not Work . 132 4.11.5 Polynomial Functions . 134 4.11.6 Inferences with Polynomials . 136 4.11.7 Polynomial Pre-Theories . 137 4.12 Strings as Typed λ-Terms . 139 4.12.1 A Simple Type Theory . 139 4.12.2 Strings as λ-Terms . 141 4.12.3 Using λ-terms for Pre-Theories . 142 4.13 Concepts and Types . 145 4.13.1 A Context of Terms . 145 4.13.2 Concept Structure and Type Structure . 146 4.13.3 Generalizing the Language-theoretic Context . 148 4.13.4 Putting Things to Work . 150 4.14 Another Order on Pre-Theories . 151 4.15 A Kind of Completeness . 152 4.16 A Kind of Incompleteness . 155 CONTENTS 5 5 The Intensional Metatheory of Language 159 5.1 Problems of the Classical Conception . 160 5.2 The Intensional Conception: Philosophical Outline . 162 5.3 The Thinking Speaker: Independent Evidence . 166 5.3.1 Preliminaries . 166 5.3.2 Language Change . 167 5.3.3 Sociolinguistic Typology: Trudgill . 169 5.3.4 Roy Harrison: The Language Makers . 169 5.3.5 Coseriu on Knowledge of Language . 170 5.4 The Mathematics of Intensional Linguistics . 171 5.4.1 Languages as Structures . 171 5.4.2 Language Definability . 173 5.4.3 Adequacy . 175 5.5 Some Notes on Intensional Linguistics . 176 6 The Finitary Metatheory of Language 179 6.1 The Finitist Position . 180 6.2 FLP, PLP and Subregular Languages . 181 6.3 Derivatives of Languages . 182 6.4 Infinitary Prefixes . 185 6.5 A Note on Learnability . 187 6.6 Conclusion . 188 7 Conclusion and Outlook 189 7.1 Things that have been done . 190 7.2 Things that should be done . 192 6 CONTENTS Chapter 1 Introduction 7 8 CHAPTER 1. INTRODUCTION 1.1 What is Metalinguistics? If we should define the goal of the science of language concisely in a sentence, most scholars would say something like: it is the study of our implicit, unconscious knowledge of language. We will see that this statement is very problematic; and most of this first chapter will be devoted to show why it is problematic. We start with some basic, uncritical observations.1 The subject of linguistics consists the study of languages and of language. Study of languages means that linguists have to look at languages, which are their primary object of study. Study of language means that what is interesting to linguists are in particular general properties of all languages, rather than properties of particular languages. In any way, linguistics is based on the observation of ourselves as a species, because after all we are interested in the \verbal behavior" of humans. This is maybe the main point which distinguishes it from a science like physics. What distinguishes it from a science like psychology is mostly the following: linguistics is typically not about what we actually say in given circumstances, but about what we can say. It does not describe our actual behavior, but rather our possible behavior. There might not be a complete agreement on this point, still it seems to guide theoretical linguistics in its current practice.2 So in the sequel, I take it for granted that if we are to describe language, what we essentially do is to provide rules for well-formed utterances, rather than providing rules which prescribe what we have to say in a given circumstance. Linguistic rules are thus rules for possible behaviors, not for actual behavior. Having said this, it is exactly this intensional character which distinguishes linguistics from most of psychology, though of course not from all of it. A next distinguishing hallmark is the fact that language is one of the main fields of human creativity: there is no upper bound to the number of utterances we can make. There are two main arguments for this claim: firstly, the old Chomskyan argument that given any sentence (say the presumably longest sentence of my finite language of English), I can construct a new, longer sentence (say by means of conjunction), which is again English. This is quite convincing, though not strictly empirical, because it already presupposes an abstract notion of \any sentence", which is not an empirical notion or object. The second argument3 is based on the frequency distribution of our observations: as a matter of fact, most sentences we observe, we observe only once. If we would have observed a considerable portion of the language in question, this would be an extremely improbable distribution; but it is very plausible under the assumption that we 1What is to follow can be read as the outlines of a theory of the science of language. It falls in this sense under the general field of theory of science, as exemplified e.g. by Kevin Kelly, [33]. However, as I lay out in the sequel, the peculiarities of linguistics seem to outweigh the common ground with the general theory of science, at least for the aspects I am focussing on. 2This is surely not the place for a complete discussion of this point. So let me just say: this position is not necessarily the correct one, but it seems to me the \working assumption" of theoretical linguistics in the canonical sense. Moreover, to me it seems to be generally unclear what theoretical linguistics would actually look like if we would think of it as a science predicting verbal behavior in given circumstances. In my view, this depends on a lot of things: for example, if we want to reconstruct canonical semantics in this view, I guess we first need a good theory of communication, in particular a notion of what successful communication means. As this and similar questions are complex and mostly open, I will just assume the standard \working assumption" of linguistics being on possible rather than actual behavior. 3Put forward by Alexander Clark, as far as I know. 1.1. WHAT IS METALINGUISTICS? 9 have only observed a small fragment of the complete language. As there is not the slightest evidence that this changes with a growing number of observations we make, this points us towards the fact that the number of possible observations is infinite.
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