Finitism in a Broader Sense

50 CHAPTER 2. FUNDAMENTALS AND PROBLEMS is insufficient. What we also need is that in theorizing, the finitist always has to check whether his theory is still adequate as new data comes in.

Note that this shows some similarity of the intensionalist and the finitist position: whereas the intensionalist considers a structure ofpossible utterances, which models the speakers growing knowledge according to new inferences, the finitist has to consider always newactual utterances, which comes to him not by construction or inference, but by experience. So, what we are the formal tools of finitary linguistics? We get the following easy observations:

Lemma 5 There is no smallest class C of languages (contained in the regular languages) which contains all finite languages and some infinite language.

Lemma 6 There is no largest class C of languages (contained in the regular languages) which does not contain the finite languages.

Both observations are quite obvious, so we omit the proof. What does that mean for us? There is no clear upper/lower bound for the class of languages we should consider. Whereas in principle, FLP does not provide us with an upper bound for the class of languages we are interested in, our finitistic philosophy does: as the regular languages are the largest class of languages which can be recognized by finitary means, there is no non-regular language which is observable, as this would necessarily involve projection anyway. But of course, the class of regular languages does not have FLP, so it is not a candidate; that even holds for smaller classes such as the star-free languages.

As in the finitary setting there is no proper construction of “language” and we always stay within the observations, there is a great gain we have: we simply avoid the entire problem of projection, circularity etc., sticking to a rigid falsificationism. We construct theories for infinite languages only considering observations. However, the price we have to pay will be considerable to most linguists.

Note that the idea to treat natural language in a subregular fashion is by no means new: see for example [55]. It has not found much support in the community, because from the point of view of formal linguistics it is not very appealing. Nonetheless, from the point of view of linguistic metatheory it is quite appealing. If we connect it with the notion of trivialization, introduced above, then it might become even a fruitful field of study in connection with the

“classical mathematics of language”, that is, the formal methods used in classical linguistics.

2.8. ONTOLOGIES OF LINGUISTICS AND THEIR CONSTRUCTION 51 What we will show here is that the two positions, though theoretically distinct, in practice often coincide, and if not, narrow finitism seems to be preferable.

The main reason is the following: assume they do not coincide. That means, we do not take a falsificationist stance based on something like FLP within broad finitism. But from the point of view of complexity, all we can really say about “language” in general is that it is regular - which it is by assumption. So this is quite a vacuous way to deal with natural language. If we want to make it more meaningful, we have to look for tighter upper bounds for “language”.

But again, this can be only achieved properties similar FLP; because if our framework contains the class of regular languages, there is no way to falsify it.

So broad finitism as well has to recur to falsificationism if it is supposed to bring us interesting generalizations.

Conversely, assumeπ(I) is a regular language. Then we can also innarrow finitism write a grammar for π(I), and try to falsify it. This is, however, not exactly what happens in broad finitism: because making a new observation would lead us to reconsider entire the projection rather than falsify the grammar – this would be the same as in the classical metatheory. So the differences between broad and narrow finitism are mainly technical; and to put it bluntly, broad finitism seems to unify the drawbacks of narrow finitism - restriction to regular languages - with those the classical metatheory, which are all about the problems and difficulties of projection. After all, the main advantage of narrow finitism is that we avoid the critical step of projection altogether. Therefore, broad finitism is not of too much interest for us now.

There is another fundamental doubt I have about broad finitism: given a projection π, some dataset I, how can I ever know that π(I) is contained in o-language? After all, o-language is an empirical notion, and even the fact that π(I) is regular does not entail it is in o-language, as should be clear. Moreover, as π(I) is supposed to be infinite, we can never test whether all its strings are acceptable/observable. So the position of broad finitism, which I have found sympathetic to many linguists, to me seems to be quite problematic and not worth elaborating at this point. I will however treat some mathematical questions which arise when we wantπ(I) to be contained in o-language, with some surprising results (see subsection “On regular projection”, chapter on classical metatheory).

As a final note, I am aware that there are still many different positions on the metatheory of language, and in fact much more than I can mention here.

But I hope the ones I have outlined are the most important, reasonable and interesting ones.

52 CHAPTER 2. FUNDAMENTALS AND PROBLEMS

Chapter 3

The Ontology of Metalinguistics

53

54 CHAPTER 3. THE ONTOLOGY OF METALINGUISTICS

Summary of the Metalinguistic Ontology

The most fundamental datum of metalinguistics are judgments of the form

` w~ ∈ L. But we here justify a more elaborate ontology; in particular, we introduce negative data. This negative data is however of a fundamentally different nature: it is not an empirical datum, but rather constructed by the argument: if we accept this, we have to accept anything. The reason we cannot use negative judgments as primitives is that this way, we would become way too much negative data: we usually want explicitly more than any speaker accepts.

The positive and negative data give rise to we callpartial languages: pairs of finite languages, their intersection being empty. The goal of metalinguistics is to complete them to full infinite languages; but for this purpose we can only use the positive data, which is given, not the negative data, which is a construction:

apart from methodological reasons, we still need the negative data to check whether our pre-theories are adequate, that is, whether they agree with our intuitions, because if we do not have this method of control, we have no other and projections become quite arbitrary.

3.1 Preliminaries

As we have said, there are different possible conceptions of what “language” is.

We will work out three fundamental positions. Each of them is distinguished by fundamental differences in the basic ontology of linguistics. In this introduction, we do not work on the particular ontologies of linguistics, but we first try to sketch an ontology formetalinguistics; that is, we want to give a formal inventory the metalinguist is given, and what he has to provide. This ontology is then basic and common for all three metatheories: so they are based on the same metalinguistic ontology, though they construct different linguistic ontologies. Whereas we already introduced the notions of “language”, “o-language”, observed language (data) etc., we will here look at more formal foundations for the mathematical

procedures we use.

For now, the most important part of metatheory is the part which (meta-)linguists are given pre-theoretically. Contrarily to the linguistic ontology, this ontology must befinite, because we do not observe infinite languages or infinite objects in general. Given this finitary ontology, it is the task to develop formal procedures which develop an adequate ontology for linguistics. In this way, we get infinite “languages”, out of our observations (or in the finitary setting, grammars describing infinite languages). Keep in mind that “language” is only a shorthand for “whatever we consider the proper subject of linguistics”, it is thus a formally underspecified notion. This is the crucial step of linguistic metatheory:

we formalize a procedure, which given some finite input, gives something infinite – namely “language” – as output. In the simplest, classical case, it is nothing but a function; in the intensional case, there are some additional non-deterministic choices to be made. So we have to determine the input of the function. But this is not all a metalinguistic ontology has to provide. We would like in addition to have somecriteria of adequacy for metalinguistic procedures. That is to say, we want some non-trivial criteria to tell whether a procedure does a good job or not. There are two kinds of criteria: there area priori criteria, which apply to the procedure regardless of any input, that is, they concern abstract intrinsic

3.2. LINGUISTIC JUDGMENTS 55