| Properly formulated our grammar ought to be a formal description of English. It may be fruitful to draw an analogy between Gödel’s proof of a certain “defect” in formal logic and similar formal descriptions. This caution carry through to any formal language used to describe another language. Suppose the investigator decides that a particular expression in English is indeed grammatical. By the above proof the mathematician would not then feel comfortable inferring that the expression would necessarily be describable by the rules of the grammar. However, it should always be possible to adjust the rules so as to include the new expression specifically. |
A table or list approach to grammar.The non-scientific grammarian may think that the task is finished when comprehensive lists are compiled of all the various linguistic forms in their various categories and classes. It will be useful to illustrate how someone with this philosophy might proceed in the analysis. One particular division of classes often proposed is between words that meet certain formal criteria (their morphology), and those that meet certain distributional criteria (their syntax). In the following paragraphs I will illustrate this methodological principle as it has been applied in three classes of words: adjectives, nouns, verbs. |
Adjectives and adjectivals.The adjective in (1a) meets all the formal criteria. This word is found in three inflected forms. The adjective in (1b) meets only some of these criteria. This word’s inflectional varieties are built on different words; the adjective is suppletive. The word in (1c) does not meet any formal criteria for being an adjective. There is but one form for this word and in the situations where other adjectives might use different forms, this word uses a modifier. Because this expression uses several words whereas a paradigmatic adjective is a single word, this is called a periphrastic expression. The word beautiful is therefore called an adjectival because its class is based on distributional criteria rather than formal ones. The formal term for a word with its various holophrastic forms is a lemma (pl. lemmata). |
| (1) | a. | dear — dearer — dearest |
| b. | good — better — best | |
| c. | beautiful — more beautiful — most beautiful |
Nouns and nominals.In a similar fashion the noun in (2a) meets all the formal criteria for its class because it is fully inflected*. The noun in (2b) meets only some of these criteria because its inflection is irregular. Since the singular and plural forms of the word in (2c) are suppletive**, there are no formal criteria, and so the classifying linguist puts this word in the class of the so-called nominals. |
| (2) | a. | apple — apple’s — apples — apples’ |
| b. | woman — woman’s — women — women’s | |
| c. | person — person’s — people — people’s |
Verbs and verbals.A further example of this principle of classification is the large and varied class of verbs and verbals. These have a larger number of different forms than adjectives or nouns and a larger number of distributions. The verb in (3c) is suppletive. The word in (3d) is a particular kind of verbal called a modal auxiliary. This word has acquired some suppletive versions that are also periphrastic. |
| (3) | a. | sin — sins — sinned — sinning — sinned |
| b. | sing — sings — sang — singing — sung | |
| c. | be — am/is/are — was/were — being — been | |
| d. | (be able) — can/could — could — (being able) — (been able) |
| Should the investigator accept periphrastic forms as grammatically equivalent to the simpler holophrastic forms? Is periphrasis a way to describe a specific distribution as though it were a simpler form? Would it not be appropriate to continue by giving the rules that describe the various relationships between these forms? At this point the investigator would need a framework in which to state descriptive rules rigorously. Do these rules handle the forms on the list? But the ultimate goal in constructing a grammar ought to be something beyond a list. The scientist should provide the grammar with the capability of handling any and all the expressions of the language. Is not the linguist’s task similar to that of the geometer who cannot personally solve every geometry problem? What the geometer does is try to provide all the tools needed to solve certain kinds of problems. The linguist will presumably want the grammar developed to be capable of describing the significant structure of creative and novel compositions and a fortiori the periphrasis found in the common idiom. |
A grammar as a framework for description.One aim of the present work is to provide a more basic and in some ways more exhaustive description of English than is usually given. Even more importantly, however, there is the goal of introducing a formal descriptive apparatus. This framework should in some sense adequately model natural languages in general. There is no dearth of models and modeling tools available. The multitudinous tools and heuristics, which we outlined in six broad classes in the first chapter, have been established for this purpose over the years by numerous mathematicians and more recently by computer scientists as well. The descriptive framework in which a modern scientist would formulate English grammar is rigorous and explicit. For these reasons I have chosen to use the symbolic system called a calculus. I have developed this system from the formal language developed thoroughly by Rudolf Carnap (1959), one of the logical positivists. |
A logical calculus.There are two mathematical uses of the term calculus. Most often this term refers to the branch of mathematics involving methods of differentiation and integration. This is not how the positivists used the term. Their calculus is a language as a formal object — an artificial language, a metalanguage. The form that a calculus in actual practice takes depends on how the modeler conceptualizes the objects to be formalized. This means that the different approaches that different linguists take to natural language each require a different calculus for their formalization. Whatever the approach, the rigor of a calculus makes it possible to evaluate whether it has been developed appropriately as a descriptive framework. |
Strengths of a formal grammar.In justifying the construction of a calculus the positivists argued that symbolic language demands clarity of thought. Ideally each symbol and each structural combination of symbols has its own meaning — it has a unique interpretation. One of the reasons philosophers cite for building an artificial language is that they might avoid the problems that come with trying to use a natural language. Natural languages are inherently ambiguous. Symbolic languages in the form of mathematical models, have already proven to be of tremendous pragmatic importance in their application in the physical sciences. I believe it is safe to say that it was when philosophers consciously began to express their ideas using mathematical constructs that science made significant advances. This has occurred in linguistics as well. I believe that it has been with the formal description of languages that real progress has come to the study of language. Grammarians must now step beyond the initial stages. Because of all the limitations and ambiguities of a natural language, they cannot continue being restricted by it, when what they really want to do is describe a natural language with rigor and clarity. |