In my paper "Thesen zum Universalienprojekt" (1976) I mention two complementary procedures for discovering language universals: 1. The investigation of the dimensions and principles whose existence is necessitated by the communicative function of language; 2. The development of a formal language in which all syntactic rules are explicitly formulated and in which all syntactic categories are defined by their relation to a minimally necessary number of syntactic categories. Since the first procedure is treated in many of the other papers of this volume, I wish to discuss the role of formal methods in the research of language universals. As an example I want to take the dimensions of determination and show how expressions denoting concepts are modified and turned into reference identifying expressions. There is a general end a specific motivation for the introduction of formal methods into linguistics. The general motivation is to make statements in linguistics as exact and verifiable as they are in the natural sciences. The specific motivation is to make the grammars of various languages comparable by describing them with the same form of rules. The form has to be flexible enough to describe the phenomena of any possible natural language. All natural languages have in common that they may potentially express any meaning. The flexibility of the form of grammatical rules may therefore be attained, if syntactic rules are not isolated from the semantic function they express and syntactic classes are not defined merely by the relative position of their elements in the sentence, but also by the communicative function their elements fulfill in their combination with elements of other classes.
Montague (1974) has shown that this flexibility may be attained by using the language of algebra combined with categorial grammar. Algebraic systems have been developed by mathematicians to model any systems whose operations are definable. Montague does not merely use the tools of mathematics for describing the features of language, but regards syntax, semantics and pragmatics as branches of mathematics. One of the advantages of this approach is that we may apply the laws developed by mathematicians to the systems constructed by linguists for the description and explanation of natural language.