Visiting Some Relatives of Peirce's

The notion of a relational grammar is extended to ternary relations and illustrated by a fragment of English. Some of Peirce's terms for ternary relations are shown to be incorrect and corrected. Binary relations have been studied extensively by Peirce and Schrr oder in the nineteenth century, and in this century by Tarski and his students. No comparable attention has been paid to ternary relations. This is surprising, for Peirce had already dealt with ternary relations on various occasions. But Schrr oder strictly connned himself to binary relations, and it is the topic of binary relations that has become the focus of interest for Tarski and his students. For notable exceptions see Carnap (1929), Copi and Harary (1953), and Aubert (1955) and the theory of relational database systems. Peirce has illustrated his ternary relational terms by natural language examples. For instance, the equation (b a) m = b (am) where b = betray, a = enemy, and m = man was explained in English as follows: \those individuals each of which stand to every man in the relation of betrayer to every enemy of his are identical with those individuals each of which is a betrayer to every enemy of a man of that man." 1 This may be hard to swallow and even Peirce himself had some problems here as we shall see. Therefore I think that Peirce's discussion of ternary relations can best be studied in a framework that is as rigorous with respect to syntactic structure as it is to semantic structure. Such a method is provided by relational grammar. Relational grammar was proposed in Suppes (1976). The purpose of this paper is therefore both to extend the notion of a relational grammar by adding ternary relations and to apply this extension to a better understanding of Peirce's writings about relations, or \relatives" to use his own term. Our focus is therefore more on the establishment of mapping natural language into the language of relation algebra than on the development of the algebra of ternary relations. The paper continues my work on relational grammar and builds especially on previous results on anaphoric The paper is organized as follows: in section 1, the notion of a relational grammar is introduced. In section 2, this notion is extended in order to account for ternary relations. In section 3, an example of a ternary relational grammar is given. In section …