Languages of similarity

Abstract

Studying the "quantum logic", one could observe that logical structures, like a language with its syntax and semantics, a consequence operation, etc., can be directly extracted from quantum theory. This way of constructing a language has no precedent in the history of formal logic. The procedure of "taking a well-known physical theory and distilling its logic", as Finkelstein [5] calls it, seems now to be sufficiently developed to apply it beyond the scope of quantum mechanics. The main characteristic feature of this procedure is that, roughly speaking, it reconstructs a syntax from a semantics: we identify some semantical objects within quantum theory, and infer the syntax of the corresponding language. It is the reverse of the common trends in formal logic, where we look for a semantics of a given syntactical structure. The main concern of this work is to find, in the "quantum-logical" manner, languages related to a "mini-theory" of similarity. It seems to be the first application of the quantum "reading off" procedure to a nonquantum theory.' We are still close to the quantum case, as a special similarity structure underlies quantum theory, too. Our considerations can be thus seen as a test of the quantum "reading off" procedure. This study of logical aspects of the notion of similarity is also of interest because this notion appears in some variants of the Kripkian modal semantics.2 After introducing basic notions of the "mini-theory" of similarity and illustrating them by examples (Section 2), we discuss in Section 3 the problem of determining a similarity structure by a set of properties. It is demonstrated that one must use vague properties to do this. This result is of crucial importance; because of it the languages of similarity we obtain have to be multi-valued. A set of objects with a similarity relation over it, together with the set of properties determining this similarity, form a structure resembling to some extent the basic structure of quantum theory. It makes it possible to apply here the same procedure as in the quantum case in order to find a language of the similarity "theory".