Automatic listing of important observational statements. II
Petr Hájek · Czech digital mathematics library · 1973
II PETR HAJEK This is a direct continuation of the first part -Problems and Solutions -of the present paper.(See the previous number of this journal.)Part II -Functor calculi 7. BASIC NOTIONSWe shall now consider the structure of sentences and the ways in which sentences take values.The notions we are going to introduce are generalizations of notions studied in the classical predicate calculus and are all essentially described in [2] -Introduction.We shall keep Church's terminology as much as possible; our deviation consists in working systematically with abstract values and in different (more detailed) notion of operators (generalized quantifiers).Recall that describing the predicate calculus one defines formulas of some language, in particular closed formulas; the meaning of closed formulas is given by giving a relational structure of an appropriate type.The meaning of a predicate is the corresponding relation on the field of the structure or -equivalently -the characteristic function of that relation, hence a two-valued function on the field of the structure.Let now V be a set of abstract values; for every set M, each mapping of M" into V (n natural) will be called an n-ary V-valued function on M. A V-valued function is understood as a generalized relation; instead of asking whether an n-tuple is in the relation (yes -no), we ask how it is in the relation.(Compare e.g. the question "are x and y related?" with the question "what is the relationship of x and y?".) 7.1.Definition, (i) Let a fixed non-empty set V of abstract values be given.A type is a finite non-empty sequence of natural numbers.A V-structure of the type <».,..., ..., nk} is a (fe + l)-tuple M = where M 4= 0 (the field of M) and each/ is an nrary V -valued function on M. (If n, = 0 then/; e V.)