144 IV. STRUCTURAL FACTORS INLANGUAGES
few of these variables, let us say x^ (say, structure), x2 (say, evaluation) ., but these variables have been found, up to the present, in all our experience and all our equations.
A most important extension of the notion of 'function' and 'propo-sitional function' has been further accomplished by Cassius J. Keyser, who, in 1913, in his discussion of the multiple interpretations of postulate systems, introduced the notion of the 'doctrinal function'. Since, the doctrinal function has been discussed at length by Keyser in his Mathematical Philosophy and his other writings, by Carmichael2, and others. Let us recall that a propositional function is defined as an oo-valued statement, containing one or more variables, such that when single values are assigned to these variables the expression becomes a one-valued proposition. A manifold of interrelated propositional functions, usually called postulates, with all the consequences following from them, usually called theorems, has been termed by Keyser a doctrinal function. A doctrinal function, thus, has no specific content, as it deals with variables, but establishes definite relations between these variables. In principle, we can assign many single values to the variable terms and so generate many doctrines from one doctrinal function. In an oo-valued -system which eliminates identity and is based on structure, doctrinal functions become of an extraordinary importance.
In an oo-valued world of absolute individuals on objective levels, our statements can always be formulated in a way that makes obvious the use of oo-valued terms (variables) and so the postulates can always be expressed by propositional function. As postulates establish relations or multi-dimensional order, a set of postulates which defines a doctrinal function gives, also uniquely, the linguistic structure. As a rule, the builders of doctrines do not start with sets of postulates which would explicitly involve variables, but they build their doctrine around some specific content or one special respective value for the variables, and so the structure of a doctrine, outside of some mathematical disciplines, has never been explicitly given. If we trace a given doctrine with specific content to its doctrinal function without content, but variable terms, then, only, do we obtain a set of postulates which gives us the linguistic structure. Briefly, to find the structure of a doctrine, we must formulate the doctrinal function of which the given doctrine is only a special interpretation. In non-mathematical disciplines, where doctrines are not traced down to a set of postulates, we have no means of knowing their structure, or whether two different doctrines originated from one doctrinal function, or from two. In other words, we have no simple means of ascertaining whether the two different doctrines have similar or differ-