A simplification of basic logic

The program of “basic logic” can be summarized as follows:I. To treat every syntactical system as a subclass of a certain fixed infinite classUof “U-expressions.” This can always be done by modifying in trivial ways the notation of each syntactical system which is not already such a subclass. As a result all syntactical systems become comparable with each other in the sense that they are merely different subclasses of a single class of expressions. The class U can be chosen in such a way as to be inductively definable thus in terms of a fixed symbol ‘σ’: (1) The symbol ‘σ’ is a U-expression. (2) The result of placing two U-expressions (or two occurrences of the same U-expression) next to each other, and enclosing this total expression within a pair of parentheses, is a U-expression.II. To formulate a particular syntactical systemKwithin which every syntactical system (and indeedKitself) is “represented.” Such a system is here said to be a “basic system,” and an appropriate interpretation of it is said to be a “basic logic.” Within such a logic every finitary logic is definable, as well as the basic logic itself. Such a logic should be of fundamental importance, especially if it is so constructed as to be the weakest such logic and so contain no theorems that are not essential to its being basic.With the above considerations in view, the system K has been defined in such a way that it is a subclass of U and is a basic syntactical system. A simpler definition of K will be given than heretofore in previous papers, and the minimum character of K will be made more clear.