The evolution of type theory in logic and mathematics

ion Principle 1.1 "If in an expression, f . . . j a simple or a compound sign has one or more occurrences and if we regard that sign as replaceable in all or some of these occurrences by something else {but everywhere by the same thing), then we call the part that remains invariant in the expression a function, and the replaceable part the argument of the function." (Begriffsschrift, Section 9) Frege put no restrictions on what could play the role of an argument. An argument could be a number (as was the situation in analysis), but also a proposition, or a function. Similarly, the result of applying a function to an argument did not necessarily have to be a number. Functions of more than one argument were constructed by a method that is very close to the method presented by Schonfinkel [109] in 1924: Abstraction Principle 1.2 "If, given a function, we think of a sign1 that was hitherto regarded as not replaceable as being replaceable at some or all of 1 We can now regard a sign that previously was considered replaceable as replaceable also in those places in which up to this point it was considered fixed. [footnote by Frege]ion Principle 1.2 "If, given a function, we think of a sign1 that was hitherto regarded as not replaceable as being replaceable at some or all of 1 We can now regard a sign that previously was considered replaceable as replaceable also in those places in which up to this point it was considered fixed. [footnote by Frege] lb Paradox threats in formal systems .19 it.s occurrences, then by adopting thi.s conception we obtain a function that ha.s a new argument in addition to tho.se it had before." (Begriff.s.schrift, Section 9) With this definition of function, two of the three possible paradox threats mentioned on p. 16 occurred: 1. The generalisation of the concept of function made the system more abstract and less intuitive. The fact that functions could have different types of arguments is at the basis of the Russell Paradox; 2. Frege introduced a formal system instead of the informal systems that were used up till then. Type theory, that would be helpful in distinguishing between the different types of arguments that a function might take, was left informal. So, Frege had to proceed with caution. And so he did, at this stage. He remarked that "if the [ ... ]letter [sign] occurs as a function sign, this circumstance [should] be taken into account." (Begriff.s.schrift, Section 11) This could be interpreted as if Frege was aware of some typing rule that does not allow to substitute functions for object variables or objects for function variables. In his paper Function and Concept [4 7], Frege more explicitly stated: " Now just as functions are fundamentally different from objects, so also functions whose arguments are and must be functions are fundamentally different from functions whose arguments are objects and cannot be anything else. I call the latter first-level, the former second-level." (Function and Concept, pp. 26-27)