Logical types from the axiomatic point of view

Abstract

The main purpose of the thesis is the establishment of an improved system of logical types. The absolutist view, according to which there exists only one hierarchy of types which has to be found out by consideration of empirical facts and by logical analysis, has been abandoned, and the attempt has been made to raise the whole problem from the material into the formal, the syntactical, sphere. Thereby such pseudo-problems disappear as usually arise when the borderline between cognoscence and stipulation is not strictly respected. From the two alternatives of treating the various branches of science either within one and the same linguistic system (Carnap's "Logischer Aufbau der Welt"; the "Unity of Science"-thesis of Physicalism), or separately in the form of axiomatic systems, I have chosen the latter; and accordingly, instead of erecting one system of types, conventions have been suggested for the erection of a special type-hierarchy for every given axiomatic system. In the course of the investigation, it has been necessary to discuss more or less independently a number of special logical problems in greater detail. Thus one chapter of the thesis has been devoted to the syntax of axiomatic systems, another to the introduction of parameters as a counterpart to constants and variables. Further, I have studied the concept of a cardinal number closely, and I have tried to make a small contribution to the theory of identity.