Probability and induction

Abstract

This thesis is based on two books by Rudolph Carnap - The Logical Foundations of Probability (abbreviated throughout to Foundations), published by the University of Chicago Press, 1950, and Continuum of Inductive Methods (abbreviated to continuum), published by the same Press, 1952. Chapter 1 states and explains seven of the 10 axioms which Carnap accepts as an axiom system for probability. The main aim of the thesis is to study some criticisms and alternatives to these seven axioms. In Chapter 2, however, we deviate a little to see how Carnap provides an interpretation, in logical terms, for this axiom system in a manner analogous to Whitehead and Russell's interpretation of Peano's arithmetic axioms. Chapter 3 returns to the main theme and discusses some subjectivist criticisms of Carnap's axioms. Chapter 4 defends Carnap from an attack by Dr Popper on his axiom system but Chapter 5 argues that Popper has, indeed, picked on an important deficiency of Carnap's treatment if it is considered as embracing the whole of inductive thought. It is of interest that our view as to the irrelevance of the first part of Popper's attck is supported by an abstract and criticism of Popper's paper by John G Kemeny in the Journal of Symbolic Logic, Vol. 20, No. 3, which unfortunately appeared after the completion of this thesis.