Infinity and verifiability in Carnap's inductive logic

摘要

Truth of sentences in infinity is discussed in the framework of Rudolf Carnap's inductive logic, which uses finite state descriptions and an asymptotic limit approach for defining probabilities in infinity. This means that Carnap's approach suits well for a semantics which is based on finite observability. However, a proper link between asymptotic probability and truth in infinity is missing from Carnap's treatment. A novel notion of truth in infinity is introduced and referred to as the extended truth. The idea is that the truth of the sentence S is extended by a particular sequence of state descriptions (where the larger one contains all of the smaller ones) iff S is true in each state description of the sequence. The corresponding notion of extended probability is introduced. Some important results are proved for extended truth and extended probability.