Implication and information: A quantitative-informational analysis of material implication

Abstract

In this chapter, we show the relationship between material implication and the notion of information, comparing it with another notion of implication called probabilistic implication. After some basic definitions are presented, a probabilistic semantic perspective is constructed, on the basis of which is defined the amount of information of a formula of the language of classical propositional calculus. Our main objective is to show that the usual material implication does not capture the notion of information as found in the mathematical theory of communication developed by thinkers such as Shannon. It is suggested that a logical system with a probabilistic implication can perhaps do so more appropriately.