Towards a Measure Theory of Semantic Information

TL;DR

This paper critiques existing semantic information measures, analyzes their limitations, and proposes a new measure based on the unit circle that resolves the Bar-Hillel-Carnap paradox and aligns with Floridi's criteria.

Contribution

It introduces a novel semantic information measure using the unit circle, successfully addressing the paradox and improving upon Floridi's previous theories.

Findings

Existing measures assign maximum informativeness to contradictions.

The new measure based on the unit circle removes the paradox.

Contradictory messages are shown to be equally informative.

Abstract

A classic account of the quantification of semantic information is that of Bar-Hiller and Carnap. Their account proposes an inverse relation between the informativeness of a statement and its probability. However, their approach assigns the maximum informativeness to a contradiction: which Floridi refers to as the Bar-Hillel-Carnap paradox. He developed a novel theory founded on a distance metric and parabolic relation, designed to remove this paradox. Unfortunately is approach does not succeed in that aim. In this paper I critique Floridi's theory of strongly semantic information on its own terms and show where it succeeds and fails. I then present a new approach based on the unit circle (a relation that has been the basis of theories from basic trigonometry to quantum theory). This is used, by analogy with von Neumann's quantum probability to construct a measure space for informativeness that meets all the requirements stipulated by Floridi and removes the paradox. In addition, while contradictions and tautologies have zero informativeness, it is found that messages which are contradictory to each other are equally informative. The utility of this is explained by means of an example.