Towards a Measure Theory of Semantic Information
This addresses a foundational problem in information theory for philosophers and logicians by proposing a novel solution to a long-standing paradox, though it appears incremental as it builds on prior critiques and analogies.
The paper critiques Floridi's theory of strongly semantic information for failing to resolve the Bar-Hillel-Carnap paradox and introduces a new approach based on the unit circle and von Neumann's quantum probability to construct a measure space for informativeness that meets Floridi's requirements and removes the paradox, with contradictions and tautologies having zero informativeness but contradictory messages being equally informative.
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.