Towards a Measure Theory of Semantic Information

2025-08-01

AI Summary

The Bar-Hillel–Carnap paradox—where contradictory propositions are assigned maximal semantic information under classical semantic information theories—undermines the intuitive and philosophical foundations of semantic information. Method: This paper critically reconstructs Floridi’s Strong Semantic Information Theory by introducing a novel geometric measurement framework based on the unit circle, analogously inspired by von Neumann’s quantum probability formalism; semantic distance is defined via angular separation, enabling a nonlinear mapping between information quantity and semantic deviation. Contribution/Results: The proposed measure strictly satisfies Floridi’s semantic informativeness requirements: contradictions and tautologies both yield zero information; mutually exclusive propositions receive symmetric, equal information values; and contingent statements yield intermediate values. Consequently, the paradox is fully resolved, yielding the first semantic information metric space that simultaneously achieves formal rigor and semantic intuitiveness.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

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.

Problem

• Critique Floridi's semantic information theory shortcomings
• Develop new measure space for informativeness using unit circle
• Resolve Bar-Hillel-Carnap paradox in semantic information quantification

Innovation

• New approach based on unit circle
• Measure space for informativeness
• Removes Bar-Hillel-Carnap paradox