Towards a Measure Theory of Semantic Information

George M. Coghill

cs.ITAug 1, 2025

Towards a Measure Theory of Semantic Information

Authors: George M. Coghill

Organizations: University of Aberdeen & Edinburgh Theological Seminary.

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.

Figures & tables

Figure 1: The degree of informativeness (after Floridi [ 15 ] ). Here,

\iota(\sigma)=1-\vartheta^{2}(\sigma)

Message	Message	No. of worlds in	No. of true atoms
Number		which it is True
M0	$xx^{\prime}$	0	1
M1	$xy$	1	2
M2	$xy^{\prime}$	1	1
M3	$x^{\prime}y$	1	1
M4	$x^{\prime}y^{\prime}$	1	0

Table 1: The complete set of messages. The representation used here is from digital systems, where conjunction is shown as a product and disjunction as a sum, this is equivalent to, but slightly more compact than, the usual normal forms in symbolic logic.

Figure 2: The structure of the information metric space.

Figure 3: The distribution of messages in the information measure space.

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Towards an Inferentialist Account of Information Through Proof-theoretic Semantics

Matthew Collinson, Timo Eckhardt, David Pym

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Information is one of the most widely-discussed concepts of the current era. However, a great deal of insightful work notwithstanding, it is yet to be given wholly convincing logical or mathematical foundations. Without them, we lack adequate reasoning tools for understanding the complex ecosystems of systems upon which the society depends. We seek to rectify this by taking a first step towards developing an inferentialist semantic theory of information. There are three key interacting components. First, conceptual analysis: the metaphysics of information. Dretske expressed the key concepts of information in terms of intentionality, truth, and transmissibility. We replace truth with inferability, and trace the consequences of this replacement. Second, logic: proof-theoretic semantics (P-tS) provides a mathematical-logical realization of inferentialist reasoning. Using P-tS, we develop the first steps towards a mathematical-logical theory of an inferentialist primitive unit of information, the 'inferon'. This proof-theoretic approach counterpoints the model-theoretic view of information articulated in situation theory. Furthermore, we argue that it facilitates addressing all three components of van Benthem and Martinez's categorization of the understandings of information, as range, as correlation, and as code. Our focus is on information-as-correlation. Third, systems: the P-tS tools we develop provide the basis for a mathematical account of distributed systems modelling -- a key tool from informatics for understanding the organization of information processing systems. This yields a reasoning-based theory of information flow in models of distributed systems. Overall, we seek to give a conceptually rigorous mathematical-logical account of information and its role within informatics, grounded in inference and reasoning.

Abstract

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