Towards a Measure Theory of Semantic Information

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.

We introduce incongruent normal form (INF), a structural representation for self-referential semantic sentences. An INF replaces a self-referential sentence with a finite family of non-self-referential sentences that are individually satisfiable but not jointly satisfiable. This transformation isolates the semantic obstruction created by self-reference while preserving classical semantics locally and is accompanied by correctness theorems characterizing when global inconsistency arises from locally compatible commitments. We then study the role of incongruence as a structural source of semantic informativeness. Using a minimal model-theoretic notion of informativeness-understood as the ability of sentences to distinguish among admissible models-we show that semantic completeness precludes informativeness, while incongruence preserves it. Moreover, incongruence is not confined to paradoxical constructions: any consistent incomplete first-order theory admits finite incongruent families arising from incompatible complete extensions. In this sense, incompleteness manifests structurally as locally realizable but globally incompatible semantic commitments, providing a minimal formal basis for semantic knowledge. Finally, we introduce a quantitative semantic framework. In a canonical finite semantic-state setting, we model semantic commitments as Boolean functions and define a Fourier-analytic notion of semantic energy based on total influence. We derive uncertainty-style bounds relating semantic determinacy, informativeness, and spectral simplicity, and establish a matrix inequality bounding aggregate semantic variance by total semantic energy. These results show quantitatively that semantic informativeness cannot collapse into a single determinate state without unbounded energy cost, identifying incongruence as a fundamental structural and quantitative feature of semantic representation.

Shannon's information theory deliberately excludes message semantics. This paper develops a rigorous framework for semantic communication that integrates formal proof systems with Shannon-theoretic tools. We introduce an axiomatic information model comprising Lsem-definable state sets linked by computable enabling maps, and define the semantic channel as a composition of Markov kernels whose supports respect the enabling structure. A fixed proof system induces an irredundant semantic core and a derivation-depth stratification, enabling four distortion measures of increasing semantic depth: Hamming, closure, depth, and a parameterized composite. Six families of computable semantic channel invariants are defined and their inter-relationships established, including a data processing bound, a semantic Fano bound, and an ideal-channel collapse theorem. The central quantitative result is a deductive compression gain: under closure-based fidelity, the minimum block length is determined by the irredundant core size rather than the full knowledge-base size. We instantiate the framework for heterogeneous multi-agent communication, introducing an overlap decomposition that yields necessary and sufficient conditions for closure-reliable communication. A semantic bottleneck phenomenon is identified in broadcast settings: vocabulary mismatch imposes irreducible fidelity limitations even over noiseless carriers. All results are verified on an explicit Datalog instance.