A Mathematical Theory of Pragmatic Information

• URL: http://arxiv.org/abs/2609.10986v1
• Date: Thu, 10 Sep 2026 02:04:44 GMT
• Title: A Mathematical Theory of Pragmatic Information
• Abstract summary: We propose a pragmatic information theory unifying communication, control, and decision-making.<n>We develop pragmatic entropy, up/down mutual information, channel capacity, and rate-distortion.<n>This framework provides a rigorous foundation for task-oriented communication, networked control, autonomous systems, and embodied AI.
• Score: 8.738344472520353
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a pragmatic information theory unifying communication, control, and decision-making. Its core is the isoteleia mapping, formalizing equifinality: distinct semantic paths leading to the same optimal action are pragmatically equivalent. This induces a three-tier hierarchy of syntactic, semantic, and pragmatic information, each abstraction discarding task-irrelevant distinctions. We develop pragmatic entropy, up/down mutual information, channel capacity, and rate-distortion, and prove three coding theorems generalizing Shannon's classical results. We introduce pragmatic value (VoI) and cost (CoI) of information as decision-theoretic duals to rate-distortion and capacity, respectively, and formulate a Lagrangian dual framework for cross-layer optimization. The pragmatic efficiency bound $\mathcal{E}_p(λ)=\sup_R[Φ_p(R)-λ\,\mathrm{CoI}_p(R)]$ quantifies the maximum net utility any resource-constrained intelligent system can extract, thereby establishing a fundamental behavioral capacity limit---generalizing Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages yield closed-form Gaussian expressions, while dynamic settings are addressed via a Bellman equation for sequential decision-making. This framework provides a rigorous foundation for task-oriented communication, networked control, autonomous systems, and embodied AI, shifting focus from symbol fidelity to the effectiveness of information in guiding actions, and offers a unified mathematical language for next-generation intelligent systems.

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