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.
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.
George M. Coghill
University of Aberdeen & Edinburgh Theological Seminary.
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 propose uncommon self-knowledge (USK) as a candidate criterion for consciousness: synergistic information a system carries about itself that exists only in the joint of its subsystems and is destroyed by decomposition. Drawing on Gottwald's partition-lattice grounding of Partial Information Decomposition (PID), where redundancy corresponds to Aumann's common knowledge and synergy to the gap between separate and joint observation, we propose the synergistic component of self-directed information as a candidate formal signature for conscious processing. If correct, the framework would (1) offer a clean separation between consciousness and metacognition (synergistic vs. redundant self-knowledge), (2) provide principled resolutions to counterexamples that challenge IIT, GWT, and HOT, (3) be operationalizable via Partial Information Rate Decomposition (PIRD) with self-targeting, and (4) generate distinctive empirical predictions, the strongest being a GWT timing dissociation (consciousness correlates with pre-broadcast synergy formation, not broadcast itself) and a specific dissociation between self-report disruption and task-performance disruption under middle-layer perturbation in LLMs. The proposal is consistent with recent empirical findings that both anaesthesia and Alzheimer's disease specifically reduce synergistic information processing while preserving or increasing redundancy.