Knowability as Continuity: A Topological Account of Informational Dependence

This paper introduces a sophisticated framework for understanding and modeling informational dependence and knowability in empirical settings, moving beyond traditional logical approaches that often assume idealized, sharp values. It achieves this by leveraging the mathematical tools of general topology and metric spaces.

The central insight is to interpret empirical “variables” (such as physical quantities like position or velocity) as functions mapping states of the world to topological spaces, where open sets represent observable, imprecise measurements or “approximations.” Similarly, “questions” are formalized as topologies on the state space, with open neighborhoods representing partial answers.

The paper’s significance lies in bridging the gap between abstract logical notions of dependence and the realities of empirical inquiry, where observations are inherently imprecise. It proposes that knowability can be understood as continuity. Specifically:
* Exact dependence (as studied in prior work like Logic of Functional Dependence, LFD) means that knowing the precise value of one variable (X) completely determines the precise value of another (Y).
* Continuous dependence (referred to as “epistemic dependence”) is introduced as a more realistic notion. If Y depends continuously on X, it means that an observer can come to know the value of Y to any desired accuracy, provided they obtain sufficiently accurate approximations of X. This is formalized as the global continuity of the function mapping X-values to Y-values in a topological setting. A locally continuous version captures “knowable epistemic dependence,” allowing for this inference within a specific neighborhood.
* Uniformly continuous dependence (referred to as “strong epistemic dependence”) represents an even stronger form of knowability, particularly relevant in metric spaces. It captures the idea of “epistemic know-how,” where the observer knows in advance precisely how accurate their measurement of X needs to be to achieve a desired accuracy for Y, uniformly across the domain.

Key innovations include:
1. Topological Semantics for Dependence: Formalizing empirical variables and questions as topological maps and spaces, respectively, allowing for the representation of imprecise observations and approximations.
2. Graded Notions of Knowability: Introducing and formally distinguishing between mere functional dependence, continuous dependence (which implies information flow and knowability), and uniformly continuous dependence (which captures prescriptive “know-how”).
3. Analysis of Independence: Defining “topological independence,” where observations of X yield no new information about Y. Crucially, the paper demonstrates the counter-intuitive finding that functional dependence and topological independence can coexist – meaning Y might be completely determined by X, yet no approximate measurement of X can reveal anything new about Y.
4. Logical Systems: The development of two new logical systems, LCD (Logic of Continuous Dependence) and LUD (Logic of Uniform Dependence). These logics are shown to be sound, complete, and decidable for their respective classes of models (topological dependence models for LCD, and metric/pseudo-locally Lipschitz models for LUD), providing rigorous tools for reasoning about these new epistemic concepts.
5. Epistemic Opacity of Point-Continuity: The paper highlights that mere point-continuity (a very local form of dependence) does not guarantee knowability in a robust sense, deeming it “epistemically opaque” due to its extreme sensitivity to small errors in observation.
6. “Epistemic Bootstrapping”: The observation that in “favorable” environments (like locally compact spaces, which include Euclidean spaces), epistemic dependence automatically “bootstraps” to locally strong (uniformly continuous) dependence, simplifying the path from scientific understanding to practical know-how.

The main prior ingredients upon which this work builds are:
* Dependence Logic (DL) and Logic of Functional Dependence (LFD): These prior logical systems established the formal study of exact, set-theoretic functional dependence between variables and questions, providing the foundational concepts that this paper then extends topologically.
* General Topology: Concepts such as topological spaces, open sets, neighborhoods, continuity, interior operators, and product spaces are fundamental for defining the empirical setting and notions of knowability.
* Metric Space Theory: Crucial for formalizing the stronger notion of uniform continuity and the concept of “know-how,” which require notions of distance and error.
* Modal Logic: The logical framework used for axiomatization, including S4 modal axioms for knowledge/knowability, and standard techniques for proving completeness and decidability (e.g., canonical models, p-morphisms).
* Epistemic Logic: Provides the philosophical and logical context for reasoning about knowledge, belief, and information, informing the epistemic interpretations of the mathematical concepts.
* Situation Theory: An older theory that viewed information flow through “constraints” between situations, providing a philosophical precursor to the idea of dependence as an information-carrying relation.