The Overlap Is Not the Bond: A Set-Theoretic and ‎‎Probabilistic Framework for Marriage, Family Structure, and ‎‎Divorce

Abstract

This article develops a theoretical and conceptual mathematical framework for representing marriage, family structure, and ‎divorce. The framework combines set-theoretic representation, normalized similarity measures, active-bond dynamics, ‎probability trees, and ideas from complex systems. Each person is modeled as a time-dependent attribute structure embedded ‎in a universal attribute space. The central modeling assumption is a non-identity principle: within a sufficiently rich ‎description, two distinct persons are represented by non-identical identity structures. Interpersonal closeness is therefore not ‎defined by complete equality, but by normalized overlap measures such as the Jaccard index and, in a more general ‎formulation, by weighted or fuzzy similarity measures. Marriage is modeled not merely as the intersection of two person-sets, ‎but as an active marital-bond structure supported by shared values, emotional attachment, cooperation, legal or symbolic ‎commitment, residence, children, shared responsibilities, and common goals. Love is represented as a time-dependent ‎reinforcement variable that may increase, decrease, stabilize, or recover according to interaction history, conflict, repair ‎capacity, and deliberate maintenance. Because relationship outcomes are high-dimensional, context-sensitive, and path-‎dependent, the article does not propose a universal deterministic equation for marriage or divorce. Instead, it formulates a ‎probability-tree representation in which branch probabilities may depend on personality, culture, socioeconomic pressure, ‎family interference, previous reactions, and timing. Divorce is defined as the collapse of the active marital-bond measure ‎rather than the disappearance of all shared attributes. The framework is extended to nuclear and extended family structures ‎and is connected to family systems theory, mathematical sociology, agent-based modeling, and complex-systems reasoning. ‎The numerical examples and Monte Carlo simulation included in the article are illustrative and are not presented as empirical ‎divorce predictions. No empirical validation is reported in the present paper. Rather, the model is proposed as a formal ‎theoretical scaffold designed for near-future empirical testing, in which the marital-bond measure and probability-tree branch ‎probabilities may be operationalized, estimated, and evaluated using longitudinal couple data.‎