A Minimal $\kappa$--$\tau$ Logic for Risk-Sensitive Abduction

Abstract

Standard approaches to abductive reasoning can retain multiple candidate explanations, but they do not generally combine explicit compositional cross-hypothesis interaction with an internal, rival-sensitive commitment judgment. This paper argues that in risk-sensitive domains -- where premature commitment carries asymmetric downside costs -- the timing of commitment is itself a governed decision that the inferential apparatus should formally represent. We present a minimal $\kappa$--$\tau$ logical framework built on two primitives: epistemic interaction among hypotheses ($\kappa$) and a normative commitment threshold ($\tau$). Hypotheses may coexist, reinforce or inhibit one another, and form emergent composite explanations, while collapse into committed conclusions is regulated by governance constraints rather than forced by inference alone. The logic is developed in two complementary modes sharing the interaction relation and the governance apparatus: a synthetic mode, in which atomic hypotheses are composed upward into emergent explanations, and an analytic mode, in which complex observed states of affairs are decomposed into causal clusters of latent factors, with commitment governed at both the cluster and the factor level. The framework provides formal machinery for domains in which the distinction between highly likely and commit-worthy is operationally consequential. The $\kappa$--$\tau$ logic is positioned as the symbolic governance layer of a neurosymbolic architecture: its epistemic parameters are naturally estimated by neural components -- semantic embeddings and generative models, as demonstrated in existing computational realizations -- while its normative parameters remain under explicit human governance, yielding transparent and auditable abductive reasoning for deployment in high-stakes settings.