Coherence Without Complementarity: Dutch Books and Paraconsistent Bayesianism

Canonical Dutch-book arguments are usually presented in Boolean settings. Building on nonclassical convex-hull work and pragmatic-load cautions, we recast coherence as an extension problem: partial ticket prices are coherent when they extend to admissible valuations on gambles. We develop an order-enriched envelope interface that separates structural no-sure-loss constraints from semantic ticket design and from precisification assumptions. For LP, we prove both a finite-fragment representation theorem and a propositional global extension theorem (from finite coherence to global representing measures via compactness and weak-* methods). We then derive LP-specific payoff consequences: the glut decomposition identity, coherence of "high--high" assignments, and a geometric result showing classical probabilism as an exposed face of LP credal geometry. The upshot is that probabilism is structural while complementarity is semantic: LP coherence supports both precise and imprecise regimes depending on ticket design and admissibility.