Abstract
A Subjective Logic (SL) binomial opinion (b, d, u, a) is exactly a neutrosophic triple (T, I, F) = (b, u, d) restricted to the plane T + I + F = 1, plus a base rate a. We show that SL is a retract of the neutrosophic space: every SL opinion embeds injectively, and a normalization map sends every neutrosophic tuple back to an SL opinion, losing precisely the information SL cannot express (incompleteness, intra-opinion contradiction, neutrality, open-world “neither”, and the provenance of uncertainty). We then treat four neutrosophic logics as successive extensions of SL. For (T, I, N, F) we prove that neutrosophic negation projects onto SL complement. For (T, I, N, U₁, …, Uₙ, F) we prove that SL trust discounting and evidence ageing are the projections of componentwise multi-uncertainty operators. For the n-valued refined neutrosophic logic we prove that Smarandache’s n-norm conjunction equals SL multiplication with base rates 0, and the n-conorm disjunction equals SL comultiplication with base rates 1 (verified numerically to 10⁻¹⁶). For the Refined Neutrosophic Evidential Logic (T, C, U, N, G, I₁, I₂, …, F) we show that SL’s cumulative fusion is the evidence-count special case, and that RNEL keeps the between-source conflict that SL fusion erases. We also apply neutrosophic probability to SL: an SL opinion is a complete neutrosophic probability whose indeterminate chance is resolved by the base rate, and a neutrosophic opinion separates ontic from epistemic indeterminacy. Finally, we extend all four logics and neutrosophic probability to Smarandache’s over/under/off values (components above 1 or below 0): signed evidence gives such values a Beta semantics compatible with SL, the operators keep their De Morgan and projection properties on off-tuples, and we define an over/under/off RNEL and a refined neutrosophic off-probability. Each section ends with a worked application: online reputation, cyber-threat-intelligence trust chains, physics and multi-class diagnosis, conflict-aware retrieval-augmented generation, an election with indeterminate ballots, and, for the off-extensions, fake reviews, a compromised relay, retracted sources and an over-vote.