Trust and Its Betrayal under Three Representational Strategies

arXiv:2608.00321v1 Announce Type: new Abstract: Trust is a propositional attitude of a distinctive kind: to trust is to rely on another under conditions where reliance could be disappointed, and the disappointment of trust---betrayal---differs qualitatively from the disappointment of a prediction. We treat trust as a \emph{subjunctive} epistemic state: $A$ trusts $B$'s competence when $A$ believes that \emph{were $P$ true, $B$ would know it}, and $B$'s integrity when $A$ believes that \emph{were $B$ to know $P$, he would disclose it to $A$}. We develop three representations of this state---as lexicographic \emph{assumption} as \emph{ordinal closeness} in a Lewis--Stalnaker sphere system , and as \emph{strong belief} in a conditional probability system and for each we ask whether the Brandenburger--Keisler impossibility on common belief survives when the assumption of rationality is replaced by an assumption of trustworthiness. The three representations agree that every \emph{finite} depth of common trust is realizable while the \emph{completed} common-trust fixed point is the locus of difficulty, but they differ sharply in \emph{how} the difficulty manifests, and---our organizing finding---in how each survives a concrete betrayal. W show that the same betrayal refutes an agent's \emph{level ordering} under the lexicographic representation, contaminates her \emph{closeness ordering} in proportion to the betrayer's deliberateness under the ordinal representation, and merely \emph{shifts her operative conditioning hypothesis} while leaving her belief structure coherent under the strong-belief representation.