It has been relatively easy to locate a natural semantics for propositional logics that involve connectives other than disjunction. Discussion of disjunction has been postponed because the situation here is more complicated. It is possible to locate a condition ‖∨‖ on models that the rules for disjunction express (Section 7.2). However, it is unfamiliar; ‖∨‖ is neither the classical condition nor the intuitionistic reading of ∨ that was introduced by Beth (Section 7.1). Furthermore, there is reason to worry whether ‖∨‖ qualifies as a recursive characterization of truth conditions (Section 7.3). So it will be necessary to try to rescue ‖∨‖ (if we can) with a new isomorphism result. In Section 7.4, a variant of the Kripke semantics will be introduced (called path semantics) that includes an additional structure used in the disjunction truth condition. An isomorphism is shown to exist between the models in path semantics and models that obey ‖∨‖ (Section 7.5). This goes part of the way towards legitimizing the condition expressed by the disjunction rules as qualifying as a semantics. However Section 7.6 demonstrates that neither functionality nor a desirable form of compositionality holds for ‖∨‖. So whether the isomorphism result goes far enough to offset this pathology is in doubt. Whether we should accept ‖∨‖ as a legitimate reading for ∨ will be something left for the reader to judge.