In order to specify databases completely at the conceptual level, conceptual database specification languages should contain a data definition (sub)language (DDL), for specifying data structures (+constraints), a data retrieval (sub)language (DRL), for specifying queries, as well as a (declarative) data manipulation (sub)language (DML), for specifying transactions. Object Role Modeling (ORM) is a powerful method for designing and querying database models at the conceptual level. By means of verbalization the application is also described in natural language as used by domain experts, for communication and validation purposes. ORM currently comprises a DDL and a DRL (ConQuer). However, the ORM-method does not yet contain an expressive DML for specifying transactions at the conceptual level. In an earlier paper we designed a syntactic extension of the ORM-method with a DML for specifying transactions at the conceptual level in a purely declarative way. For all transactions we proposed syntaxes, verbalizations, and diagrams. However, we did not give a formal semantics then. The purpose of this paper is to add a clear, formal and purely declarative semantics to the proposed ORM-transactions. The paper also formally defines rollbacks and illustrates everything with examples (including a solution to a well-known transaction specification problem). The extension of ORM with an expressive set of completely declaratively specified transactions makes ORM complete as a database specification method at the conceptual level. (C) 2016 Elsevier Ltd. All rights reserved.