This theoretical article explores the affordances and challenges of Euler diagrams as tools for supporting undergraduate introduction-to-proof students to make sense of, and reason about, logical implications. To theoretically frame students' meaning making with Euler diagrams, we introduce the notion of logico-spatial linked structuring (or LSLS). We argue that students' use of Euler diagrams as representations of logical statements entails a conceptual linking between spatial and non-spatial representations, and the LSLS framework provides a tool for modeling this conceptual linking. Moreover, from our Piagetian epistemological perspective, reasoning with Euler diagrams entails engaging in spatial mental operations and making a logical conclusion from the result. We illustrate the utility of the LSLS framework through examples with two undergraduate students as they reasoned about the truth of the converse and contrapositive of a given logical implication, and we identify specific spatial operations that they used and coordinated in their problem solving.