Relation prediction in knowledge graphs (KGs) aims at predicting missing relations in incomplete triples, whereas the dominant embedding paradigm has a restriction on handling unseen entities during testing. In the real-world scenario, the inductive setting is more common because entities in the training process are finite. Previous methods capture an inductive ability by implicit logic in KGs. However, it would be challenging to preciously acquire entity-independent relational semantics of compositional logic rules and to deal with the deficient supervision of logic caused by the scarcity of relational semantics. To this end, we propose a novel graph convolutional network (GCN)-based model LogCo with logical reasoning by contrastive representations. LogCo firstly extracts enclosing subgraphs and relational paths between two entities to supply the entity-independence. Then a contrastive strategy for relational path instances and the subgraph is proposed for the issue of deficient supervision. The contrastive representations are learned for a joint training regime. Finally, prediction results and logic rules for reasoning are attained. Comprehensive experiments on twelve inductive datasets show that LogCo achieves outstanding performance comparing with state-of-the-art inductive relation prediction baselines.