Multiparty dialogue question answering (QA) in machine reading comprehension (MRC) is a challenging task due to its complex information flow interactions and logical QA inference. Existing models typically handle such QA tasks by decoupling dialogue information at both speaker and utterance levels. However, few of them consider the logical inference relations in multiparty dialogue QA, leading to suboptimal QA performance. To address this issue, this paper proposes a memory network with logical inference (LIMN) for extractive QA in multiparty dialogues. LIMN introduces an inference module, which is pretrained by incorporating plain QA articles as external knowledge. It generates logical inference-aware representations from latent space for multiparty dialogues. To further model complex interactions among logical dialogue contexts, questions and key-utterance information, a key-utterance-based interaction method is proposed for leverage. Moreover, a multitask learning strategy is adopted for robust MRC. Extensive experiments were conducted on Molweni and FriendsQA benchmarks, which included 25k and 10k questions, respectively. Comparative results showed that LIMN achieves state-of-the-art results on both benchmarks, demonstrating the enhancement of logical QA inference in multiparty dialogue QA tasks.