Natural Language Inference (NLI) is the task of determining whether a premise entails a hypothesis. NLI with temporal order is a challenging task because tense and aspect are complex linguistic phenomena involving interactions with temporal adverbs and temporal connectives. To tackle this, temporal and aspectual inference has been analyzed in various ways in the field of formal semantics. However, a Japanese NLI system for temporal order based on the analysis of formal semantics has not been sufficiently developed. We present a logic-based NLI system that considers temporal order in Japanese based on compositional semantics via Combinatory Categorial Grammar (CCG) syntactic analysis. Our system performs inference involving temporal order by using axioms for temporal relations and automated theorem provers. We evaluate our system by experimenting with Japanese NLI datasets that involve temporal order. We show that our system outperforms previous logic-based systems as well as current deep learning-based models.