On the logical skills of large language models: evaluations using arbitrarily complex first-order logic problems

We present a method of generating first-order logic statements whose complexity can be controlled along multiple dimensions. We use this method to automatically create several datasets consisting of questions asking for the truth or falsity of first-order logic statements in Zermelo-Fraenkel set theory. While the resolution of these questions does not require any knowledge beyond basic notation of first-order logic and set theory, it does require a degree of planning and logical reasoning, which can be controlled up to arbitrarily high difficulty by the complexity of the generated statements. Furthermore, we do extensive evaluations of the performance of various large language models, including recent models such as DeepSeek-R1 and OpenAI's o3-mini, on these datasets. All of the datasets along with the code used for generating them, as well as all data from the evaluations is publicly available at https://github.com/bkuckuck/logical-skills-of-llms.