Abstract: Large language models (LLMs) are capable of solving a wide range of tasks, yet they have struggled with reasoning.To address this, we propose $\textbf{Additional Logic Training (ALT)}$, which aims to enhance LLMs' reasoning capabilities by program-generated logical reasoning samples.We first establish principles for designing high-quality samples by integrating symbolic logic theory and previous empirical insights.Then, based on these principles, we construct a synthetic corpus named $\textbf{Formal} \ \textbf{Logic} \ \textbf{\textit{D}eduction} \ \textbf{\textit{D}iverse}$ (FLD$^{\times2}$), comprising numerous samples of multi-step deduction with unknown facts, diverse reasoning rules, diverse linguistic expressions, and challenging distractors.Finally, we empirically show that ALT on FLD$^{\times2}$ substantially enhances the reasoning capabilities of state-of-the-art LLMs, including LLaMA-3.1-70B.Improvements include gains of up to 30 points on logical reasoning benchmarks, up to 10 points on math and coding benchmarks, and 5 points on the benchmark suite BBH.Case analyses demonstrate that LLMs successfully integrate their knowledge acquired during pre-training with reasoning capabilities acquired through ALT.