First-order logic (FOL) is a fundamental formalism for factual reasoning over knowledge graphs (KGs), e.g. in researches of KG-based fact verification and logical consistency or reasoning of large language models (LLM).However, existing benchmarks and approaches insufficiently capture many claims that require comparison or counting, and lack support for several FOL quantifiers and connectives.To address these challenges and expand the expressive capacity of FOL for KG-based reasoning, we introduce FOLX-KG, a novel extended FOL σ-structure over KGs that incorporates comparison predicates and counting quantifiers.Using this extended logic, we construct Fact-FOLX-KG, a fact verification dataset consisting of 43,821 KG-based claim-formula pairs designed to enable systematic study of richer logical forms and reasoning types.We further propose FOLX Prover, an executable program-guided logic reasoning pipeline adapted for KG-based factual reasoning under the extended FOL.Experimental results show that our method achieves state-ofthe-art performance on Fact-FOLX-KG, while previous methods experience performance drop on claims requiring comparison and counting.These findings demonstrate the importance of extended logical expressiveness for robust factual reasoning over KGs. 1