Large language models (LLM) are being increasingly applied to tasks requiring commonsense reasoning. Despite their outstanding potential, the reasoning process of LLMs is prone to errors and hallucinations that hinder their applicability, especially in high-stakes scenarios. Several works have attempted to enhance commonsense reasoning performance of LLMs by (i) using prompting styles that elicit more accurate reasoning, (ii) utilizing the LLM as a semantic parser for a symbolic reasoner, or (iii) enforcing the LLM to simulate a logical inference rule. However, all these solutions have critical limitations: they are unable to leverage the internal commonsense knowledge of the LLM in tandem with an axiomatic knowledge base, they lack a mechanism to reliably repair erroneous inference steps, and their application is restricted to small knowledge bases that fit the context limit of the LLM. In this work, we present LLM-based Typed Hyperresolution (LLM-TH), a logical commonsense reasoning framework that leverages "theory resolution", a concept from classical logical inference which enables integrating LLMs into the "resolution" inference rule, thus mitigating reasoning errors and hallucinations and enabling verification of the reasoning procedure. LLM-TH is also equipped with a mechanism for repairing erroneous inference steps supported by theoretical guarantees. Using "Hyperresolution" and "Typed inference" schemes, we show that LLM-TH can efficiently reason over large knowledge bases consisting of tens of thousands of rules with arbitrary predicate arities. Our experiments on three diverse language-based reasoning tasks—preference reasoning, multi-domain deductive reasoning, and geographical question answering—showcase that LLM-TH, using merely a BART 406M parameter NLI entailment model, significantly reduces reasoning errors compared to baselines using Llama3-70B, Gemini1.5-Flash, GPT-3.5-Turbo, and Mixtral-46.7B.