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Ontologies are one of the richest sources of knowledge. Real-world ontologies often contain thousands of axioms and are often human-made. Hence, they may contain inconsistency and incomplete information which may impair classical reasoners to compute entailments that are considered as useful. To overcome these two challenges, we propose FALCON, a Fuzzy Ontology Neural reasoner to approximate reasoning over ALC ontologies. We provide an approximate technique for the model generation step in classical ALC reasoners. Our approximation is not guaranteed to construct exact logical models, but can approximate arbitrary models, which is notably faster for some large ontologies. Moreover, by sampling multiple approximate logical models, our technique supports approximate entailment also over inconsistent ontologies. Theoretical results show that more models generated lead to closer, i.e., faithful approximation of entailment over ALC entailments. Experimental results show that FALCON enables approximate reasoning and reasoning in the presence of inconsistency. Our experiments further demonstrate how ontologies can improve knowledge base completion in biomedicine by incorporating knowledge expressed in ALC.