  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.
