In this work, we show the pre-trained language models return distinguishable generation probability and uncertainty distribution to unfaithfully hallucinated texts, regardless of their size and structure. By examining 24 models on 6 data sets, we find out that 88-98% of cases return statistically significantly distinguishable generation probability and uncertainty distributions. Using this general phenomenon, we showcase a hallucination-reducing training algorithm. Our algorithm outperforms other baselines by achieving higher faithfulness metrics while maintaining sound general text quality measures.