In this note, we focus on labeled Petri nets and formalize two properties, language divergence and marking divergence, discussing their relevance for diagnosability analysis. In particular, we review the results for diagnosability and K-diagnosability presented in an article entitled "A new approach for diagnosability analysis of Petri nets using verifier nets" that we coauthored. We show that these results apply to nets that are language divergence-free, an assumption that was not explicitly mentioned in that article. In addition, we also provide an alternative structural assumption-which does not require testing the behavioral property of divergence-freeness-under which the above results hold.