In this paper we consider perspectives of application of coinductive and corecursive methods of non-well-founded mathematics to philosophical logic. So, it is shown that the problem of analysis can be solved by using greatest fixed points. Means of well-founded mathematics are enough only for an explication of the trivial analysis. We claim that the nontrivial analysis should be explicated by means of non-well-funded mathematics. Further, we build a non-well-founded propositional logic with syntax and semantics whose objects are defined by coinduction as streams. We also survey perspectives of relationship between non-well-founded logics and unconventional computing.