Reductive Reasoning Rough and Fuzzy Sets as Frameworks for Reductive Reasoning

Abstract

AbstractReductive reasoning, in particular inductive reasoning, Bocheński [9], Łukasiewicz [30], is concerned with finding a proper p satisfying a premise p⇒q for a given conclusion q. With some imprecision of language, one can say that its concern lies in finding a right cause for a given consequence. As such, inductive reasoning does encompass many areas of research like Machine Learning, see Mitchell [37], Pattern Recognition and Classification, see Duda et al. [16], Data Mining and Knowledge Discovery, see Kloesgen and Zytkow [26], all of which are concerned with a right interpretation of data and a generalization of findings from them. The matter of induction opens up an abyss of speculative theories, concerned with hypotheses making, verification and confirmation of them, means for establishing optimality criteria, consequence relations, non–monotonic reasoning etc. etc., see, e.g., Carnap [12], Popper [55], Hempel [22], Bochman [10].Our purpose in this chapter is humble; we wish to give an insight into two paradigms intended for inductive reasoning and producing decision rules from data: rough set theory and fuzzy set theory.We pay attention to structure and basic tools of these paradigms; rough sets are interesting for us, as forthcoming exposition of rough mereology borders on rough sets and uses knowledge representation in the form of information and decision systems as studied in rough set theory. Fuzzy set theory, as already observed in Introduction, is to rough mereology as set theory is to mereology, a guiding motive; in addition, main tools of fuzzy set theory: t–norms and residual implications are also of fundamental importance to rough mereology, as demonstrated in following chapters.KeywordsDecision RuleInductive ReasoningDecision SystemInductive Logic ProgrammingFuzzy PartitionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.