ParFuzGenQ · Paraconsistent Fuzzy Logic with Generalized Quantifiers

Paraconsistent Fuzzy Logic with Generalized Quantifiers

The research project ParFuzGenQ (Project No 297799) proceeded according to the research plan, as displayed in the Annex of the Grant Agreement. During the two year period, researcher Esko Turunen focused on paraconsistent logics, t-norm based fuzzy logics, in particular Pavelka style logics, generalized quantifiers and their real life applications, in particular medical expert systems. • Classical logic, being mainly the logic of mathematical reasoning, is two valued and therefore difficult to apply in real-world situations, where vagueness and ill-definable phenomena are often present. In fuzzy logics and many-valued logics the truth value of a proposition can also be partial, something between true and false. An important many valued logic was invented already in 1920s by Jan Lukasiewicz. After decades of silence, its fundamental importance was discovered. Logics can be studied by algebraic methods; the algebraic counterpart of Lukasiewicz’ infinite valued logics are MV-algebras, further developed by Jan Pavelka to fuzzy logic direction. Complete MV-algebras and their logics are quite well known, while other MV-algebras and their logic are less known. Turunen focused on perfect MV-algebras and their logic. Perfect MV-algebras are structures, where below absolute truth there are infinite many quasi truths and, symmetrically, above absolute false there are infinite many quasi falsehoods. These two sets are separate. In a joint work Perfect Pavelka Logic with Mirko Navara, Turunen studied Pavelka style logic in perfect MV. This logic has a special feature that quasi true premises never lead to conclusions that would be false. Turunen also started to write a book entitled The Logic of Quasi Truth – an Algebraic Treatment jointly with Antonio di Nola and Revaz Grigolia. This monograph will be published in the book series Studies in Fuzziness and Soft Computing (Springer Verlag) in autumn 2015. • In classical logic we are well off with the quantifiers ‘For all’ and ‘There exists’ while in real life applications they will not do. Generalized quantifiers such as ‘Almost all’, ‘Most’, ‘Many’, ‘Above average’ etc in fuzzy logic framework are presently a hot research topic. In his paper An Algebraic study of Peterson’s Intermediate Syllogisms, Turunen approached the problem from syllogistic point of view; generalized syllogisms are special instances of generalized quantifiers. An example of the 120 valid (of 4000 possible) generalized syllogism is the following Almost all jokes are old (premise 1) Many jokes are funny (premise 2) --------------------------------------------- Some old jokes are funny (conclusion) Turunen proved that Peterson’s intermediate syllogisms obey algebraically an MV-structure; this paves the way for the ongoing follow-up studies. • The general aim of data mining is to extract knowledge from big data matrices, where data is often unstructured. Typical matrices are size of tens of thousands of rows and hundreds of columns; columns may contain, say, information about patients’ symptoms, illnesses and diagnosis and rows particular patients. GUHA data mining method is a logic based approach to extract knowledge from such big data matrices. The principles of GUHA are implemented to software called LISpMiner. The main goal of GUHA method is to automatically answer to questions ‘Does the given data contain interesting and meaningful dependences?’ By GUHA methodology we can formally express, what are called analytic questions; e.g. ‘What short of symptoms are almost always related to which diagnosis?’ and find answers automatically. Thus GUHA logic is also closely related to generalized quantifiers. However, by definition, GUHA only offers hypothesis supported by the given data, their statistical relevance is not automatically tested. To overcome this deficiency, Turunen published a joint paper Bayesian analysis of GUHA hypothesis with Robert Piche, Marko Järvenpää and Milan Simunek. This research paper, whose results are now implemented also to LISpMiner software, connects Bayesian statistics to non-classical GUHA logics; it gives a concrete tool to interpret GUHA data mining results to Bayesian statistical language. • Classical logic as well as most non-classical logics are explosive in the sense that from a one single inconsistent statement A, i.e. that both A and non-A are simultaneously true, anything can be inferred. Such a feature is, however, unrealistic in certain real life application; e.g. Judge can give a fair judgment, even if the evidence would be partially contradictory. Para consistent logics challenge the consistency demand of classical logic; even if we are in certain circumstances where the available information is inconsistent, the inference relation does not explore into triviality. Paraconsistent logics are particular applicable in solving decision making problems. Turunen showed in two join paper with Tinguaro Rodriguez entitled Another paraconsistent algebraic semantics for Lukasiewicz-Pavelka Logic and Two Consistent Many Valued Logics for Paraconsistent Phenomena how Pavelka style fuzzy logic can be equipped with several paraconsistent semantics reflecting real life situations where inconsistent information is present.

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