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A Philosophical Treatise on the Connection of Scientific Reasoning

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A Philosophical Treatise on the Connection of Scientific Reasoning mathematics Review A Philosophical Treatise on the Connection of Scientific Reasoning with Fuzzy Logic Evangelos Athanassopoulos 1 and Michael Gr. Voskoglou 2,* 1 Independent Researcher, Giannakopoulou 39, 27300 Gastouni, Greece; [email protected] 2 Department of Applied Mathematics, Graduate Technological Educational Institute of Western Greece, 22334 Patras, Greece * Correspondence: [email protected] Received: 4 May 2020; Accepted: 19 May 2020; Published:1 June 2020 Abstract: The present article studies the connection of scientific reasoning with fuzzy logic. Induction and deduction are the two main types of human reasoning. Although deduction is the basis of the scientific method, almost all the scientific progress (with pure mathematics being probably the unique exception) has its roots to inductive reasoning. Fuzzy logic gives to the disdainful by the classical/bivalent logic induction its proper place and importance as a fundamental component of the scientific reasoning. The error of induction is transferred to deductive reasoning through its premises. Consequently, although deduction is always a valid process, it is not an infallible method. Thus, there is a need of quantifying the degree of truth not only of the inductive, but also of the deductive arguments. In the former case, probability and statistics and of course fuzzy logic in cases of imprecision are the tools available for this purpose. In the latter case, the Bayesian probabilities play a dominant role. As many specialists argue nowadays, the whole science could be viewed as a Bayesian process. A timely example, concerning the validity of the viruses’ tests, is presented, illustrating the importance of the Bayesian processes for scientific reasoning. Keywords: inductive and deductive reasoning; fuzzy logic (FL); scientific method; probability and statistics; Bayesian probabilities “Doubt is not a pleasant situation, but the excessive certainty is an unreasonable situation” Voltaire (1694–1778) 1. Introduction It is hard to deny that the rapid and impressive advances of the last 100–150 years in science and technology have been greatly based on the principles of the bivalent logic of Aristotle, which has played (and plays) a dominant role for more than 23 centuries for the progress of the Western civilization. On the contrary, ideas of multi-valued logics were traditionally connected to the culture and customs of the Eastern countries of Asia, being inherent in the philosophy of Buddha Siddhartha Gautama, who lived in India during the fifth century BC. Nevertheless, the formalization of those logics was performed in the West with the Zadeh’s Fuzzy Logic (FL), an infinite-valued logic based on the mathematical theory of fuzzy set (FS) [1]. The instantaneous switch from truth (1) to falsity (0) can easily distinguish propositions of classical logic from those in FL with values lying in the interval [0, 1]. As it usually happens in science with such radical ideas, FL, when first introduced during the 1970s, was confronted with distrust and reserve by most mathematicians and other positive scientists. However, it has been eventually proved to be an effective generalization and complement of the classical logic and has found many and important applications to almost all sectors of human activity, e.g., [2]: Chapters 4–8, [3], etc. Mathematics 2020, 8, 875; doi:10.3390/math8060875 www.mdpi.com/journal/mathematics Mathematics 2020, 8, x FOR PEER REVIEW 2 of 16 Mathematicsand complement2020, 8, 875 of the classical logic and has found many and important applications to almost all2 of 15 sectors of human activity, e.g., [2]: Chapters 4–8, [3], etc. Human reasoning is characterized by inaccuracies and uncertainties, which stem from the Human reasoning is characterized by inaccuracies and uncertainties, which stem from the nature nature of humans and the world. In fact, none of our senses and observation instruments allows us of humans and the world. In fact, none of our senses and observation instruments allows us to reach to reach an absolute precision in a world which is based on the principle of continuity, as opposed to an absolute precision in a world which is based on the principle of continuity, as opposed to discrete discrete values. Consequently, FL, introducing the concept of membership degree that allows a values.condition Consequently, to be in a state FL, other introducing than true the or conceptfalse, provides of membership a flexibility degree to formalize that allows human a reasoning. condition to beAnother in a state advantage other than of trueFL is or that false, rules provides are set ain flexibility natural language to formalize with the human help reasoning.of linguistic, Another and advantagetherefore offuzzy, FL is variables. that rules In are this set way, in natural the po languagetential ofwith FL becomes the help strong of linguistic, enough and to enable therefore fuzzy,developing variables. a model In this of way, human the potentialreasoning ofproceeding FL becomes in natural strong enoughlanguage. to Furthermore, enable developing since at a modelthe of humanbasis of reasoning in proceeding expert systems in natural are human language. notions Furthermore, and concepts, sincethe success at the of basis these of systems reasoning in expertdepends systems upon the are correspondence human notions between and concepts, human the reasoning success and of these their systems formalization. depends Thus, upon FL the correspondenceappears as a powerful between theoretical human reasoning framework and for their studying formalization. not only human Thus, FLreasoning, appears but as aalso powerful the theoreticalstructure framework of expert systems. for studying not only human reasoning, but also the structure of expert systems. However,However, only only a a limited limited number number ofof reportsreports appear in in the the literature literature connecting connecting FL FLto tohuman human reasoning;reasoning; e.g., e.g., see [see4–6 [4–6].]. This This was ourwas mainour motivationmain motivation for performing for performing the present the present study by study continuing by ourcontinuing previousanalogous our previous efforts. analogous In fact, efforts. in a recent In fact book, in writtena recent in book the Greekwritten language, in the Greek the first language, author of thisthe article first studiesauthor of the this inductive article studies reasoning the inductiv under thee reasoning light of FL under and criticizesthe light of the FL excessive and criticizes accusations the excessive accusations of the Philosophy of Science against it [7]. Additionally, in an earlier work [8], of the Philosophy of Science against it [7]. Additionally, in an earlier work [8], the second author has the second author has introduced a model for analyzing human reasoning by representing its steps introduced a model for analyzing human reasoning by representing its steps as fuzzy sets on a set of as fuzzy sets on a set of linguistic labels characterizing the individual’s performance in each of those linguistic labels characterizing the individual’s performance in each of those steps. steps. TheThe target target of of the the article article at at hand hand isis toto analyzeanalyze the scientific scientific method method of of reasoning reasoning under under the the light light of FL.of FL. The The rest rest of of the the article article is is organized organized asas follows:follows: Section 2 examines the the inductive inductive and and deductive deductive reasoningreasoning from from the the scope scope of FL.of SectionFL. Section3 explains 3 expl howains thesehow these two fundamentaltwo fundamental components components of human of reasoninghuman reasoning act together act fortogether creating, for creating, improving improv anding expanding and expanding the scientific the scientific knowledge knowledge and presents and a graphicalpresents a representationgraphical representation of the scientific of the scientific progress progress through throug theh centuries. the centuries. Section Section4 studies 4 studies the unreasonablethe unreasonable effectiveness effectiveness of mathematics of mathematics in the naturalin the natural sciences sciences (Winger’s (Winger’s enigma), enigma), while Sectionwhile 5 quantifiesSection 5 the quantifies degree ofthe truth degree of of the truth inductive of the in andductive deductive and deductive arguments. argume Thents. article The closesarticle closes with the generalwith conclusionsthe general conclusions presented inpresen Sectionted6 .in A Section schematic 6. A diagramschematic of diagram the proposed of the studyproposed is presented study is in Figurepresented1. in Figure 1. FigureFigure 1. 1.Schematic Schematic diagram of of the the proposed proposed study. study. TheThe novelty novelty of of the the present present article article isis thatthat it brings together together philosophy, philosophy, mathematics mathematics and and fuzzy fuzzy logic,logic, thus thus becoming becoming a challenge a challenge for for further further scientific scientif discussionic discussion on on the the subject. subject. This This is theis the added added value of thisvalue work. of this work. 2. Human2. Human Reasoning Reasoning and and Fuzzy Fuzzy Logic Logic InductionInduction and and deduction deduction are are the the two two fundamentalfundamental mechanisms mechanisms of of human human reasoning. reasoning. Roughly Roughly speaking,speaking, induction induction is is the the process process of ofgoing going fromfrom
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