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Lukasiewicz and Many-Valued Logics

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Lukasiewicz and Many-Valued Logics 1 Article 2 Lukasiewicz and Many-Valued Logics 3 Angel Garrido 1*, Piedad Yuste 2 4 1 Faculty of Sciences; [email protected] 5 2 Faculty of Philosophy; [email protected] 6 * Correspondence: [email protected]; Tel.: +34-91-398-72-37 7 Academic Editor: Angel Garrido 8 Received: date; Accepted: date; Published: date 9 Abstract: Our initial purpose is contributing to the search for the origins of many-valued logics 10 (MVLs, by acronym), and, within them, as a special case that of “Fuzzy Logic”, also called by 11 different manners, as Diffuse Logic, either Heuristic Logic, or `logique floue´ (in French), etc. Such 12 origins departing to Aristotle goes to our Lvov-Warsaw School, with an essential role played by the 13 philosopher and logician Jan Lukasiewicz. 14 Keywords: Mathematical Logic; Non-Classical Logics; Fuzzy Logic; Uncertainty; Vagueness; 15 Artificial Intelligence. 16 17 1. Introduction 18 The theory of “vague sets” (today so-called Fuzzy Sets) proceeds from the quantum 19 physicist and philosopher Max Black (1937), which also analyzes the problem of modeling 20 vagueness: “It is a paradox that the most developed and useful theories are expressed in terms of 21 objects never encountered in experience. While the mathematician constructs a theory in terms of 22 “perfect” objects, the experimental scientist observes of which the properties demanded by theory 23 are only approximately true. 24 25 2. Results about Vagueness 26 There are at least two fundamental approaches to Fuzzy Logic. The first of them may be 27 considered related with Multi-Valued Logical tradition; properly, the Petr Hájek´s school, centered 28 at Charles University of Prague group, or the group of Ostrava University, with Vilém Novák, Irina 29 Perfilieva, and other researchers [56-60], jointly with important ramifications on nearest countries, as 30 Poland, with Urzsula Wybraniec Skardowska, or Roman Murawski; Austria; Hungary, with Janos 31 Fodor; Belgium, with Etienne E. Kerre et al., or Italy, with Giangiacomo or Brunella Gerla, and also 32 Arianna Betti, for instance, etc. The procedure must be to fix a set of designed values, and then, to 33 define an entailment relation. We may define a suitable set of axioms and Inference Rules, as engine 34 for its deductive apparatus. 35 The second line of advance would be headed by Jan Pavelka, Goguen, Belohlavek, and 36 others, being directed on to provide a deductive apparatus in which approximate reasoning may be 37 admitted. It will be reached by a suitable fuzzy subset of logical axioms, and by Fuzzy Inference 38 Rules. 39 Between both approaches, it will be very different the logical consequence operator. In 40 the first case, it gives the set of logical consequences of a given axiomatic structures. While in the 41 second case it gives the fuzzy subset of logical consequences, into a given fuzzy subset of 42 hypotheses. 43 According to L. A. Zadeh, “More often than not, the classes of objects encountered in 44 the real physical world do not have precisely defined criteria of membership. Clearly, the “class of Axioms 2016, 5, x; doi:10.3390/ www.mdpi.com/journal/axioms Axioms 2016, 5, x 2 of 5 45 all real numbers which are much greater than 1”, or “the class of beautiful women” . , or “the class 46 of tall men”, do not constitute classes or sets in the usual mathematical sense of these terms. Yet, the 47 fact remains that such imprecisely defined “classes” play an important role in human thinking, 48 particularly in the domains of pattern recognition, communication of information, and abstraction”. 49 With the increasing of complexity our skill to make accurate diminishes. Real world problems are 50 too complex, and the difficulty involves the degree of vagueness. 51 Bertrand Russell wisely said that “there are two kinds of science: the old, which is the 52 official, and a new science, that most of the old look with horror. The result is a constant battle 53 between the old minds who admire science for its older and younger women, who appreciate the 54 value of the work of their peers. To some extent, this struggle is useful, but beyond a certain point, it 55 becomes disastrous.” 56 [B. R., Autobiography, p. 722]. 57 58 This lucid commentary, very typical of this British mathematician and philosopher, 59 seems tailored to the controversy that we study, as has rarely been given in the history of the new 60 scientific theories such bitter and unjust insults as those that occurred with the development and 61 deployment of Non-Classical Logics, and particularly of Fuzzy Logic, Fuzzy Logic or also called 62 Heuristic Logic. 63 The logic is a well-constructed set of formulas, with a ratio of inference. However, 64 classical logic is bivalent, therefore, is very limited when it comes to solving problems with 65 uncertainty in the data, which are most common in the real world. It is well known that artificial 66 intelligence logic requires. Because its classic shows too many shortcomings, it is necessary to 67 introduce more sophisticated tools, such as non-classical logics, including fuzzy logic, modal logic, 68 non-monotonic logic, the order-consistent, and so successively. 69 All in the same line: against dogmatism and dualistic view of the world: absolutely true 70 vs. false absolutely, black against white, good or bad by nature, Yes vs. No, 0 vs. 1, full vs. empty, 71 and so on. For this reason, we try to analyze here some of them, being very interesting both classical 72 and modern non-classical logics, focusing in particular on the theme of your reception, which was 73 quite different between countries. 74 At first (like almost all new ideas), the work of Lotfi A. Zadeh [72-73, 85], and other 75 precursors, had no scientific enthusiastic reception, but over time, these ideas gradually gained 76 acceptance in many cases, with the emergence of inspired followers. So these theories have grown 77 considerably in a way, being accepted by scientists today most professional. 78 One of the sharpest and persistent controversies was promoted by the so-called old 79 “patriarch” of the probabilistic Italian Academy, Bruno de Finetti (1906-1985), and his school. 80 However, some minor authors have followed repeating several objections, which are often based on 81 ignorance, prejudice and various misunderstandings. 82 Many statisticians were persuaded-and still are, by the influential work of the author 83 before mentioned about only one type of treatment of uncertainty, from a mathematical point of 84 view is needed, and therefore the fuzzy logic would be superfluous. It's kind of jealous rage after 85 finding that a potentially strong rival invades our “little plot”. 86 Let's look here not only the characteristics of each of this non-classical logic, jointly 87 with their interrelationships and possible applications to various fields, but also and mainly the lines 88 of resistance observed for its implementation and use in our universities to better understand its 89 development and progress. That is, Controversies and Disputes have arisen before its appearance, as 90 a remarkable example of the questions that often accompany any new scientific theories. But even 91 so, in this particular case have shown a fiercer fury, especially in Western countries, instead being 92 very well accepted, and naturally, in Asian countries, especially in Japan. Has the typical mental 93 blindness, some inertia opposed to change, or has been a problem of cultural tradition, imbued as we 94 are a strong Aristotelian tradition in our school systems...? Are all them different issues of great 95 interest for the advancement of science and the historical-philosophical analysis? 96 Axioms 2016, 5, x 3 of 5 97 The concept of validity is the more essential key to logic, because when we affirm the 98 validity of an argument, we are saying it is impossible that its conclusion is false if its premises are 99 true. This leads to two questions still controversial. So, what are the “bearers of truth”? The answers 100 are multiple judgments, statements, sentences, propositions. And what is truth? An issue far more 101 radical and it is no easy answer. Intuitively, we tend to assign to the truth two fundamental 102 properties, namely universality and objectivity. 103 We can all tell the truth, regardless of who, or what, we are. But both properties alone 104 do not guarantee anything, not even differentiate the mere opinion or belief of the truth. But we 105 cannot address an argument on the concept of truth, if we do not decide first what things can be true 106 or false. We say that propositions are those who can be true or false. Propositions are descriptions of 107 the world, are statements or denials of events in possible worlds. 108 The future contingencies were dealt with by Aristotle, as interpreted by Jan 109 Lukasiewicz, pushes us toward fatalism [5-40]. Accepting that a proposition about a future event is 110 true or false becomes necessary or impossible (respectively) the event expressed by the proposition. 111 The solution proposed by himself in his classic work is the acceptance of logic with three truth 112 values, which in addition to true and false, accepting an indeterminate truth value. Naturally the 113 laws of excluded and not stop working contradiction with the effects resulting therefrom as will be 114 seen later. 115 To search for the origins could lead too far and eventually disperse, which, as we know 116 is not very convenient for a job pretending to be research. So we will refer to these first signs that 117 appear in the East (China, India…), and then we may analyze the problem of “future contingents”, 118 treated by Aristotle in Peri hermeneias, or De Interpretatione (in Latin).
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