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Home , Andrzej Mostowski, Berlin Circle, Emil Leon Post, Helena Rasiowa

1 Article 2 Lukasiewicz and Many-Valued

3 Angel Garrido 1*, Piedad Yuste 2

4 1 Faculty of Sciences; [email protected] 5 2 Faculty of ; [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 ”, also called by 11 different manners, as Diffuse Logic, either Heuristic Logic, or `logique floue´ (in French), etc. Such 12 origins departing to goes to our Lvov- School, with an essential role played by the 13 philosopher and logician Jan Lukasiewicz.

14 Keywords: ; 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 , 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 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

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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 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 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 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 , 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-? 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 ”? 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 . 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 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 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). 119 That issue would then central in medieval times, as during the Scholasticism, with 120 , and Duns Scotus, or Richard of Levenham, among others, looked at from 121 different point of views, for his relationship with and `Divine Foreknowledge´. Then, 122 this issue is taken up by Spanish Jesuits Luis de Molina or Francisco Suarez, and even the great 123 polymath G. W. Leibniz dedicated his time [68]. 124 Even then there is a dark time for the logic, and reappearing in the nineteenth century, 125 philosophers and mathematicians such as George Cantor, Augustus De Morgan and George Boole, 126 , C. S. Peirce, ... There was born the new set theory, now called “classic”, but then also 127 had terrible enemies, as the then almighty Leopold Kronecker, who from his professorship in Berlin 128 did everything possible to hinder the work of Cantor, and the rise of those new ideas. 129 Parallel to this, there arises a kind of school of thought and way of seeing must be the 130 act of philosophizing. This is happening like tributaries of the great `river´ and sub-tributaries, from 131 masters to disciples. Starting in the past with G. W. Leibniz, it continues by Bernard Bolzano, 132 which influence-much about his intellectual heir, Franz Brentano. This, in turn, greatly influence on 133 all his subsequent scholars [56, 62-63]. 134 Among these students of Franz Brentano will be one that particularly interested us. 135 This was the Polish philosopher Kazimierz Twardowski, who shared many characteristics with his 136 teacher: love for precision and clarity of ideas, charisma among those who treated him, preference 137 for the spoken to the written word, etc… From his chair in the city of Lvov spread many of the ideas 138 of Franz Brentano, adding their own. So, until to reach a circle of people very interested in the 139 renewal of philosophical studies, especially from the logical point of view. In a certain sense, served 140 a function similar (but independent) to the Circle (Wiener Kreis), or later the Berlin Circle, 141 because very different and singular characteristics. It's called Lvov-Warsaw School. Its members 142 took the logical-philosophical and mathematical studies at Poland to the forefront of global world 143 research. It was during the “interbellum”, or period between the two World Wars, i.e. ranging from 144 1918-1939. Then, rouse the Diaspora, after the war and by the strong communist dictatorship. 145 Many notable names among the members of this school of logic, but could cite to Jan 146 Lukasiewicz, Stanislaw Lesniewski, Alexius Meinong, Kazimierz Ajdukiewicz, Tadeusz 147 Kotarbinski, Mordechai Wajsberg, , Jerzy Slupecki, or . Also must Axioms 2016, 5, x 4 of 5

148 be cited Jan Wolenski (as vindicator of the LWS´ memory), or , , 149 Urszula Wybraniec-Skardowska, and many others [81-83]. 150 The beginning of modern philosophy in Poland was possibly in 1985, when 151 Kazimierz Twardowski (1866-1938) arrived to Lvov train station, to take possession of his Chair at 152 Lvov University. This philosopher belonged to the so-called `Austrian School´, leaded by Franz 153 Brentano. 154 Jan Lukasiewicz (1878-1956), and Wladyslaw Witwicki (1878-1948), are some of the 155 very Twardowski´s oldest disciples. In the case of Lukasiewicz, the first o be fascinated by logic, and 156 then excelled in abstract reasoning; in the case of W. Witwicki, was recognised in Psychology, and 157 good translating to . It was the authentic mentor of Alfred Tarski, very famous after its arrival 158 to California [69-71]. 159 If someone wants to know something about the Lvov-Warsaw School, or in particular 160 on Jan Lukasiewicz, you should read the works of Jan Wolenski, the champion of the memory of this 161 great group of brilliant thinkers. But it is not the professor Wolenski the only valid reference when 162 studying what was the Lvov-Warsaw School and each of its components [64-69, 81-83]. Thus, we 163 have the work of Barry Smith [68], Arianna Betti, Roman Murawski [56], Roger Pouivet, etc. 164 And all this without forgetting the members of the more mathematical branch of the 165 same group, with such notable figures as Stefan Banach, the creator of functional analysis, or the 166 great topologist , among others. In addition, the own Alfred Tarski, who once 167 was the assistant of Jan Lukasiewicz, is at the key intersection of logic and : set theory, 168 topology, model theory, undecidability [70, 71], etc. It is also worthy to be mentioned also the Polish 169 logician Zdzislaw I. Pawlak (1906-2006), another good mathematician, who devised a new approach 170 to the problem of vagueness, introducing his Rough Sets, which can coexist and be a complementary 171 structure to that of Fuzzy Sets. 172 Many European countries have had and still have great specialists in MVLs. Like, for 173 example, this would be the case in Belgium, Etienne E. Kerre, Bernard De Baets, etc. Or in France, 174 Henri Prade or Didier Dubois, as well as Elie Sanchez, and Bernadette Bouchon-Meunier. Also in the 175 Czech Republic, Petr Hájek or Vilem Novák [57, 58, 64, 65]. Or in Germany, with the recently 176 deceased Siegfried Gottwald , or also Hans-Jürgen Zimmermann. 177 One of the most interesting cases in the history of AI is the country of Romania. We have 178 the most landmark in the mathematician Grigore C. Moisil (1906-1973), who introduced Computer 179 Science in their country [42-45], and after he left a very brilliant school of researchers from Romania 180 devoted to mathematics and AI, many of them scattered around the world by the `economic 181 diaspora´, after the Communist period. The study of Lukasiewicz-Moisil Algebras was followed by 182 Gheorge Georgescu and Afrodita Iorgulescu, from Bucharest; also are very relevant Cristian Calude, 183 George Paun (about membrane computing), and some others, in different areas. Also worthy of 184 mention is the figure of Solomon Marcus (1925-), an inspired disciple of Moisil, because Marcus has 185 made great contributions to many fields of Mathematics, such as Logic, Analysis, or Computational 186 Linguistics, of which is one of the founder and principal contributors. 187 Antonio Monteiro (1907-1980), mathematician born in Portuguese Angola, showed that 188 for every monadic Boolean-algebra we can a 3-valued Łukasiewicz-algebra, and that any 189 3-valued Ł-algebra is isomorphic to a Ł-algebra thus derived from a monadic B-algebra. 190 In Poland they follow the great tradition of the LWS of logic and mathematics, and with 191 contributions to research the uncertainty topic through the Rough Sets, by Zdislaw Pawlak 192 (1926-2006), and continued by Andrzej Skowron, among others. 193 194 3. Lukasiewicz and MVLS Axioms 2016, 5, x 5 of 5

195 The roots of the Lvov-Warsaw School (LWS, by acronym) can be traced back to Aristotle 196 himself. But in later times we better put them into thinking GW Leibniz and who somehow inherited 197 many of these ways of thinking, such as the philosopher and mathematician Bernhard Bolzano.

198 Since he would pass the key figure of Franz Brentano, who had as one of his disciples to 199 Kazimierz Twardowski, which starts with the brilliant Polish school of mathematics and philosophy 200 dealt with. Among them, one of the most interesting thinkers must be Jan Lukasiewicz, the father of 201 many-valued logic [68, 80-82].

202 Jan Lukasiewicz (1878-1956) began teaching at the University of Lvóv (now Lwiw; former 203 Lemberg, but also Leópolis), and then at Warsaw, but after World War II must to continue in Dublin. 204 Some questions may be very astonishing in the CV of Lukasiewicz [4-39]. For instance, that a firstly 205 Polish Minister of Education in Paderewski cabinet, into the new Polish Republic, and also Rector for 206 two times at Warsaw University, was awarded with a Doctorate `Honoris Causa´ in spring 1936, at 207 University of Münster, into the maximum of effervescency of Nazism in Germany. The explanation 208 must be their good relation with a very good friend, the former theologian, and then logician, 209 Heinrich Schölz, which was the first Chairman of Mathematical Logic in German universities.

210 Lukasiewicz firstly studied Law, and then Mathematics and Philosophy in Lvov (then 211 Lemberg). His doctoral supervisor was Kazimierz Twardowski, and in 1902 he obtain his Ph. D. title 212 with a very special mention: `sub auspiciis Imperatoris´ (i.e., under the auspices of the Kaiser). Also 213 he received a doctorate ring with diamonds from the Kaiser of the Austro-Hungarian Empire, Franz 214 Joseph I.

215 From 1902, Lukasiewicz was employed as a private teacher, and also as a desk in the 216 Universitary Library of Lvov. So it was until 1904 when he obtained a scholarship to study abroad. 217 He defends his `Habilitationschrift´ in 1906, entitled “Analysis and construction of the concept of 218 cause”. This permits to give university courses. His first lectures were on the Algebra of Logic, 219 according to the recent translation to Polish of this book of the French logician Louis Couturat.

220 Between 1902 and 1906, Lukasiewicz continued his studies in the universities of Berlin and 221 Leuwen (Lovaina). In 1906, by his `Habilitationschrift´, he obtain the qualification as university 222 professor at Lvov. And then, in 1911, he was appointed as associate professor in his `alma mater´ 223 (Lemberg).

224 Jan Lukasiewicz was also very active in historical research on logic. According to Scholz, 225 the better pages on are due to him. And also, as Arianna Betti says, “Jan Lukasiewicz 226 is first and foremost associated with the rejection of the Principle of Bivalence and the discovery of 227 Many-Valued Logic.”

228 The discovery of MVL by Lukasiewicz was in 1918, a little earlier than . 229 According to Jan Wolenski, “although Post´s remarks were parenthetical and extremely condensed, 230 Lukasiewicz explained his intuitions and motivations carefully and at length. He was guided by 231 considerations about future contingents and the concept of possibility”. So, he introduces, firstly, 232 three-valued logic, then four-valued logic, generalized to logics with an arbitrary finite number of 233 veritative values, and finally, to logics with a countably infinite-valued number of such values [56, 234 57, 63, 64, 83].

235 Very noteworthy is his treatment of the history of logic in the light of the new formal 236 logic (then called Logistics). Thus, not only he addressed the issue of future contingents departing 237 from Aristotle, but also put in value logic of the Stoics, at least so far taken. In fact, Heinrich Scholz 238 said, rightly, that Lukasiewicz had written the most lucid pages on all the history of logic [45-52, 239 73-98]. Axioms 2016, 5, x 6 of 5

240 But the writings of Jan Lukasiewicz suffered after a long slumber, which took care to leave 241 an engineer and mathematician Azeri, Lotfi Asker Zadeh [45, 54, 65, 84], who had studied in Tehran, 242 he prosecuted studies at MIT and eventually made landfall as a professor at the University of 243 California, Berkeley. The one who would see their potential utility in 1965, firstly obtaining a 244 generalized version of the classical theory of sets, now denoted by FST, acronym of the so-called 245 “Fuzzy Set Theory”, and later, its application to logic, creating the “Fuzzy Logic”, particularly with 246 the “fuzzy” proposition´s modifiers and fuzzy rule-based systems, very useful for instance, on 247 expert systems, such as the Mamdani, either the Takagi-Sugeno-Kang, or Yatsumoto´s method. 248 Although a number of critics -as ever- many times simply malicious and misinformed, received him 249 with all the “heavy artillery”.

250 And over time, was in Eastern countries where these ideas came to fruition, creating a 251 powerful technological “boom”, with new techniques based on “fuzzy” concepts. This trend was 252 particularly strong in Japan, and then it spread to other countries close to the Japanese country, such 253 as South Korea, China or India.

254 More later those ideas, even more applications came to Western countries, both European 255 and American, and today it will admit, with brilliant studies both from a mathematical point of view 256 and its philosophical implications, as always connected therewith. Some emerging countries, such as 257 Brazil or Turkey, are currently dumped in the investigation of all these theories and associated 258 methods.

259

260 4. A study of case: Reception of many-valued logics in Spain

261 There are certain groups, mostly centered around a “hub”, core or accumulation point, 262 from which new ideas and impulses radiate growing, in the center of each of these “core engine” is 263 usually a -more or less-veteran researcher, well connected and with prestige.

264 One of the first Hispanic scholars giving notice of the new currents was Juan David Garcia 265 Bacca, who in 1936 published his Introduction to modern logic, a work praised by I. M. Bochenski 266 and Heinrich Scholz [45].

267 Later try so eminent teachers, between them Alfredo Deaño (editor by Spanish translation 268 of Lukasiewicz´s selected papers), Miguel Sánchez-Mazas Ferlosio (studying and interpreting the 269 logico-mathematical works of Leibniz, its `characteristic universalis´, etc.) [46, 47, 67], either Jesús 270 Mosterín or Manuel Sacristán [66], very often clashing against a most conservative and nothing good 271 context to innovative ideas.

272 But one good initiative has been the creation in the old mining town of Mieres, and by the 273 Government of Asturias, named the ECSC (`European Center of Soft Computing´, i.e. `Research 274 Center for Artificial Intelligence and Soft Computing´), initially around someone well-known as 275 Enric Trillas, which can be considered the father of the introduction of Fuzzy Logic in the Spanish 276 University curricula [71, 72]. This center has attracted many of the most famous international 277 researchers, such as well-known Japanese Professor Michio Sugeno. His topics of research are very 278 broad working, but revolve around fuzzy methods, as well as philosophical implications these carry.

279 Although I have left for last, a name should not be omitted landmark, from those that 280 appear only from time to time in Spain. We are referring to the Father Pablo Domínguez Prieto 281 (1966-2009), Spanish philosopher and theologian [11, 46], who wrote the first major book in Spain on 282 the Lvov-Warsaw School, starting for that of his doctoral thesis in Philosophy, who had come to the 283 Complutense University at Madrid (1993). Such work is so-called Indeterminación y Verdad. La Axioms 2016, 5, x 7 of 5

284 polivalencia lógica en la Escuela de Lvov-Varsovia (Indeterminacy and Truth), and was published in 1995, 285 with a foreword by Archbishop J. M. Zyzinski, and showing a very strong influence by Jan 286 Wolenski.

287 Father Pablo can be considered as one of the Spanish forerunner in the study of MVLs, from 288 the philosophical point of view and in particular of the great Polish contribution (LWS) to logic and 289 mathematical fields. A romantic `halo´ comes to close its brief existence, because his passion for the 290 mountain climbing made him want to do in the snow Moncayo mountain, after giving lectures to the 291 nuns of the monastery of Tulebras. And that was his last top headlong he died, leaving orphans 292 these Spanish studies again. Left this short comment must be a tribute to his memory.

293 Another interesting Spanish author who has been reporting these new streams of logic is 294 Prof. Julián Velarde [73-78]; fro instance, with its paper “Polyvalent Logic”, or his book Formal Logic, 295 a volume II belonging to its History of Logic, all them around the University of Oviedo and its service 296 publications, or later, to the Editorial Pentalfa. Also of great interest may be his work Gnoseology of 297 Fuzzy Systems, which analyzes the deep philosophical connections of these issues.

298 New research groups have been formed in recent times, as the Spanish CSIC (Consejo 299 Superior de Investigaciones Científicas), in Barcelona, led by Lluis Godo and Francesc Esteva. Or the 300 group that belongs to the UPNA (Public University of Navarra), headed by Humberto Bustince; 301 either in the University of Granada (lead by Miguel Delgado Calvo-Flores), or in the University of 302 Zaragoza, with Tomasa Calvo; even, found some valuable researchers in our own city, Madrid, but 303 also may be cited Malaga, Santiago de Compostela, Oviedo, Almería, etc. [45-52, 66, 67].

304

305 Acknowledgements

306 This work was supported by the MICINN´s Research Project and Investigation Group of our Spanish 307 University (UNED) that we belong to, as a Project entitled

308 Polemics and Controversies

309 (or “El papel de las controversias en la producción de las prácticas teóricas y en el fortalecimiento de la sociedad 310 civil”), being its former Principal Researcher the Professor Quintin Racionero, whom recently has passsed 311 away, and then it´s our IP Prof. Cristina De Peretti.

312

313

314 Conflicts of Interest: Declare conflicts of interest or state “The authors declare no conflict of interest." “The 315 founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in 316 the writing of the manuscript, and in the decision to publish the results”. 317 318 319 320 321 322 323 324 Axioms 2016, 5, x 8 of 5

325 References 326 327 1. - Aristotle, Tratados de Lógica I y II (Organon). Contienen el De Interpretatione. Madrid, Biblioteca Clásica, 328 Madrid, Editorial Gredos, 2000. 329 2. - N. Rescher, Topics in . Kluwer Academic Press, Basel, 1968. Springer, Synthèse 330 Library, 2013. 331 3. - Black, M., “Vagueness: an exercise in logical analysis”. , 4, pp. 427-455, 1937. 332 4. - Ibid., “Reasoning with loose concepts”. Dialogue, 2, pp. 1-12, 1963. 333 5. - Borkowski, L., and Slupecki, J., “The Logical Works of J. Lukasiewicz”, Studia Logica, 8, pp. 7-56, 1958. 334 6. - Berka, K. “Lukasiewicz on Aristotle's Syllogistic”, Ruch Filozoficzny, XXXVI, 1, 1978. 335 7. - Betti, A., “The Incomplete Story of Lukasiewicz and Bivalence”. LOGICA 2001 Conference. 336 8. - Coniglione, F., “Filosofia e scienza in Jan Lukasiewicz”, Epistemologia, 17, 1, pp. 73-100, 1994. 337 9. - Deaño, A., Introducción a la Lógica Formal. Alianza Universidad, Madrid, 2004. 338 10. - Ibid., Jan Łukasiewicz. Estudios de lógica y filosofía. Edición e introducción del profesor Deaño. Madrid, 339 Revista de Occidente. 340 11. - Ibid., 1980, Las concepciones de la lógica. Madrid, Taurus, 1980. 341 12. - Domínguez Prieto, P., Indeterminación y Verdad. La polivalencia lógica en la Escuela de Lvov-Varsovia. 342 Nossa y Jara Editores, Móstoles (Madrid), 1995. 343 13. Lukasiewicz, J., Aristotle´s Syllogistic. Oxford, The Clarendon Press, 1957. 344 14. - Ibid., Elements of Mathematical Logic (in Polish). Warsaw, PWN (Państwowe Wydawnictwo 345 Naukowe), 1963. 346 15. - Ibid., O zasadzie sprzecznosci u Arystotelesa (On the Principle of Contradiction in Aristotle) (1910), 347 PWN, Warszawa 1987. Germ. transl., Über den Satz des Widerspruchs bei Aristoteles, Olms Verlag, 1994 348 (J. Barski). Eng. transl. by Owen LeBlanc in progress. French translation, París, L´Eclat. 349 16. - Ibid., Selected Works. L. Borkowski, as editor. Amsterdam, North Holland C. Publ. Co., 1970. 350 17. - Ibid., Para una historia de la Lógica de Enunciados. Valencia, Cuadernos Teorema, 1974. 351 18. - Ibid., La silogística de Aristóteles desde el punto de vista de la lógica formal moderna. Madrid, Editorial 352 Tecnos, 1977. English edition, Aristotle´s Syllogistic from the Standpoint of Modern Formal Logic. Clarendon 353 Press, Oxford, 1951. 354 19. - Marshall, D., “Lukasiewicz, Leibniz and the arithmetization of the syllogism”. Notre Dame Journal of 355 Formal Logic, 18 (2), pp. 235-242, 1977. 356 20. - Kotarbinski, K., “Jan Lukasiewicz´s works on the history of logic”, Studia Logica, 8, pp. 57-62, 1958. 357 21. - Mostowski, A., “L'oeuvre scientifique de Jan Lukasiewicz dans le domaine de la logique 358 mathématique”, Fundamenta mathematicae, 44, pp. 1-11, 1957. 359 22. - Kwiatkowski, T., “Jan Lukasiewicz - A historian of logic”, Organon, 16-17, pp. 169-188, 1980-1981. 360 23. - Lukasiewicz, J., Aristotle´s Syllogistic. Oxford (UK): The Clarendon Press, 1957. 361 24. - Ibid., Elements of Mathematical Logic (in Polish). Warsaw, PWN (Państwowe Wydawnictwo 362 Naukowe), 1963. 363 25. - Ibid., O zasadzie sprzecznosci u Arystotelesa (On the Principle of Contradiction in Aristotle) (1910), PWN, 364 Warszawa 1987. Germ. transl., Über den Satz des Widerspruchs bei Aristoteles, Olms Verlag, 1994 (J. Barski). 365 Eng. transl. by Owen LeBlanc in progress. French translation, París, L´Eclat. 366 26. - Ibid., Selected Works. L. Borkowski, as editor. Amsterdam, North Holland C. Publ. Co., 1970. 367 27. - Ibid., Para una historia de la Lógica de Enunciados. Valencia, Cuadernos Teorema, 1974. 368 28. - Ibid., La silogística de Aristóteles desde el punto de vista de la lógica formal moderna. Madrid, Editorial 369 Tecnos, 1977. English edition, Aristotle´s Syllogistic from the Standpoint of Modern Formal Logic, Clarendon 370 Press, Oxford, 1951. 371 29. - Ibid., Logika i metafizyka. Wydzial Filozofii i Socjologii Uniwersytetu Warszawskiego, Warsawa, 1998 372 (J. J. Jadacki ed.), including a precious collection of bio-bibliographical information, transcriptions of 373 letters and an extended selection of photos. 374 30. - Ibid., Z zagadnien logiki i filozofii. Pisma wybrane. Warszawa, PWN ((Państwowe Wydawnictwo 375 Naukowe; J. Slupecki as ed.), 1961. 376 31. Trzesicki, K., “Lukasiewiczian logic of tenses and the problem of determinism”, in K. Szaniawski (ed.), op. 377 cit., pp. 293-312. Axioms 2016, 5, x 9 of 5

378 32. - Trzesicki, K. "Lukasiewicz on Philosophy and determinism", in F. Coniglione, R. Poli, J. Wolenski 379 (eds.), Polish Scientific Philosophy, Rodopi Verlag, Amsterdam, pp. 251-297, 1993. 380 33. - Schiaparelli, A., “Aspetti della critica di Jan Lukasiewicz al principio aristotelico di non 381 contraddizione”, Elenchos, 1, pp. 43-77, 1994. 382 34. - Scholz, H., “In memoriam Jan Lukasiewicz”, Arch. Math. Logik Grundlagenforsch., 3, pp. 3-18, 1957. 383 35. - Seddon, F., Aristotle & Lukasiewicz on the Principle of Contradiction: On the Principle of Contradiction. 384 Modern Logic Pub., 1996. 385 36. - Simons, P., “Lukasiewicz, Meinong, and Many-Valued Logic”, in K. Szaniawski (ed.), The Vienna 386 Circle and the Lvov-Warsaw School. Kluwer Ac. Publ., Dordrecht, pp. 249-259, 1989. 387 37. - Slupecki, J., “Jan Lukasiewicz” (Polish). Wiadomosci matematyczne, (2) 15, pp. 73-78, 1972. 388 38. - Wójcicki, R. (ed.), Selected papers on Lukasiewicz's sentential calculus. Ossolineum Verlag, Wroclaw, 389 1977. 390 39. Wolenski, J., “Jan Lukasiewicz on the Liar Paradox, Logical Consequence, Truth and Induction”. Modern 391 Logic, 4, 394-400, 1994. 392 40. - Marshall, D., “Lukasiewicz, Leibniz and the arithmetization of the syllogism”. Notre Dame Journal of 393 Formal Logic, 18 (2), pp. 235-242, 1977. 394 41. Rudeanu, S., et al., “G. C. Moisil memorial issue”. Multiple-Valued Logic, 6, no. 1-2. Gordon and Breach, 395 2001. 396 42. - Marcus, S., “Grigore C. Moisil: A life becoming a myth”. IJCCC, vol. 1, no. 1, 73–79, 2006. 397 43. - Moisil, G., Încercări vechi și noi în logica neclasică. Bucuresti, 1965. 398 44. - Ibid., Elemente de logică matematică și teoria mulțimilor. Bucuresti, 1968. 399 45. - Ibid., “Recherches sur les Logiques Non-Chrysipiennes”. Annales Scientifiques de l'Université de Jassy, 400 26, 431–466, 1940. 401 46. – García Bacca, J. D., Introducción a la Lógica Moderna. Madrid, Ediciones Labor, 1936. 402 47. - Garrido, A., “Special functions in Fuzzy Analysis”. Opuscula Mathematica, vol. 26(3), pp. 457-464. 403 AGH University of Science and Technology, Krakow, 2006. 404 48. - Ibid., “Searching the arcane origins of Fuzzy Logic”. BRAIN, vol. 2(2), pp. 51-57, 2011. 405 49. - Ibid., Filosofía y Matemáticas de la Vaguedad y de la Incertidumbre, PhD. Thesis in Mathematical Logic, 406 qualified with Summa Cum Laude. First Extraordinary Doctorate Award. Madrid, UNED, 2013. 407 50. - Ibid., Lógicas de nuestro tiempo. Madrid, Editorial Dykinson, 2014. 408 51. - Ibid., Lógica Aplicada. Vaguedad e Incertidumbre. Madrid, Editorial Dykinson, 2014. 409 52. - Ibid, Lógica Matemática e Inteligencia Artificial. Madrid, Editorial Dykinson, 2015. 410 53. - Ibid., Complejidad. Madrid, Editorial Dykinson, 2016. 411 54. - Klaua, D., Allgemeine Mengenlehre. Ein Fundament der Mathematik. Leipzig, Akademie-Verlag, 1964. 412 55. - Kleene, S. C., Introduction to . Amsterdam, North-Holland Publ. Co., and New York, 413 Van Nostrand, 1952. 414 56. - Murawski, R., Essays in the Philosophy and History of Logic and Mathematics. Poznan Studies in the 415 Philosophy of the Sciences & the . Rodopi Verlag, 2010. 416 57. - Novák, V., et al., Mathematical Principles of Fuzzy Logic. Kluwer Acad. Press, Dordrecht, Boston, 1999. 417 58. - Ibid., and Perfilieva, I. (eds.), Discovering the World With Fuzzy Logic. Springer-Verlag, Heidelberg 418 2000. 419 59. - Pavelka, J., “On Fuzzy Logic I Many‐valued rules of inference”. Mathematical Logic Quarterly, 25 (3‐6), 420 pp. 45-52, 1979. 421 60. - Ibid., “On Fuzzy Logic II. Enriched residuated lattices and semantics of propositional calculi”. 422 Mathematical Logic Quarterly, 25 (7‐12), pp. 119-134, 1979. 423 61. - Ibid., “On Fuzzy Logic III. Semantical completeness of some many‐valued propositional calculi”. 424 Mathematical Logic Quarterly, 25 (25‐29), pp. 447-464, 1979. 425 62. - Prior, A. N., “Three-valued logic and future contingents”. Philosophical Quarterly, 3, pp. 317-326, 1953. 426 63. - Ibid., “Lukasiewicz's contribution to logic”, in AA.VV., Philosophy in the mid-century, a survey, ed. by 427 R. Klibanski, vol. I, Logic and philosophy of science, La Nuova Italia, Firenze, pp. 53-55, 1958. 428 64. - Rasiowa, H., An Algebraic Approach to Non-Classical Logics. Warsaw, PWN, and Amsterdam, 429 North-Holland Publ. Co., 1974. 430 65. - Ibid., “Toward fuzzy logic”. In L. A. Zadeh and J. Kacprzyk, as editors, Fuzzy Logic for the Management 431 of Uncertainty. New York, Wiley, pp. 5-25, 1992. Axioms 2016, 5, x 10 of 5

432 66. - Rosser, J. B., and Turquette, A. R., Many-Valued Logics. North-Holland Publ. Co, Amsterdam, 1952. 433 67. - Sacristán, M., Introducción a la Lógica y al Análisis Formal. Círculo Universidad, Círculo de Lectores, 434 Barcelona, 1990. 435 68. - Sánchez-Mazas Ferlosio, M., Obras Escogidas. En dos volúmenes. Coordinados por A. Ibarra. 436 UPV/EHU, Cátedra Miguel Sánchez-Mazas, 2003 y 2004. 437 69. - Smith, B., Austrian Philosophy: The Legacy of Franz Brentano. La Salle and Chicago. Open Court, 1994. 438 70. - Tarski, A., Introduction to Logic and to the Methodology of Deductive Sciences. Dover, New York, 1994 439 (first edition, 1941). 440 71. - Tarski, A., Logic, Semantics, Metamathematics. 2nd ed., Hackett Publ., 1983. 441 72. - Trillas, E., Alsina, C., and Terricabras, J. M., Introducción a la Lógica Borrosa. Ediciones Ariel, Barcelona, 442 1995. 443 73. - Trillas, E., “Lotfi A. Zadeh: On the man and his work”. Scientia Iranica, D 18(3), pp. 574-579, 2011. 444 74. - Velarde, J., Gnoseología de los Sistemas Difusos. Servicio de Publicaciones de la Universidad de Oviedo, 445 1990. 446 75. - Ibid., Historia de la Lógica. Servicio de Publicaciones de la Universidad de Oviedo, 1989. 447 76. - Ibid., “Análisis gnoseológico de la Teoría de los Sistemas Difusos”. El Basilisco: Revista de filosofía, 448 ciencias humanas, teoría de la ciencia y de la cultura, 11, pp. 28-45, 1992. 449 77. - Ibid., “Panorama de la Lógica en España”. Theoria: an international journal for theory, history and 450 foundations of science, 7 (16-18), 1992. 451 78. - Ibid., “Pensamiento difuso, pero no confuso: de Aristóteles a Zadeh y vuelta”. Psicothema, vol. 8, issue 452 2, pp. 435-446. Publicación de la Universidad de Oviedo. 453 79. - Ibid., “Incertidumbre y grados de creencia”. Teorema: Revista internacional de filosofía, 24, pp. 27-41, 454 2005. 455 80. - Wajsberg, M., Logical Works. S. J. Surma as editor. Wroclaw, Ossolineum Verlag, 1977. 456 81. - Wolenski, J., Logic and Philosophy in the Lvov-Warsaw School. Kluwer Academic Publishers, Dordrecht, 457 1989. 458 82. - Ibid., with J. Wolenski, as ed. Philosophical Logic in Poland. Kluwer Academic Publishers, Dordrecht, 459 1994. 460 83. - Ibid., Historico-Philosophical Essays, vol. I. Copernicus Center, Krakow, 2013. 461 84. - Wybraniec-Skardowska, U., Polish Logic. Some lines from a personal perspective. Stanford presentation, 462 July 2009. 463 85. - Zadeh, L. A., “Fuzzy Sets”. Information and Control, 8, pp. 338-353, 1965.

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