This is a collection of new investigations and discoveries on the history of a great tradition, the Lvov-Warsaw School of logic , philosophy and mathematics, by the best specialists from all over the world. The papers range from historical considerations to new philosophical, logical and mathematical developments of this impressive School, including applications to Computer Science, Mathematics, Metalogic, Scientific and Analytic Philosophy, Theory of Models and Linguistics.

ABSTRACT It is well-known that Artificial Intelligence requires Logic. But its Classical version shows too many insufficiencies. So, it is very necessary to introduce more sophisticated tools, as may be Fuzzy Logic, Modal Logic, Non-Monotonic Logic, and so on. When you are searching the possible precedent of such new ideas, we may found that they are not totally new, because some ancient thinkers have suggested many centuries ago similar concepts, certainly without adequate mathematical formulation, but in the same line: against (...) the dogmatism and the dualistic vision of the world: absolutely true vs. absolutely false, black vs. white, good or bad by nature, 0 vs.1, etc. We attempt to analyze here some of these greatly unexplored, and very interesting early origins. (shrink)

ABSTRACT Usually, the problems in AI may be many times related to Philosophy of Mind, and perhaps because this reason may be in essence very disputable. So, for instance, the famous question: Can a machine think? It was proposed by Alan Turing [16]. And it may be the more decisive question, but for many people it would be a nonsense. So, two of the very fundamental and more confronted positions usually considered according this line include the Connectionism and the Computational (...) Theory of Mind. We analyze here its content, with their past disputes, and current situation. (shrink)

The roots of the Lvov-Warsaw School can be traced back to Aristotle himself. But in later times we better put them into thinking GW Leibniz and who somehow inherited many of these ways of thinking, such as the philosopher and mathematician Bernhard Bolzano. Since he would pass the key figure of Franz Brentano, who had as one of his disciples to Kazimierz Twardowski, which starts with the brilliant Polish school of mathematics and philosophy dealt with. Among them, one of the (...) most interesting thinkers must be Jan Lukasiewicz, the father of many-valued logic.Jan Łukasiewicz began teaching at the University of Lvov, and then at Warsaw, but after World War II must to continue in Dublin. Some questions may be very astonishing in the CV of Łukasiewicz. For instance, that a firstly Polish Minister of Education in Paderewski cabinet, into the new Polish Republic, and also Rector for two times at Warsaw University, was awarded with a Doctorate ‘Honoris Causa’ in spring 1936, at University of Münster, into the maximum of effervescency of Nazism in Germany. The explanation must be their good relation with a very good friend, the former theologian, and then logician, Heinrich Schölz, who was the first Chairman of Mathematical Logic in German universities.Łukasiewicz firstly studied Law, and then Mathematics and Philosophy in Lvov. His doctoral supervisor was Kazimierz Twardowski, and in 1902 he obtain his Ph. D. title with a very special mention: ‘sub auspiciis Imperatoris’. Also he received a doctorate ring with diamonds from the Kaiser of the Austro-Hungarian Empire, Franz Joseph I.From 1902, Łukasiewicz was employed as a private teacher, and also as a desk in the Universitary Library of Lvov. So it was until 1904 when he obtained a scholarship to study abroad. He defends his ‘Habilitationschrift’ in 1906, entitled “Analysis and construction of the concept of cause”. This permits to give university courses. His first lectures were on the Algebra of Logic, according to the recent translation to Polish of this book of the French logician Louis Couturat.Between 1902 and 1906, Lukasiewicz continued his studies in the universities of Berlin and Leuwen. In 1906, by his ‘Habilitationschrift’, he obtain the qualification as university professor at Lvov. And the, in 1911, he was appointed as associate professor in his ‘alma mater’.Jan Łukasiewicz was also very active in historical research on logic, giving a new and up-to-date interpretation of Aristotle’s syllogism and of the Stoics’ propositional calculus. According to Scholz, the better pages on history of logic are due to him. And also, as Arianna Betti says, “Jan Lukasiewicz is first and foremost associated with the rejection of the Principle of Bivalence and the discovery of Many-Valued Logic.”The discovery of MVL by Lukasiewicz was in 1918, a little earlier than Emil Leon Post. According to Jan Wolenski, “although Post’s remarks were parenthetical and extremely condensed, Lukasiewicz explained his intuitions and motivations carefully and at length. He was guided by considerations about future contingents and the concept of possibility”. So, he introduces, firstly, three-valued logic, then four-valued logic, generalized to logics with an arbitrary finite number of veritative values, and finally, to logics with a countably infinite-valued number of such values.Very noteworthy is his treatment of the history of logic in the light of the new formal logic. Thus, not only he addressed the issue of future contingents departing from Aristotle, but also put in value logic of the Stoics, at least so far taken. In fact, Heinrich Scholz said, rightly, that Lukasiewicz had written the most lucid pages on the history of logic. (shrink)