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Jean-Yves Béziau Federal University of Rio de Janeiro Brazilian Research Council

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What is logic?

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We are logical (rational) animals

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Relation

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Logic and logic Logic : reasoning logic : the theory of reasoning History : the series of events history : the science which studies History

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Logics or logics? Are there different logics? Are there different Logics?

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Is Aristotle the creator of logic? Aristotle was maybe the first to have a logic, a theory of reasoning But he was not the first person to have a Logic, to reason (not the first logical animal)

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Pythagoras Before Aristotle, the Greeks introduced a new way of reasoning, a new Logic, based on the reduction to the absurd - Irrationality Some people consider that this was the birth of Mathematics Mathematicians have never used Aristotles theory of reasoning

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The Paradox of Descartes Descartes was against logic But he was very logical

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DESCARTES 4 PRECEPTS Clarity Never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt. DivisionTo divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution. Ascension To conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence. ExhaustivityTo make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.

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PASCAL 8 RULES Rules for Definitions Not to undertake to define any of the things so well known of themselves that clearer terms cannot be had to explain them. Not to leave any terms that are at all obscure or ambiguous without definition. Not to employ in the definition of terms any words but such as are perfectly known or already explained. Rules for Axioms Not to omit any necessary principle without asking whether it is admitted, however clear and evident it may be. Not to demand, in axioms, any but things that are perfectly evident of themselves. Rules for Proofs Not to undertake to demonstrate any thing that is so evident of itself that nothing can be given that is clearer to prove it. To prove all propositions at all obscure, and to employ in their proof only very evident maxims or propositions already admitted or demonstrated. To always mentally substitute definitions in the place of things defined, in order not to be misled by the ambiguity of terms which have been restricted by definitions.

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TARSKI: Introduction to logic and the methodology of deductive sciences - VI On the Deductive Method 36 Fundamental constituents of a deductive theoryprimitive and defined terms, axioms and theorems (Sur la méthode déductive, in Travaux du IXe Congrès International de Philosophie, VI, Paris: Hermann, pp ) Ideas which are closely related to those presented in this section can be found in earlier literature. See, for instance, the opusculum (posthumously published), De I'esprit geometrique et de I'art de persuader, of the great French philosopher and mathematician B. PASCAL ( ).

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Logic: the laws of thought KANT BOOLE Logic and logic are eternal Logic is eternal, logic is changing

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Modern Logic

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Different names for modern logic Formal logic Symbolic logic Algebra of logic Logistic Metamathematics Methodology of deductive sciences Mathematical logic Logic

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Different systems Classical logic Intuitionistic logic Many-valued logic Modal logic Non monotonic logic Fuzzy logic Substructural logic Linear logic Paraconsistent logic …

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Universal Logic A general theory of logics, of the different theories of reasoning, of the different logical structures Not a universal system of Logic Not a Logic, not a system that is the description of the right way of reaoning

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Languages and Linguistics There are many languages They have something in common despite very strong differences, i.e. chinese, english, arabic This thing in common is not a language itself, the essence of language is not a language, it is the object of linguistics Linguistics is not a universal language but the study of the universal features of languages

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Ferdinand de Saussure The structure of language The originator of structuralism

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Universal Algebra J.J.Sylvester A.N.Whitehead Garrett Birkhoff

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But Universal Algebra is different from Universal Logic Different structures, differents objects, differents tools Logics are structures but not necessarily algebraic structures

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Structure = Lattice

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To be is to be an element of a structure (a class of structures) 4 does not exist by itself

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Algebra Mu ammad ibn Mūsā al-Khwārizmī (780 – 850 Persia) Algorithm Algarismo = digit

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Two reasons to reject axioms Theoretical reasons Practical reasons

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Anti-classical logic A simple example of a logic not obeying any standard axioms Non-reflexive, non-monotonic, non-transitive, non- structural But proof theory and semantics

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Special Issue of Logica Universalis Vol4 n2 2010

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1. Do all human beings have the same capacity of reasoning? Do a man, a woman, a child, a papuan, a yuppie, reason in the same way? 2.Does reasoning evolve? Did human beings reason in the same way two centuries ago? In the future will human beings reason in the same way? Did computers change our way to reason? Is a mathematical proof independent of time and culture ? 3.Do we reason in different ways depending on the situation? Do we use the same logic for everyday life, physics, economy? 4.Do the different systems of logic reflect the diversity of reasonings? 5.Is there any absolute true way of reasoning

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Logic and logic are relative Nevertheless logic as a science can be universal

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(1) science is not a private business, it is objective, not subjective, not a question of taste (2) science explains not the idiosyncrasies of a particular phenomenon, but some general patterns of phenomena

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Science is concerned with a double ALL, ALL minds and ALL objects. Chuaqui and Suppes (1995) have shown that classical physics can be described with a first-order logic theory with only universal quantifiers

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logic as a science is universal (physics as a science is universal) There is no universal system of logic (there is no universal theory of the universe)

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Louis Rougier ( ) The relativity of logic 1941 With the discovery of the conventional and relative character of logic, human spirit has burned his last idol.

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Haskell Curry ( ) Leçons de logique algébrique 1952 Translated and presented by Jonathan Seldin

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Leon Henkin La structure algébrique des théories mathématiques 1956

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DEVIATION/EXPANSION DeviationsIntuitionistic logic Relevant logic ExpansionsModal logic Causal logic

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GRADES SubsystemsPositive classical propositional logic Full classical propositional logic SupersystemsMany-sorted classical first-order logic Second order classical logic

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TECHNIQUES ProofHilbert systems Sequents systems SemanticsLogical matrices Kripke structures

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