The Relativity and Universality of Logic

1. The Relativity and Universality of Logic Jean-Yves Béziau FederalUniversity of Rio de Janeiro BrazilianResearch Council

2. Whatislogic?

3. We are logical (rational) animals

4. The Brainaccording to Aristotle

5. Relation

6. Logicis not logic

7. Logic and logic • Logic : reasoning • logic : the theory of reasoning • History : the series of events • history : the science whichstudiesHistory

8. Logics or logics? • Are theredifferentlogics? • Are theredifferentLogics?

9. Is Aristotle the creator of logic? • Aristotlewasmaybe the first to have a logic, a theory of reasoning • But hewas not the first person to have a Logic, to reason (not the first logical animal)

10. Pythagoras • BeforeAristotle, the Greeksintroduced a new way of reasoning, a new Logic, based on the reduction to the absurd - Irrationality • Some people considerthatthiswas the birth of Mathematics • Mathematicians have never usedAristotle’stheory of reasoning

13. The Paradox of Descartes • Descartes wasagainstlogic • But hewasverylogical

16. TARSKI: Introduction to logic and the methodology of deductive sciences - VI On the Deductive Method36 Fundamental constituents of a deductivetheory—primitive and defined terms, axioms and theorems (Sur la méthode déductive‘, in Travaux du IXe Congrès International de Philosophie, VI, Paris: Hermann, pp.95-103) • 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'espritgeometrique et de I'art de persuader, of the great French philosopher and mathematician B. PASCAL (1623-1662).

17. Logic: the laws of thought KANT BOOLE • Logic and logic • are eternal • Logiciseternal, • logicischanging

18. Modern Logic

19. Different names for modern logic • Formallogic • Symboliclogic • Algebra of logic • Logistic • Metamathematics • Methodology of deductive sciences • Mathematicallogic • Logic

20. Differentsystems • Classicallogic • Intuitionisticlogic • Many-valuedlogic • Modal logic • Non monotoniclogic • Fuzzylogic • Substructurallogic • Linearlogic • Paraconsistentlogic …

21. Universal Logic • A generaltheory of logics, of the differenttheories of reasoning, of the differentlogical structures • Not a universal system of Logic • Not a Logic, not a system thatis the description of the right way of reaoning

22. 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

23. Ferdinand de Saussure • The structure of language • The originator of structuralism

24. UniversalAlgebra • J.J.Sylvester • A.N.Whitehead • Garrett Birkhoff

25. But Universal Algebra is different from Universal Logic • Different structures, differentsobjects, differentstools • Logicsare structures but not necessarilyalgebraic structures

26. Structure = Lattice

29. To beis to be an element of a structure (a class of structures) • 4 does not exist by itself

30. Whatis a logicalstructure?

37. Algebra • Muḥammad ibn Mūsā al-Khwārizmī (780 – 850 Persia) • Algorithm • Algarismo = digit

38. Axiomaticemptiness

39. Two reasons to reject axioms • Theoretical reasons • Practical reasons

40. Practicalreasons

41. 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

42. Is LogicUniversal?

43. Special Issue ofLogicaUniversalisVol4 n2 2010

44. 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? • 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 ? • Do we reason in different ways depending on the situation?Do we use the same logic for everyday life, physics, economy? • Do the different systems of logic reflect the diversity of reasonings? • Is thereanyabsolutetrueway of reasoning

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

46. (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

47. 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

48. logic as a science isuniversal (physics as a science isuniversal) • There is no universal system of logic (thereis no universaltheory of the universe)

49. Louis Rougier (1889- 1982) • The relativity of logic 1941 • With the discovery of the conventional and relative character of logic, human spirit has burned his last idol.