1. The Relativity and Universality of Logic
Jean-Yves Béziau
Federal University
of Rio de Janeiro
Brazilian Research
Council

2. What is logic?

3. We are logical (rational) animals

4. The Brain according to Aristotle

5. Relation

6. Logic is not logic

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

8. Logics or logics? Are there different logics?

9. 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)

10. 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 Aristotle’s theory of
reasoning
Pythagoras

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

14. 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.
Division
To 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.
Exhaustivity
To make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.

15. PASCAL 8 RULES Rules for Definitions Axioms Proofs
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.
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.
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.

16. TARSKI: Introduction to logic and the methodology of deductive sciences - VI On the Deductive Method 36 Fundamental constituents of a deductive theory—primitive 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 ( ).

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

18. Modern Logic

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

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

21. 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

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

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

26. Structure = Lattice

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

30. What is a logical structure?

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

38. Axiomatic emptiness

39. Two reasons to reject axioms
Theoretical reasons
Practical reasons

40. Practical reasons

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 Logic Universal?

43. Special Issue of Logica Universalis Vol4 n2 2010

44. Do all human beings have the same capacity of reasoning
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 there any absolute true way 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 is universal
(physics as a science is universal)
There is no universal system of logic
(there is no universal theory 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.

51. Haskell Curry (1900 - 1982) Leçons de logique algébrique 1952
Translated and presented by
Jonathan Seldin

52. Leon Henkin
La structure algébrique des théories mathématiques
1956

53. The Universe of Logics (The world of possible logics)

54. DEVIATION/EXPANSION
Deviations
Intuitionistic logic
Relevant logic
Expansions
Modal logic
Causal logic

55. GRADES
Subsystems
Positive classical propositional logic
Full classical propositional logic
Supersystems
Many-sorted classical first-order logic
Second order classical logic

56. TECHNIQUES
Proof
Hilbert systems
Sequents systems
Semantics
Logical matrices
Kripke structures

60. 4th World Congress and School on U N I V E R S A L L O G I C