GROUP 37

1. GROUP 37 INTUITIONIST VS CLASSSICAL NATURAL DEDUCTION

2. NATURAL DEDUCTION Logical Reasoning by Inference Rules

3. NATURAL DEDUCTION Modus Ponens in summer, it’s warm

4. Difference Classicism Conclusion Intuitionism

5. Difference Classicism Conclusion Intuitionism

6. Classicist Proposition Based on Premise

7. Aristotle Syllogistic Reasoning Focus not on content but Structure

8. N = 2M + 1 3 is an even number

9. mathematician mathematician mathematician mathematician mathematician mathematician mathematician mathematician mathematician mathematician mathematician Rational Thinking! mathematician mathematician mathematician mathematician mathematician mathematician mathematician mathematician mathematician mathematician

12. Classical 322BC 384BC 1874

13. Classical 322BC 384BC 1874

14. Classical 322BC 384BC 1874

15. Classical 322BC 384BC 1874 1895

16. Classical 322BC 384BC 1874 1895

17. Classical 322BC 384BC 1874 1895 mid 19s

18. Classical 322BC 384BC 1874 1895 mid 19s

19. Classical 322BC 384BC 1874 1895 mid 19s early 20s

20. Classical 322BC 384BC 1874 1895 mid 19s early 20s

21. Classical 322BC 384BC 1874 1895 mid 19s early 20s recently

23. Difference Classicism Conclusion Intuitionism

24. Intuitionist Mathematics Logics

25. Intuitionist Mind Consciousness

26. All men are mortal Socrates is man. Therefore, Socrates is mortal

27. Objects Predicates B C D E F G H Socrates Man I Mortal

28. 1st Premise Objects Predicates B C D E F G H Socrates Man I Mortal

29. 1st Premise Objects Predicates B C D E F G H Socrates Man I Mortal

30. Socrates is mortal Objects Predicates B C D E F G H Socrates Man I Mortal

31. Discussion between Cantor and Kronecker 1800 1850 1900 1950

32. Discussion between Frege and Russell 1800 1850 1900 1950

33. Brouwer’sdoctoral thesis 1800 1850 1900 1950

34. Difference Classicism Conclusion Intuitionism

35. Differences Philosophy

36. Differences Existence of an object Intuitionist Classicist X

37. Differences Existence of an object De Morgan’s Law

38. Differences Law of Excluded Middle

39. Differences Law of Excluded Middle Proof that there exist two irrational numbers a and b such that a x b is rational. Classicist Consider By Law of Excluded Middle, it is either rational or irrational. If it is rational, the proof is complete and a = and b = If is irrational, then we let a = and b = Then = = = 2, which is rational. QED

40. Differences Law of Excluded Middle Proof that there exist two irrational numbers a and b such that a x b is rational. Intuitionist Object the proof. Proper existence proof should provide an explicit, unconditional description of the object it asserts to exist. It requires further justification.

41. Differences Double Negation Elimination

42. Differences Double Negation Elimination Classicist: It's not the case that it's not raining, implies that it’s raining. Constructive: Disagree. There must be a proof that it’s raining to conclude that it’s raining.

43. Differences Double Negation Elimination Constructive: However, If it’s raining, we can conclude that it’s contradictory that it is not raining. The statement It’s raining is stronger than the statement It’s not the case that it’s not raining.

44. Differences Double Negation Elimination Constructive

45. Differences Double Negation Elimination 4 Streets

46. Difference Classicism Conclusion Intuitionism

47. Conclusion • Intuitionistic logic is a restriction of Classical Logic • Classical Logic is the mainstream • Recently, rise of the computer gave rise to increased interest in intuitionistic logic