三个逻辑学家走进酒吧，侍者问：“每个人都要来杯啤酒吗？”第一个逻辑学家说：“我不知道。”第二个说：“我也不知道。”第三个说：“是的！”

1. 逻辑

2. 三个逻辑学家走进酒吧，侍者问：“每个人都要来杯啤酒吗？”第一个逻辑学家说：“我不知道。”第二个说：“我也不知道。”第三个说：“是的！”

3. 信息系一共就三个班。今天他对我说，你是3班的么？我说，原来你是2班的啊！他说，原来你是1班啊！信息系一共就三个班。今天他对我说，你是3班的么？我说，原来你是2班的啊！他说，原来你是1班啊！

4. 能否设计一套推理系统，判断所有命题？ • 能否有一门计算机语言，能够描述自身？

5. 布尔逻辑

6. 常量：真，假 • 运算：n元组到常量的映射 • -a=真 • a*b=真 • a*b*c…=假

7. 一元运算 • 相当于 布尔常量到布尔常量的映射 • 可以理解为一种二元关系。 • 例如：Not

8. Not in ES • In EcmaScript • Logical not is: ! • Bitwise not is:~ alert(!1); //false alert(~1);//-2

9. Unary Op • 始点集合的每个元素可以指向终点集合的任意元素；每个元素有二种指法。 • 总共有2*2种二元关系 T F T F

10. 一元运算

11. BiOp • 二元运算 • 二元组到常量的映射 <F,F> <F,T> <T,F> <T,T> T F

12. 总共有多少这样的二元运算？（? • 每个二元组可以有两种映射方法，总共有4个二元组， • 2^4 <F,F> <F,T> <T,F> <T,T> T F

13. and alert(true && false); //false //bitwise alert(3 & 2); //2 alert(-3 & -2); //?

14. or alert(true || false); //true //bitwise alert(-3 | -2); //?

15. nor

16. nand

17. Xor

18. Implication （ if |x|>1，then x>1 ） holds if |x|<=1 x>1 other wise, doesn’t hold

19. iif A B

20. People on an island understand English but speak their native language. In Da and ja, one means yes and the other no; but whether da means yes or no is unknown. • If you want to know if KadaKata is in the south, and you’re allowed to ask only one question to a person, what should you ask?

21. Da means yes  K is in the south, right? • If da means yes and k is in the south, • Da • If da means yes and k is not in the south • Ja • If da means no and k is in the south • Da • If da means no and k is not in the south • ja

22. uttered True Da K is in the south Not uttered (Ja) False K is in the south Da uttered

23. BiOp’s • …

24. UnaOp represented by BiOp • 一元元算

25. n元运算符 • n元函数 • n+1元关系。 • 可以用二元运算表示 • F(x1,x2,…x[n])=F(x1,G(x2,…x[n]))=…

26. 完备组 • 有一些运算可以表达成其它运算的复合 • 比如：a^b= (a||b) && ! (a&&b) • 如果有些运算可以代替所有的运算，则这些运算是完备的 • ! ,||, && • !, || • ！， && In most languages such as ES/natural,…

27. 最小完备组 • 如果完备运算组合中，去掉任何一个都不行，则称之为最小的 • ! ,||, && ？ • ! ,|| ？ • ! , && ？

28. 最小完备组 • 就用一个运算符可能不可能呢？ • Nor，或者Nand! • Nor: • Not a=a nor a, a& b=(a nor a) nor (b nor b) • a ||b =(a nor b) nor ( a nor b) • Nand: • Not a=a nand a, a && b=(a nand b) nand (a nand b), a || b= (a nand a) nand (b nand b)

29. 命题逻辑

30. 命题逻辑 • 命题： • 陈述句，有对错。真假必居其一, 且只居其一. • 5>3 • 我正在撒谎. —— 悖论,不是命题 • 命题变量 • 有对错，但不知道 • 命题运算仿照bool逻辑

31. Normal Form • And Form • A and !B and (C or !D) • Or Form • A Or !B or (C and !D)

32. Predicate

33. 谓词逻辑 • 带变量的陈述句 • X>3 • 返回值为bool类型的函数 • Arguments 可以任意多个

34. First Order Logic

35. First Order Logic • Predicate + Quantifier (∀,∃) • ∀x P(x) ∧ Q(x) • P:x>-1, Q x>0

36. Free and Bound Variables • An occurrence of variable in a sentence is free if it is not in the scope of any quantifier with the same variable. A non-free variable occurrence is bound. • ∀x P(x) ∧ Q(x) both occurrences of x bound by ∀x • (∀x P(x)) ∧ Q(x) first occurrence of x bound by ∀x, second free • ∀x (R(x, z) ∧ ∃y S(x, y)) both occurrences of x bound by ∀x, occurrence of y bound by ∃y, occurrence of z free • A sentence is open if it contains free variables; it is closed otherwise

37. Some Op/Func can be regarded as predicate • > • == • != • … • Many Theorems/Axioms are represented in First Order Logic alert(3>4); alert(“”==“a”);

38. Godel Theorem

39. Godel’s incomplete theorem • A System cannot be complete if consistent

40. Example • This sentence is not true

41. Significance

42. Application • A system cannot be complete, if consistent • A system can only upgraded from outside. • Never expect a self-evolving model • Otherwise it’s like expecting a perpetual motion machine

43. Never expect a system • Omni-potent

44. Self-Describing

45. Application • A system can only be described from external, higher order language • Care must be taken to separate standard terms to the terms of the language itselfE.g., • type in C# standard and the Type instance in C# language • function in ES standard and the Function instance in ES lang. • Confusion about the two will make your head spin

46. Example • LiteralVar • Expression • The rules of expression • Language • Production Rules • Courses About Production Rules • Facts • Statistics • Conclusion • Proposition Logic • First Order Logic • …

47. Second order

48. Any • Some

49. Halting Problem

50. The Halting problem • Given a program P, and input I, will the program P ever terminate? • Meaning will P(I) loop forever or halt? • Can a computer program determine this? • Can a human? • First shown by Alan Turing in 1936 • Before digital computers existed!