Introduction to Symbolic Logic

1. IntroductiontoSymbolic Logic San Diego Math Circle David W. Brown

2. Sudoku Minesweeper Mathematics Science Engineering Computers Medical diagnosis Law Contracts Debate Philosophy Politics? Policy? Why learn about logic?

3. Why study symbolic logic? • Symbolic logic excels in separating the formal structure of logical relationships from the material content of the statements being related. • With the material content of statements out of the way, we can discover how to “calculate” the properties of logical relationships.

4. Understand: Truth value Propositions Connectives Truth Tables Theorems Inference Fallacy No time now for: Boolean Algebra Sets Predicates Quantifiers Venn Diagrams Logic gates Etc. Goals

5. What is Truth? • A philosophical, not mathematical matter • Mathematically, “true” and “false” are simply the two mutually exclusive values that can be taken by a well-formed formula. • True: 1, T, +, up, on, Republican • False: 0, F, -, down, off, Democrat

6. What is a“Well-formed formula” ? A “well-formed formula” is any construction that has definite truth value, whether that value is “true” or “false”. Synonymous alternative terms: • Statement • Proposition • Sentence • Wff ( Well-formed formula )

7. What is an atomic formula? Not something from science fiction or a cold war thriller … Propositions may be compound, constructed from simpler propositions. An “atomic formula” or “atomic sentence”, etc. is one that cannot be decomposed into simpler wffs.

8. Terminology • The symbols such as p or q used to represent statements, whether atomic or compound, may be called variously • Statement letters • Literals • Variables (provided the context is clear; in predicate logic variables are used differently)

9. Language • We speak in natural language or informal language, which is imprecise, flexible, and expressive. • Logic requires formal language, which is precise, rigid, and constructive. • Logic can connect natural language statements, but only if those statements have definite truth value.

10. Natural language Why? Two’s company. Three’s a crowd. I love you. OMG! Time flies. I voted. Obama rules! Formal language Why? Today is Saturday. All men are created equal. True or false 2 x 2 = 5 1 + 1 = 1 McCain lost. Natural vs. Formal Language

12. Roses are red, Violets are blue, Sugar is sweet, and so are you. Translating Natural Language into Formal Language

13. Roses are red, Violets are blue, Sugar is sweet, and so are you. R = “Roses are red” V = “Violets are blue” S = “Sugar is sweet” Y = “You are sweet” Translating Natural Language into Formal Language

14. Roses are red, Violets are blue, Sugar is sweet, and so are you. R = “Roses are red” V = “Violets are blue” S = “Sugar is sweet” Y = “You are sweet” Translating Natural Language into Formal Language Roses are red and Violets are blue and Sugar is sweet and You are sweet.

15. Roses are red, Violets are blue, Sugar is sweet, and so are you. R = “Roses are red” V = “Violets are blue” S = “Sugar is sweet” Y = “You are sweet” Translating Natural Language into Formal Language Roses are red and Violets are blue and Sugar is sweet and You are sweet. R and V and S and Y

16. Roses are red, Violets are blue, Sugar is sweet, and so are you. R = “Roses are red” V = “Violets are blue” S = “Sugar is sweet” Y = “You are sweet” Translating Natural Language into Formal Language Roses are red and Violets are blue and Sugar is sweet and You are sweet. R and V and S and Y  = “and”

17. Roses are red, Violets are blue, Sugar is sweet, and so are you. R = “Roses are red” V = “Violets are blue” S = “Sugar is sweet” Y = “You are sweet” Translating Natural Language into Formal Language Roses are red and Violets are blue and Sugar is sweet and You are sweet. R and V and S and Y  = “and” R  V  S  Y

18. Red sky at night, Sailor’s delight, Red sky at morning, Sailor take warning. Translating Natural Language into Formal Language

19. Red sky at night, Sailor’s delight, Red sky at morning, Sailor take warning. N = “The sky is red at night.” G = “Good weather is ahead.” M = “The sky is red at morning” B = “Bad weather is ahead.” Translating Natural Language into Formal Language

20. Red sky at night, Sailor’s delight, Red sky at morning, Sailor take warning. N = “The sky is red at night.” G = “Good weather is ahead.” M = “The sky is red at morning” B = “Bad weather is ahead.” Translating Natural Language into Formal Language If the sky is red at night, then Good weather is ahead and If the sky is red at morning, then Bad weather is ahead.

21. Red sky at night, Sailor’s delight, Red sky at morning, Sailor take warning. N = “The sky is red at night.” G = “Good weather is ahead.” M = “The sky is red at morning” B = “Bad weather is ahead.” Translating Natural Language into Formal Language If the sky is red at night, then Good weather is ahead and If the sky is red at morning, then Bad weather is ahead. (if N then G) and (if M then B)

22. Red sky at night, Sailor’s delight, Red sky at morning, Sailor take warning. N = “The sky is red at night.” G = “Good weather is ahead.” M = “The sky is red at morning” B = “Bad weather is ahead.” Translating Natural Language into Formal Language If the sky is red at night, then Good weather is ahead and If the sky is red at morning, then Bad weather is ahead. (if N then G) and (if M then B)  = “and” → = “if … then”

23. Red sky at night, Sailor’s delight, Red sky at morning, Sailor take warning. N = “The sky is red at night.” G = “Good weather is ahead.” M = “The sky is red at morning” B = “Bad weather is ahead.” Translating Natural Language into Formal Language If the sky is red at night, then Good weather is ahead and If the sky is red at morning, then Bad weather is ahead. (if N then G) and (if M then B)  = “and” → = “if … then” (N → G)  (M → B)

24. Two wrongs don’t make a right. W1 = “Act1 is wrong.” W2 = “Act2 is wrong.” R = “Act3 is right.” Translating Natural Language into Formal Language It is not true that a wrong act and another wrong act are equivalent to a right one. not ( W1 and W2 equivalent R ) ~ = “not”  = “and” ↔ = “equivalent” ~( W1  W2 ↔ R )

25. All work and no play makes Jack a dull boy. W = “Jack always works” P = “Jack sometimes plays” D = “Jack is a dull boy” Translating Natural Language into Formal Language If Jack always works and it is not true that Jack sometimes plays, then Jack is a dull boy. If W and not P, then D. ~ = “not”  = “and” → = “if … then” ( W  ~P ) → D

26. The Propositional Calculus By: • carefully translating natural language statements into formal propositions, • replacing statements with literals, and • replacing relational language with symbols representing precisely-defined logical operations, it becomes possible to calculate the truth value of complex statements; i.e., to reduce argumentation and proof to calculation.

27. Functions,Truth Tables,and Connectives

28. uh … not quite

29. Functions in Logic The concepts of domain and range familiar from algebra apply to functions in logic as well. In propositional logic, however, the elements of the domain and range are truth values “T” and “F” rather than numbers. This greatly limits the possibilities.

30. Functions Algebraic Functions Logical Functions

31. Simple Functions in Algebra We often denote functions in algebra “f(x)”, But some functions are so simple that we don’t use function notation for them. For example, The negation function n(x)=-x we write as “-x”. The identity function i(x)=x we write as “x”. Constant functions c(x)=c we write as “c”.

32. Unary Functions in Logic Unary functions in logic are functions of a single variable much like the functions f(x) of algebra except for the fact that the “variables” represent propositions and the domain and range are extremely small: Domain = { T, F } , Range  { T, F } This means that there can exist only 4 unary functions in logic.

33. Identity g(p) = p Negation g(p) = ~p (or ¬p) Constant True g(p) = T Constant False g(p) = F That these four are the only ones that can exist can be seen using an organizing tool called a truth table that explicitly tabulates all possibilities. The four unary functions

34. Truth Table of the Identity Function

35. Truth Table of the Identity Function

36. Truth Table of the Identity Function

37. Truth Table of the Negation Function

38. Truth Table of the Negation Function

39. Truth Table of the Negation Function

40. Truth Table of the Constant-True Function

41. Truth Table of the Constant-True Function

42. Truth Table of the Constant-True Function

43. Truth Table of the Constant-False Function

44. Truth Table of the Constant-False Function

45. Truth Table of the Constant-False Function

46. Binary Functions In the same sense that a unary function is a function of only one variable, a binary function is a function of two variables. For example: Algebra: f(x,y) = x2 + y2 Logic: g(p,q) = p  q

47. The domain of a binary function is a Cartesian product of unary domains; i.e., the set of all possible ordered pairs of truth values: D = { T, F }  { T, F } = { (T,T), (T,F), (F,T), (F,F) } On the other hand, the range of a binary function still contains just the simple truth values T and F because the output of a function is a single value. R  { T, F } Domain and Range for Binary Functions

48. How many binary functions are there? That is, how many truth tables?

49. How many binary functions are there? That is, how many truth tables?

50. How many binary functions are there? That is, how many truth tables? 2 x 2 x 2 x 2 = 16 Thus, there exist exactly 16 binary functions in propositional logic.