Chapter 3. The Logic of Quantified Statements

Transcription

1 Chapter 3. The Logic of Quantified Statements

2 3.1. Predicates and Quantified Statements I

3 Predicate in grammar Predicate refers to the part of a sentence that gives information about the subject. Example: James is a student at Bedford College. Subject: James Predicate: a student at Bedford College.

4 Remark 3x + 5 = 1: equation x > 5 : inequality We cannot say whether the above equation and inequality are true or false before any number which can replace the x is given. However, when we replace x by any number, then the above equation or inequality become a statement.

5 Predicate in logic A predicate is sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. The domain of a predicate variable is the set of all values that may be substituted in place of the variable.

6 Example P(x) is the predicate x 2 > x with domain the set R of all real numbers. Remark: (meaning of predicate) P(x): x 2 > x is NOT a statement. That is just an inequality. However, when we replace x by any real number. It becomes a statement because we know whether that is true or false.

7 Example Let P(x) be the predicate x 2 > x with domain the set R of all real numbers. Write P 2, P 1, and P 1, and indicate which of 2 2 these statements are true and which are false.

8 Truth Set If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x. Notation: (Truth Set) {x D P x }

9 Example Let Q(n) be the predicate `n is a factor of 8. Find the truth set of Q(n) if 1. The domain of n is the set of all positive integers. 2. The domain of n is the set of all integers.

10 Key words so far. Predicate: P(x) Domain Truth Set

11 Quantifier Quantifiers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true. Universal Quantifier: for all ( Notation: ) Existential Quantifier: there exists ( Notation: )

12 Universal Quantifier for all for every for any for arbitrary for each given any

13 Example (Universal Quantifier) All human beings are mortal. Symbolize the sentence: Human beings x, x is mortal. x H, x is mortal, where H denote the set of all human beings.

14 Universal Statement, Truth, counterexample Let Q(x) be a predicate and D the domain of x. A universal statement is a statement of the form x D, Q(x) It is defined to be true if, and only if, Q(x) is true for every x in D. It is defined to be false if, and only if, Q(x) is false for at least one x in D. A value for x for which Q(x) is false is called a counterexample to the universal statement.

15 Example (Truth and Falsity of Universal Statements) 1. Let D = 1,2, 3, 4, 5, and consider the statement x D, x 2 x. Show that this statement is true. 2. Consider the statement x R, x 2 x. Find a counterexample to show that this statement is false.

16 Existential Quantifier There exists There is a We can find a There is at least one For some For at least one

17 Existential Statement, True, False Let Q(x) be a predicate and D the domain of x. An existential statement is a statement of the form x D such that Q(x). It is defined to be true if, and only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false for all x in D.

18 Example

19 Universal Conditional Statement Example

20 Example Rewrite each of the following statements in the form, if, then. a. If a real number is an integer, then it is a rational number. b. All bytes have eight bits. c. No fire trucks are green.

21 Example Rewrite each of the following statements in the form, if, then. a. If a real number is an integer, then it is a rational number. b. All bytes have eight bits. c. No fire trucks are green. a. real number x, if x is an integer, then x is a rational number. b. x, if x is byte, then x has eight bits. c. x, if x is a fire truck, then x is not green.

22 Relationship between predicates Let P(x) and Q(x) be predicates which have the common domain D.

23 Example Let Q(n) be `n is a factor of 8, R(n) be `n is a factor of 4, S(n) be `n < 5 and n 3, and suppose the domain of n is Z +, the set of positive integers. Use the and symbols to indicate true relationship among Q(n), R(n), and S(n).

24 Example Let Q(n) be `n is a factor of 8, R(n) be `n is a factor of 4, S(n) be `n < 5 and n 3, and suppose the domain of n is Z +, the set of positive integers. Use the and symbols to indicate true relationship among Q(n), R(n), and S(n). Truth set of Q(n)={1, 2, 4, 8} Truth set of R(n)={1,2,4} Truth set of S(n) = {1,2,4}

25 3.2. Predicates and Quantified Statements II

26 Logical equivalence for quantified statements The statements are logically equivalent if the statements always have identical truth values 1. no matter what predicates are substituted for the predicate symbols and 2. no matter what sets are used for the domains of the predicate variables.

27 The meaning of `negate in dictionary 1. to cancel or destroy the effect of something. 2. to deny the existence of something.

28 Example Statement: `All mathematicians wear glasses What is the negation of the statement? Answer: 1. Some mathematicians don t wear glasses. 2. There is at least one mathematician who doesn t wear glasses.

29 Negation of a Universal Statement The negation of a universal statement (`all are ) is logically equivalent to an existential statement (`some are not or `there is at least one that is not ) Symbolically, ~ x D, Q x x D such that ~Q(x)

30 Example Write formal negation for the statement primes p, p is odd. Answer: a prime p such that p is not odd.

31 Example Statement: Some snowflakes are the same. What is negation for the statement? Answer: No snowflakes are the same. All snowflakes are different.

32 Negation of an Existential Statement The negation of an existential statement ( some are ) is logically equivalent to a universal statement ( none are or all are not ). Symbolically, ~ x D such that Q x x D, ~Q(x)

33 Example Write formal negations for the statement a triangle T such that the sum of the angles of T equals 200 degree. Answer: triangles T, the sum of the angles of T does not equal 200 degree.

34 Example Statement: No politicians are honest. 1. Write the statement formally. 2. Write the formal negation. 3. Write the informal negation. Answer: 1. Hint: Combine the universal quantifier with the predicate. politicians x, x is not honest. 2. a politician x such that x is honest. 3. Some politicians are honest.

35 Example Write informal negations for the following statements: All computer programs are finite. Answer: There is a computer program that is not finite. Some computer programs are infinite.

36 Example Write informal negations for the following statements: Some computer hackers are over 40. Answer: All computer hackers are 40 or under. No computer hackers are over 40.

37 Negation of Universal Conditional Statements ~ x, P x Q x x such that ~(P x Q x ) x such that (P x ~Q x )

38 Example Write a formal negation for statement people p, if p is blond then p has blue eyes. Answer: Hint: Use x such that P Q. a person p such that p is blond and p does not have blue eyes.

39 Example Write the informal negation for statement: If a computer program has more than 100, 000 lines, then it contains a bug. Answer: Hint: Interpret the given statement as for all computer program, if it has more than. There is at lease one computer program that has more than 100,000 lines and does not contain a bug.

40 Variants of Universal Conditional Statements Consider a statement of the form: x D, if P x, then Q x. 1. Contrapositive statement: x D, if ~Q x, then ~P x. 2. Converse: x D, if Q x, then P x. 3. Inverse: x D, if~p x, then~q x.

41 Example Write a formal statement for the following statement: If a real number is greater than 2, then its square is greater than 4. Answer: x R, if x > 2, then x 2 > 4.

42 Example Write a formal and an informal contrapositive for the following statement: x R, if x > 2, then x 2 > 4. Answer: x R, if x 2 4, then x 2. If the square of a real number is less than or equal to 4, then the number is less than or equal to 2.

43 Example Write a formal and an informal converse for the following statement: x R, if x > 2, then x 2 > 4. Answer: x R, if x 2 > 4, then x > 2. If the square of a real number is grater than 4, then the number is greater than 2.

44 Example Write a formal and an informal inverse for the following statement: x R, if x > 2, then x 2 > 4. Answer: x R, if x 2, then x 2 4. If a real number is less than or equal to 2, then the square of the umber is less than or equal to 4.

45 Remark 1. Universal conditional statement is logically equivalent to its contrapositive: x D, if P x, then Q x. x D, if ~Q x, then ~P x. 2. Universal statement is NOT logically equivalent to its convers or inverse.

46 Necessary and Sufficient Conditions x, if s(x), then n(x). s(x) is a sufficient condition for n(x). n(x) is a necessary condition for s(x).

47 Only if x, r x only if s x. means x, if r(x), then s(x).

48 Example Rewrite the following statement as quantified conditional statement. Do not use the word necessary or sufficient. Squareness is a sufficient condition for rectangularity. Answer: If a figure is a square, then it is a rectangle. x, if x is a square, then x is a rectangle.

49 Example Rewrite the following statement as quantified conditional statement. Do not use the word necessary or sufficient. Being at least 35 years old is a necessary condition for being President of the United States. Answer: people x, if x is President of the United States, then x is at least 35 years old.

50 Example Rewrite the following as a universal conditional statement: A product of two numbers is 0 only if one of the numbers is 0. Answer: If a product of two numbers is 0, then one of the numbers is 0.

51 Remark( Contrapositive ) 1. people x, if x is President of the United States, then x is at least 35 years old. people x, if x is younger than 35 years old, then x cannot be President of the United States. 2. If a product of two numbers is 0, then one of the numbers is 0. If neither of two numbers is 0, then the product of the numbers is not 0.

52 3.4. Arguments with Quantified Statements

53 Watch YouTube clips Modus Ponens Modus Tollens

54 Rule of inference Universal modus ponens ( direct form of logical argument ) Contrapositive type of logical argument Universal modus tollens ( logical argument using contrapositive )

55 The rule of instantiation If some property is true of everything in a set, then it is true of any particular thing in the set.

56 Example Rule of instantiation If some property is true of everything in a set, then it is true of any particular thing in the set. Example All human are mortal. : True to all human. Thus, everybody in this class is mortal ( because everybody in this class is human.)

57 The rule of instantiation If some property is true of everything in a set, then it is true of any particular thing in the set. The valid form of argument: Rule of instantiation + modus ponens This type of argument is called a Universal Modus Ponens.

58 Example Naïve Argument All human are mortal. : True to all human. Thus, everybody in this class is mortal ( because everybody in this class is human.) Rule of instantiation + modus ponens All human are mortal. Everybody in this class is human. Therefore, everybody in this class is mortal.

59 Mathematical Reasoning Induction Small facts to Bigger Conclusion ( Particular to General) Example: 1 = 1 = = 3 = = 6 = 3 4 and 2 conclude n = n (n + 1) 2 Deduction General info applied to Particular case to get conclusion( General to Particular) Example: If a student registered in MATH 321 class, then the student must take two mid-term exams. Jenna registered in MATH 321. Therefore, she must take two mid-term exams.

60 Universal Modus Ponens Modus Ponens is a logic we use when we do deductive reasoning: General rule is applied to particular case(s). Formal Version x, if P(x), then Q(x). P(a) for a particular a. Q(a). Informal Version If x makes P(x) true, then x makes Q(x) true. a makes P(x) true. a makes Q(x) true.

61 Example Rewrite the following argument using quantifiers, variables, and predicate symbols. Is this argument valid? Why? If an integer is even, then its square is even. K is a particular integer that is even. k 2 is even.

62 Example Rewrite the following argument using quantifiers, variables, and predicate symbols. If an integer is even, then its square is even. k is a particular integer that is even. k 2 is even. Symbolize: E(x): x is an even integer. S(x): x 2 is even. k: particular integer x, if E(x), then S(x). E(k) for a particular k. S(k)

63 Example Is this argument valid? Why? This argument is valid because it uses the universal modus ponens. If an integer is even, then its square is even. K is a particular integer that is even. k 2 is even.

64 Example Write the conclusion that can be inferred using universal modus ponens: If T is any right triangle with hypotenuse c and legs a and b, then c 2 = a 2 + b 2. The triangle shown at the right is a right triangle with both legs equal to 1 and hypotenuse c.

65 Example( Answer) Universal modus ponens If T is any right triangle with hypotenuse c and legs a and b, then c 2 = a 2 + b 2. The triangle shown at the right is a right triangle with both legs equal to 1 and hypotenuse c. Explanation General Rule The triangle shown at the right is a particular case of right triangle. c 2 = = 2. Thus, the general rule is applied to the given particular case.

66 Universal Modus Tollens Validity form of Argument: Universal instantiation + modus tollens

67 Universal Modus Tollens Formal Version x, if P(x), then Q(x). ~Q a, for a particular a. ~P(a) Informal Version If x makes P(x) true, then x makes Q(x) true. a does not make Q(x) true. a does not make P(x) true.

68 Example Rewrite the following argument using quantifiers, variables, and predicate symbols. Write the major premise in conditional form. Is this argument valid? Why? All human beings are mortal. Zeus is not mortal. Zeus is not human.

69 Example Problem Rewrite the following argument using quantifiers, variables, and predicate symbols. All human beings are mortal. Zeus is not mortal. Zeus is not human. Answer H(x): x is human. M(x): x is mortal. Z: Zeus. x, if H x, then M x. ~M Z ~H(Z)

70 Example Problem 1. Write the major premise in conditional form. 2. Is this argument valid? Why? All human beings are mortal. Zeus is not mortal. Zeus is not human. Answer 1. x, if x is human, then x is mortal. 1. This argument has the form of universal modus tollens. Thus it is valid.

71 Example Write the conclusion that can be inferred using universal modus tollens. All professors are absent-minded. Tom Hutchins is not absent-minded. Answer: Tom Hutchins is not a professor.

72 Argument form is Valid An argument form is valid if No matter what particular predicates are substituted for the predicate symbols in its premises, if the resulting premise statements are all true, then the conclusion is also true. An Argument is called valid if, and only if its form is valid.

73 Using Diagrams to Test Validity

74 Example All integers are rational numbers. Rational Numbers

75 Example Problem Use diagrams to show the validity of the following syllogism. All human beings are mortal. Zeus is not mortal. Zeus is not a human being. Analyze the syllogism Major premise: All human beings are mortal. Minor premise: Zeus is not mortal. Conclusion: Zeus is not a human being.

76 Example Problem Use diagrams to show the validity of the following syllogism. Diagram of Major premise All human beings are mortal. Zeus is not mortal. Zeus is not a human being.

77 Example Problem Use diagrams to show the validity of the following syllogism. Diagram of minor premise All human beings are mortal. Zeus is not mortal. Zeus is not a human being.

78 Example Problem Use diagrams to show the validity of the following syllogism. Conclusion All human beings are mortal. Zeus is not mortal. Zeus is not a human being.

79 Example( An Argument with No ) Use diagrams to test the following argument for validity: No polynomial functions have horizontal asymptotes. This function has a horizontal asymptote. This function is not a polynomial function.

80 Example( An Argument with No ) Polynomial functions Functions with horizontal asymptotes this function

81 Key Words Predicate Domain Codomain Truth Set Universal quantifier Existential quantifier Counterexample If and only if Only if Necessary, sufficient condition Modus ponens Modus tollens Universal modus ponens Universal modus tollens Rule of instantiation Induction, deduction Argument form is valid Argument is valid Using diagrams to test validity Diagram of major premise Diagram of minor premise

82 Negation ( Summarize the formal negation to each type of statement) Universal statement Existential statement Universal conditional statement Universal existential statement

83 Variation of Universal conditional statement What are contrapositive, converse and inverse of universal conditional statement?