Chapter 3. The Logic of Quantified Statements

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "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?

DISCRETE MATH: LECTURE Chapter 3.3 Statements with Multiple Quantifiers If you want to establish the truth of a statement of the form

DISCRETE MATH: LECTURE Chapter 3.3 Statements with Multiple Quantifiers If you want to establish the truth of a statement of the form DISCRETE MATH: LECTURE 5 DR. DANIEL FREEMAN 1. Chapter 3.3 Statements with Multiple Quantifiers If you want to establish the truth of a statement of the form x D, y E such that P (x, y) your challenge

More information

THE LOGIC OF QUANTIFIED STATEMENTS. Predicates and Quantified Statements I. Predicates and Quantified Statements I CHAPTER 3 SECTION 3.

THE LOGIC OF QUANTIFIED STATEMENTS. Predicates and Quantified Statements I. Predicates and Quantified Statements I CHAPTER 3 SECTION 3. CHAPTER 3 THE LOGIC OF QUANTIFIED STATEMENTS SECTION 3.1 Predicates and Quantified Statements I Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Predicates

More information

Chapter 2: The Logic of Quantified Statements. January 22, 2010

Chapter 2: The Logic of Quantified Statements. January 22, 2010 Chapter 2: The Logic of Quantified Statements January 22, 2010 Outline 1 2.1- Introduction to Predicates and Quantified Statements I 2 2.2 - Introduction to Predicates and Quantified Statements II 3 2.3

More information

Discrete Structures for Computer Science

Discrete Structures for Computer Science Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #4: Predicates and Quantifiers Based on materials developed by Dr. Adam Lee Topics n Predicates n

More information

5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction.

5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction. Statements Compounds and Truth Tables. Statements, Negations, Compounds, Conjunctions, Disjunctions, Truth Tables, Logical Equivalence, De Morgan s Law, Tautology, Contradictions, Proofs with Logical Equivalent

More information

DISCRETE MATH: FINAL REVIEW

DISCRETE MATH: FINAL REVIEW DISCRETE MATH: FINAL REVIEW DR. DANIEL FREEMAN 1) a. Does 3 = {3}? b. Is 3 {3}? c. Is 3 {3}? c. Is {3} {3}? c. Is {3} {3}? d. Does {3} = {3, 3, 3, 3}? e. Is {x Z x > 0} {x R x > 0}? 1. Chapter 1 review

More information

CSE 20 DISCRETE MATH. Fall

CSE 20 DISCRETE MATH. Fall CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs

More information

A. Propositional Logic

A. Propositional Logic CmSc 175 Discrete Mathematics A. Propositional Logic 1. Statements (Propositions ): Statements are sentences that claim certain things. Can be either true or false, but not both. Propositional logic deals

More information

Section 3.1: Direct Proof and Counterexample 1

Section 3.1: Direct Proof and Counterexample 1 Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion

More information

Propositional Logic Not Enough

Propositional Logic Not Enough Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks

More information

Section Summary. Predicate logic Quantifiers. Negating Quantifiers. Translating English to Logic. Universal Quantifier Existential Quantifier

Section Summary. Predicate logic Quantifiers. Negating Quantifiers. Translating English to Logic. Universal Quantifier Existential Quantifier Section 1.4 Section Summary Predicate logic Quantifiers Universal Quantifier Existential Quantifier Negating Quantifiers De Morgan s Laws for Quantifiers Translating English to Logic Propositional Logic

More information

WUCT121. Discrete Mathematics. Logic. Tutorial Exercises

WUCT121. Discrete Mathematics. Logic. Tutorial Exercises WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the

More information

MAT 243 Test 1 SOLUTIONS, FORM A

MAT 243 Test 1 SOLUTIONS, FORM A t MAT 243 Test 1 SOLUTIONS, FORM A 1. [10 points] Rewrite the statement below in positive form (i.e., so that all negation symbols immediately precede a predicate). ( x IR)( y IR)((T (x, y) Q(x, y)) R(x,

More information

Chapter 1: The Logic of Compound Statements. January 7, 2008

Chapter 1: The Logic of Compound Statements. January 7, 2008 Chapter 1: The Logic of Compound Statements January 7, 2008 Outline 1 1.1 Logical Form and Logical Equivalence 2 1.2 Conditional Statements 3 1.3 Valid and Invalid Arguments Central notion of deductive

More information

[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 400 lecture note #2. 1) Basics

[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 400 lecture note #2. 1) Basics 400 lecture note #2 [Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 1) Basics An argument is a sequence of statements ( s1, s2,, sn). All statements in an argument, excet for

More information

CSCE 222 Discrete Structures for Computing. Predicate Logic. Dr. Hyunyoung Lee. !!!!! Based on slides by Andreas Klappenecker

CSCE 222 Discrete Structures for Computing. Predicate Logic. Dr. Hyunyoung Lee. !!!!! Based on slides by Andreas Klappenecker CSCE 222 Discrete Structures for Computing Predicate Logic Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Predicates A function P from a set D to the set Prop of propositions is called a predicate.

More information

MACM 101 Discrete Mathematics I. Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class)

MACM 101 Discrete Mathematics I. Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class) MACM 101 Discrete Mathematics I Exercises on Predicates and Quantifiers. Due: Tuesday, October 13th (at the beginning of the class) Reminder: the work you submit must be your own. Any collaboration and

More information

CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS

CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS 1.1 - Logical Form and Logical Equivalence Definition. A statement or proposition is a sentence that is either true or false, but not both. ex. 1 + 2 = 3 IS a statement

More information

First order Logic ( Predicate Logic) and Methods of Proof

First order Logic ( Predicate Logic) and Methods of Proof First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating

More information

Readings: Conjecture. Theorem. Rosen Section 1.5

Readings: Conjecture. Theorem. Rosen Section 1.5 Readings: Conjecture Theorem Lemma Lemma Step 1 Step 2 Step 3 : Step n-1 Step n a rule of inference an axiom a rule of inference Rosen Section 1.5 Provide justification of the steps used to show that a

More information

Tutorial 2 Logic of Quantified Statements

Tutorial 2 Logic of Quantified Statements Tutorial 2 Logic of Quantified Statements 1. Let V be the set of all visitors to Universal Studios Singapore on a certain day, T (v) be v took the Transformers ride, G(v) be v took the Battlestar Galactica

More information

Section 2.1: Introduction to the Logic of Quantified Statements

Section 2.1: Introduction to the Logic of Quantified Statements Section 2.1: Introduction to the Logic of Quantified Statements In the previous chapter, we studied a branch of logic called propositional logic or propositional calculus. Loosely speaking, propositional

More information

CPSC 121: Models of Computation

CPSC 121: Models of Computation CPSC 121: Models of Computation Unit 6 Rewriting Predicate Logic Statements Based on slides by Patrice Belleville and Steve Wolfman Coming Up Pre-class quiz #7 is due Wednesday October 25th at 9:00 pm.

More information

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications Chapter 1: Foundations: Sets, Logic, and Algorithms Discrete Mathematical Structures: Theory and Applications Learning Objectives Learn about sets Explore various operations on sets Become familiar with

More information

Proofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction

Proofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction Introduction I Proofs Computer Science & Engineering 235 Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu A proof is a proof. What kind of a proof? It s a proof. A proof is a proof. And when

More information

Logic and Propositional Calculus

Logic and Propositional Calculus CHAPTER 4 Logic and Propositional Calculus 4.1 INTRODUCTION Many algorithms and proofs use logical expressions such as: IF p THEN q or If p 1 AND p 2, THEN q 1 OR q 2 Therefore it is necessary to know

More information

HANDOUT AND SET THEORY. Ariyadi Wijaya

HANDOUT AND SET THEORY. Ariyadi Wijaya HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics

More information

Argument. whenever all the assumptions are true, then the conclusion is true. If today is Wednesday, then yesterday is Tuesday. Today is Wednesday.

Argument. whenever all the assumptions are true, then the conclusion is true. If today is Wednesday, then yesterday is Tuesday. Today is Wednesday. Logic and Proof Argument An argument is a sequence of statements. All statements but the first one are called assumptions or hypothesis. The final statement is called the conclusion. An argument is valid

More information

Rules Build Arguments Rules Building Arguments

Rules Build Arguments Rules Building Arguments Section 1.6 1 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments Rules of Inference for Quantified Statements Building Arguments for Quantified

More information

CS 2336 Discrete Mathematics

CS 2336 Discrete Mathematics CS 2336 Discrete Mathematics Lecture 3 Logic: Rules of Inference 1 Outline Mathematical Argument Rules of Inference 2 Argument In mathematics, an argument is a sequence of propositions (called premises)

More information

1) Let h = John is healthy, w = John is wealthy and s = John is wise Write the following statement is symbolic form

1) Let h = John is healthy, w = John is wealthy and s = John is wise Write the following statement is symbolic form Math 378 Exam 1 Spring 2009 Show all Work Name 1) Let h = John is healthy, w = John is wealthy and s = John is wise Write the following statement is symbolic form a) In order for John to be wealthy it

More information

DISCRETE MATHEMATICS BA202

DISCRETE MATHEMATICS BA202 TOPIC 1 BASIC LOGIC This topic deals with propositional logic, logical connectives and truth tables and validity. Predicate logic, universal and existential quantification are discussed 1.1 PROPOSITION

More information

CITS2211 Discrete Structures Proofs

CITS2211 Discrete Structures Proofs CITS2211 Discrete Structures Proofs Unit coordinator: Rachel Cardell-Oliver August 13, 2017 Highlights 1 Arguments vs Proofs. 2 Proof strategies 3 Famous proofs Reading Chapter 1: What is a proof? Mathematics

More information

Proposition logic and argument. CISC2100, Spring 2017 X.Zhang

Proposition logic and argument. CISC2100, Spring 2017 X.Zhang Proposition logic and argument CISC2100, Spring 2017 X.Zhang 1 Where are my glasses? I know the following statements are true. 1. If I was reading the newspaper in the kitchen, then my glasses are on the

More information

Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models

Chapter 2. Mathematical Reasoning. 2.1 Mathematical Models Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................

More information

Predicate Calculus lecture 1

Predicate Calculus lecture 1 Predicate Calculus lecture 1 Section 1.3 Limitation of Propositional Logic Consider the following reasoning All cats have tails Gouchi is a cat Therefore, Gouchi has tail. MSU/CSE 260 Fall 2009 1 MSU/CSE

More information

3.6. Disproving Quantified Statements Disproving Existential Statements

3.6. Disproving Quantified Statements Disproving Existential Statements 36 Dproving Quantified Statements 361 Dproving Extential Statements A statement of the form x D, P( if P ( false for all x D false if and only To dprove th kind of statement, we need to show the for all

More information

Discrete Mathematics: An Open Introduction

Discrete Mathematics: An Open Introduction Discrete Mathematics: An Open Introduction Course Notes for Math 228 at the University of Northern Colorado Oscar Levin, Ph.D. Spring 2013 Contents 1 Logic 2 1.1 Propositional Logic................................

More information

Proposi'onal Logic Not Enough

Proposi'onal Logic Not Enough Section 1.4 Proposi'onal Logic Not Enough If we have: All men are mortal. Socrates is a man. Socrates is mortal Compare to: If it is snowing, then I will study discrete math. It is snowing. I will study

More information

The Logic of Compound Statements cont.

The Logic of Compound Statements cont. The Logic of Compound Statements cont. CSE 215, Computer Science 1, Fall 2011 Stony Brook University http://www.cs.stonybrook.edu/~cse215 Refresh from last time: Logical Equivalences Commutativity of :

More information

Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012

Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012 March 5, 2012 Webwork Homework. The handout on Logic is Chapter 4 from Mary Attenborough s book Mathematics for Electrical Engineering and Computing. Proving Propositions We combine basic propositions

More information

Math 55 Homework 2 solutions

Math 55 Homework 2 solutions Math 55 Homework solutions Section 1.3. 6. p q p q p q p q (p q) T T F F F T F T F F T T F T F T T F T F T F F T T T F T 8. a) Kwame will not take a job in industry and not go to graduate school. b) Yoshiko

More information

Logic. Propositional Logic: Syntax

Logic. Propositional Logic: Syntax Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about

More information

Proof. Theorems. Theorems. Example. Example. Example. Part 4. The Big Bang Theory

Proof. Theorems. Theorems. Example. Example. Example. Part 4. The Big Bang Theory Proof Theorems Part 4 The Big Bang Theory Theorems A theorem is a statement we intend to prove using existing known facts (called axioms or lemmas) Used extensively in all mathematical proofs which should

More information

LECTURE 1. Logic and Proofs

LECTURE 1. Logic and Proofs LECTURE 1 Logic and Proofs The primary purpose of this course is to introduce you, most of whom are mathematics majors, to the most fundamental skills of a mathematician; the ability to read, write, and

More information

CSE 1400 Applied Discrete Mathematics Proofs

CSE 1400 Applied Discrete Mathematics Proofs CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4

More information

Logical Reasoning. Chapter Statements and Logical Operators

Logical Reasoning. Chapter Statements and Logical Operators Chapter 2 Logical Reasoning 2.1 Statements and Logical Operators Preview Activity 1 (Compound Statements) Mathematicians often develop ways to construct new mathematical objects from existing mathematical

More information

University of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura. February 9, :30 pm Duration: 1:50 hs. Closed book, no calculators

University of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura. February 9, :30 pm Duration: 1:50 hs. Closed book, no calculators University of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura February 9, 2010 11:30 pm Duration: 1:50 hs Closed book, no calculators Last name: First name: Student number: There are 5 questions and

More information

1.3 Predicates and Quantifiers

1.3 Predicates and Quantifiers 1.3 Predicates and Quantifiers INTRODUCTION Statements x>3, x=y+3 and x + y=z are not propositions, if the variables are not specified. In this section we discuss the ways of producing propositions from

More information

Discrete Mathematics

Discrete Mathematics Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 9, 2009 Overview of ( 1.5-1.7, ~2 hours) Methods of mathematical argument (i.e., proof methods) can be formalized

More information

Logic Review Solutions

Logic Review Solutions Logic Review Solutions 1. What is true concerning the validity of the argument below? (hint: Use a Venn diagram.) 1. All pesticides are harmful to the environment. 2. No fertilizer is a pesticide. Therefore,

More information

Unit I LOGIC AND PROOFS. B. Thilaka Applied Mathematics

Unit I LOGIC AND PROOFS. B. Thilaka Applied Mathematics Unit I LOGIC AND PROOFS B. Thilaka Applied Mathematics UNIT I LOGIC AND PROOFS Propositional Logic Propositional equivalences Predicates and Quantifiers Nested Quantifiers Rules of inference Introduction

More information

Logic and Set Notation

Logic and Set Notation Logic and Set Notation Logic Notation p, q, r: statements,,,, : logical operators p: not p p q: p and q p q: p or q p q: p implies q p q:p if and only if q We can build compound sentences using the above

More information

Logical Form 5 Famous Valid Forms. Today s Lecture 1/26/10

Logical Form 5 Famous Valid Forms. Today s Lecture 1/26/10 Logical Form 5 Famous Valid Forms Today s Lecture 1/26/10 Announcements Homework: --Read Chapter 7 pp. 277-298 (doing the problems in parts A, B, and C pp. 298-300 are recommended but not required at this

More information

MATH CSE20 Homework 5 Due Monday November 4

MATH CSE20 Homework 5 Due Monday November 4 MATH CSE20 Homework 5 Due Monday November 4 Assigned reading: NT Section 1 (1) Prove the statement if true, otherwise find a counterexample. (a) For all natural numbers x and y, x + y is odd if one of

More information

Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9

Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9 Unit 1. Propositional Logic Reading do all quick-checks Propositional Logic: Ch. 2.intro, 2.2, 2.3, 2.4. Review 2.9 Typeset September 23, 2005 1 Statements or propositions Defn: A statement is an assertion

More information

Discrete Mathematics

Discrete Mathematics Department of Mathematics National Cheng Kung University 2008 2.4: The use of Quantifiers Definition (2.5) A declarative sentence is an open statement if 1) it contains one or more variables, and 1 ) quantifier:

More information

Proofs. Chapter 2 P P Q Q

Proofs. Chapter 2 P P Q Q Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,

More information

ECOM Discrete Mathematics

ECOM Discrete Mathematics ECOM 2311- Discrete Mathematics Chapter # 1 : The Foundations: Logic and Proofs Fall, 2013/2014 ECOM 2311- Discrete Mathematics - Ch.1 Dr. Musbah Shaat 1 / 85 Outline 1 Propositional Logic 2 Propositional

More information

A Little Deductive Logic

A Little Deductive Logic A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that

More information

CS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:

CS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic: x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which

More information

Rules of Inference. Arguments and Validity

Rules of Inference. Arguments and Validity Arguments and Validity A formal argument in propositional logic is a sequence of propositions, starting with a premise or set of premises, and ending in a conclusion. We say that an argument is valid if

More information

Formal Logic. Critical Thinking

Formal Logic. Critical Thinking ormal Logic Critical hinking Recap: ormal Logic If I win the lottery, then I am poor. I win the lottery. Hence, I am poor. his argument has the following abstract structure or form: If P then Q. P. Hence,

More information

CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)

CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1) CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bio-conditional Converse Inverse Contrapositive Laws of

More information

Group 5. Jeremy Gutierrez. Jesus Ochoa Perez. Alvaro Gonzalez. MATH 170: Discrete Mathematics. Dr. Lipika Deka. March 14, 2014.

Group 5. Jeremy Gutierrez. Jesus Ochoa Perez. Alvaro Gonzalez. MATH 170: Discrete Mathematics. Dr. Lipika Deka. March 14, 2014. Gutierrez, Perez, & Gonzalez 1 Group 5 Jeremy Gutierrez Jesus Ochoa Perez Alvaro Gonzalez MATH 170: Discrete Mathematics Dr. Lipika Deka March 14, 2014 Project Part 1 Gutierrez, Perez, & Gonzalez 2 Hello

More information

Direct Proof and Counterexample I:Introduction. Copyright Cengage Learning. All rights reserved.

Direct Proof and Counterexample I:Introduction. Copyright Cengage Learning. All rights reserved. Direct Proof and Counterexample I:Introduction Copyright Cengage Learning. All rights reserved. Goal Importance of proof Building up logic thinking and reasoning reading/using definition interpreting statement:

More information

Logical Operators. Conjunction Disjunction Negation Exclusive Or Implication Biconditional

Logical Operators. Conjunction Disjunction Negation Exclusive Or Implication Biconditional Logical Operators Conjunction Disjunction Negation Exclusive Or Implication Biconditional 1 Statement meaning p q p implies q if p, then q if p, q when p, q whenever p, q q if p q when p q whenever p p

More information

! Predicates! Variables! Quantifiers. ! Universal Quantifier! Existential Quantifier. ! Negating Quantifiers. ! De Morgan s Laws for Quantifiers

! Predicates! Variables! Quantifiers. ! Universal Quantifier! Existential Quantifier. ! Negating Quantifiers. ! De Morgan s Laws for Quantifiers Sec$on Summary (K. Rosen notes for Ch. 1.4, 1.5 corrected and extended by A.Borgida)! Predicates! Variables! Quantifiers! Universal Quantifier! Existential Quantifier! Negating Quantifiers! De Morgan s

More information

Mathematics 220 Midterm Practice problems from old exams Page 1 of 8

Mathematics 220 Midterm Practice problems from old exams Page 1 of 8 Mathematics 220 Midterm Practice problems from old exams Page 1 of 8 1. (a) Write the converse, contrapositive and negation of the following statement: For every integer n, if n is divisible by 3 then

More information

HOMEWORK 1: SOLUTIONS - MATH 215 INSTRUCTOR: George Voutsadakis

HOMEWORK 1: SOLUTIONS - MATH 215 INSTRUCTOR: George Voutsadakis HOMEWORK 1: SOLUTIONS - MATH 215 INSTRUCTOR: George Voutsadakis Problem 1 Make truth tables for the propositional forms (P Q) (P R) and (P Q) (R S). Solution: P Q R P Q P R (P Q) (P R) F F F F F F F F

More information

9/5/17. Fermat s last theorem. CS 220: Discrete Structures and their Applications. Proofs sections in zybooks. Proofs.

9/5/17. Fermat s last theorem. CS 220: Discrete Structures and their Applications. Proofs sections in zybooks. Proofs. Fermat s last theorem CS 220: Discrete Structures and their Applications Theorem: For every integer n > 2 there is no solution to the equation a n + b n = c n where a,b, and c are positive integers Proofs

More information

Why Proofs? Proof Techniques. Theorems. Other True Things. Proper Proof Technique. How To Construct A Proof. By Chuck Cusack

Why Proofs? Proof Techniques. Theorems. Other True Things. Proper Proof Technique. How To Construct A Proof. By Chuck Cusack Proof Techniques By Chuck Cusack Why Proofs? Writing roofs is not most student s favorite activity. To make matters worse, most students do not understand why it is imortant to rove things. Here are just

More information

LING 501, Fall 2004: Quantification

LING 501, Fall 2004: Quantification LING 501, Fall 2004: Quantification The universal quantifier Ax is conjunctive and the existential quantifier Ex is disjunctive Suppose the domain of quantification (DQ) is {a, b}. Then: (1) Ax Px Pa &

More information

Discrete Structures for Computer Science

Discrete Structures for Computer Science Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #6: Rules of Inference Based on materials developed by Dr. Adam Lee Today s topics n Rules of inference

More information

Introduction to Metalogic

Introduction to Metalogic Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)

More information

Predicate Logic. CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo

Predicate Logic. CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo Predicate Logic CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 22 Outline 1 From Proposition to Predicate

More information

Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

More information

Packet #1: Logic & Proofs. Applied Discrete Mathematics

Packet #1: Logic & Proofs. Applied Discrete Mathematics Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should

More information

Discrete Structures CRN Test 3 Version 1 CMSC 2123 Autumn 2013

Discrete Structures CRN Test 3 Version 1 CMSC 2123 Autumn 2013 . Print your name on your scantron in the space labeled NAME. 2. Print CMSC 223 in the space labeled SUBJECT. 3. Print the date 2-2-203, in the space labeled DATE. 4. Print your CRN, 786, in the space

More information

Collins' notes on Lemmon's Logic

Collins' notes on Lemmon's Logic Collins' notes on Lemmon's Logic (i) Rule of ssumption () Insert any formula at any stage into a proof. The assumed formula rests upon the assumption of itself. (ii) Double Negation (DN) a. b. ( Two negations

More information

First Order Logic (1A) Young W. Lim 11/18/13

First Order Logic (1A) Young W. Lim 11/18/13 Copyright (c) 2013. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software

More information

Section 1.3: Valid and Invalid Arguments

Section 1.3: Valid and Invalid Arguments Section 1.3: Valid and Invalid Arguments Now we have developed the basic language of logic, we shall start to consider how logic can be used to determine whether or not a given argument is valid. In order

More information

More examples of mathematical. Lecture 4 ICOM 4075

More examples of mathematical. Lecture 4 ICOM 4075 More examples of mathematical proofs Lecture 4 ICOM 4075 Proofs by construction A proof by construction is one in which anobjectthat proves the truth value of an statement is built, or found There are

More information

Resolution (14A) Young W. Lim 8/15/14

Resolution (14A) Young W. Lim 8/15/14 Resolution (14A) Young W. Lim Copyright (c) 2013-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

More information

Predicate Logic combines the distinctive features of syllogistic and propositional logic.

Predicate Logic combines the distinctive features of syllogistic and propositional logic. Predicate Logic combines the distinctive features of syllogistic and propositional logic. The fundamental component in predicate logic is the predicate, which is always symbolized with upper case letters.

More information

Formal Logic: Quantifiers, Predicates, and Validity. CS 130 Discrete Structures

Formal Logic: Quantifiers, Predicates, and Validity. CS 130 Discrete Structures Formal Logic: Quantifiers, Predicates, and Validity CS 130 Discrete Structures Variables and Statements Variables: A variable is a symbol that stands for an individual in a collection or set. For example,

More information

1. Propositions: Contrapositives and Converses

1. Propositions: Contrapositives and Converses Preliminaries 1 1. Propositions: Contrapositives and Converses Given two propositions P and Q, the statement If P, then Q is interpreted as the statement that if the proposition P is true, then the statement

More information

Deductive and Inductive Logic

Deductive and Inductive Logic Deductive Logic Overview (1) Distinguishing Deductive and Inductive Logic (2) Validity and Soundness (3) A Few Practice Deductive Arguments (4) Testing for Invalidity (5) Practice Exercises Deductive and

More information

Introduction to Isabelle/HOL

Introduction to Isabelle/HOL Introduction to Isabelle/HOL 1 Notes on Isabelle/HOL Notation In Isabelle/HOL: [ A 1 ;A 2 ; ;A n ]G can be read as if A 1 and A 2 and and A n then G 3 Note: -Px (P x) stands for P (x) (P(x)) -P(x, y) can

More information

Contribution of Problems

Contribution of Problems Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions

More information

Natural deduction for truth-functional logic

Natural deduction for truth-functional logic Natural deduction for truth-functional logic Phil 160 - Boston University Why natural deduction? After all, we just found this nice method of truth-tables, which can be used to determine the validity or

More information

MA103 STATEMENTS, PROOF, LOGIC

MA103 STATEMENTS, PROOF, LOGIC MA103 STATEMENTS, PROOF, LOGIC Abstract Mathematics is about making precise mathematical statements and establishing, by proof or disproof, whether these statements are true or false. We start by looking

More information

Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007)

Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007) Department of Computer Science University at Albany, State University of New York Solutions to Sample Discrete Mathematics Examination II (Fall 2007) Problem 1: Specify two different predicates P (x) and

More information

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

More information

Lecture : Set Theory and Logic

Lecture : Set Theory and Logic Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Contrapositive and Converse 1 Contrapositive and Converse 2 3 4 5 Contrapositive and Converse

More information

3 The Semantics of the Propositional Calculus

3 The Semantics of the Propositional Calculus 3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical

More information

Proof by Contradiction

Proof by Contradiction Proof by Contradiction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Proof by Contradiction Fall 2014 1 / 12 Outline 1 Proving Statements with Contradiction 2 Proving

More information

PHI Propositional Logic Lecture 2. Truth Tables

PHI Propositional Logic Lecture 2. Truth Tables PHI 103 - Propositional Logic Lecture 2 ruth ables ruth ables Part 1 - ruth unctions for Logical Operators ruth unction - the truth-value of any compound proposition determined solely by the truth-value

More information

Topics in Logic and Proofs

Topics in Logic and Proofs Chapter 2 Topics in Logic and Proofs Some mathematical statements carry a logical value of being true or false, while some do not. For example, the statement 4 + 5 = 9 is true, whereas the statement 2

More information

Propositions and Proofs

Propositions and Proofs Propositions and Proofs Gert Smolka, Saarland University April 25, 2018 Proposition are logical statements whose truth or falsity can be established with proofs. Coq s type theory provides us with a language

More information