Predicate Logic combines the distinctive features of syllogistic and propositional logic.
|
|
- Emily Welch
- 6 years ago
- Views:
Transcription
1 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.
2 Symbolic predicates may be used to translate three kinds of statements: Singular Statements: Socrates is dead. Universal Statements: All S are P or No S are P. Particular Statements: Some S are P or Some S are not P.
3 Translating singular statements requires lower case letters called individual constants and the logical operators of propositional logic. Socrates is mortal is symbolically translated as Ms. If Paris is beautiful, then André told the truth, becomes Bp Ta.
4 Translating universal statements requires universal quantifiers, individual variables, and the horseshoe operator: All S are P is translated as x (Sx Px), meaning, for any x, if x is an S, then x is a P. No S are P is translated as x (Sx ~Px), meaning, for any x, if x is an S, then x is not a P. The Venn diagrams responding to these statements are:
5 Translating particular statements requires existential quantifiers, individual variables, and the dot operator. Some S are P is translated as (Ǝx) (Sx Px) or There exists such an x such that x is an S and x is a P. Some S are not P is translated as (Ǝx) (Sx ~Px) or There exists such an x such that x is an S and x is not a P. The Venn diagrams responding to these statements are:
6 For both universal and particular statements, we need to distinguish between free and bound variables. Free variables are not bound by any quantifier. Bound variables are.
7 Four rules govern the removal and introduction of quantifiers: Universal Instantiation (UI): This allows universal quantifiers to be removed. It requires using either a constant or a variable. Universal Generalization (UG): This allows universal quantifiers to be introduced. Universal Generalization cannot be performed if the instantial letter is a constant.
8 Existential Generalization allows existential quantifiers to be introduced. There are no restrictions on Existential Generalization. Existential Instantiation allows for existential quantifiers to be removed. Existential instantiation requires that the existential name must be a new name that does not occur on the line that indicates the conclusion to be derived. This name must be new and many not occur earlier in the proof.
9 Common Misapplications: All four of these rules of inference require that the rules be applied only to whole lines in the proof.
10 As long as negation signs or ~s precede quantifiers, statements cannot be instantiated; thus, conclusions cannot be derived. The Quantifier Negation Rule allows us to remove the negation signs. The Quantifier Negation Rule (QN) is expressed as: (x) Fx :: ~(Ǝx) ~Fx ~(x) Fx :: (Ǝx) ~Fx (Ǝx) Fx :: ~(x) ~Fx ~(Ǝx) Fx :: (x) ~Fx
11 One type of quantifier can be replaced by the other type if and only if it appears immediately before and after the new quantifier: 1. Tilde operators that were originally present are deleted. 2. Tilde operators that were not originally present are inserted.
12 Many arguments with conclusions that are either difficult or impossible to derive by the conventional method can be handled with ease in using either conditional or indirect proof. 1. (x) (Hx Ix) / (Ǝx)Hx (Ǝx) Ix 2. (Ǝx) Hx ACP 3. Ha 2, EI 4. Ha Ia I, UI 5. Ia 3, 4, MP 6. (Ǝx) Ix 5, EG 7. (Ǝx)Hx (Ǝx) Ix 2-6, CP
13 UG must not be used within the scope of an indented sequence if the instantial variable y occurs free (unbound by any quantifier) in the first line of that sequence. The following defective proof shows why: 1. (x)rx (x)sx / (x) (Rx Sx) 2. Rx ACP 3. x(rx) 2, UG (invalid) 4. x(sx) 1, 3, MP 5. Ia 4, UI 6. Rx Sx 2-5, CP 7. (x) (Rx Sx) 6, UG
14 If Rx means x is a rabbit, and Sx means x is a snake, then the premise translates as, If everything in the universe is a rabbit, everything in the universe is snake. The statement is true because the antecedent is false: not everything in the universe is a rabbit. However, the conclusion is false because it asserts that all rabbits are snakes. Thus, the argument is invalid.
15 The Counterexample Method consists of finding a substitution instance of a given invalid argument for or a given invalid symbolized argument that has true premises and a false conclusion: Some animals are not mammals. All cats are mammals. Therefore some cats are not mammals. (Ǝx) (Ax ~Bx) (x) (Cx Bx) / (Ǝx) (Cx ~Ax)
16 The Finite Universe Method can be used to establish the validity of an invalid argument expressed in terms of a single variable. Conditional Logical Equivalence: We shrink the universe from billions of things to one thing, named Abigail. Everything in the universe is perfect is equivalent to Abigail is perfect. Something in the universe is perfect is also equivalent to Abigail is perfect because Abigail is something as well as everything. (x)px :: Pa (Ǝx)Px :: Pa
17 The method for proving an argument invalid consists in translating the premises and conclusion into singular statements, then testing the result with an indirect truth table. A universe of one is tried. If it is possible for the premises to be true and the conclusion false in this universe, the argument is immediately identified as invalid. If a contradiction results from this assumption, a universe of two is tried. If, in this second universe, it is possible for the premises to be true and the conclusion false, the argument is invalid. If not, a universe of three is tried, and so on.
18 Monadic (one-place) predicates assign attributes to individual things while Relational (N-place) Predicates establish connections between or among individuals. Translating Relational Statements: The Willis Tower is taller than the Empire State Building = Twe. However, as in monadic predicates, Tew = The Empire State Building is taller than the Willis Tower (which is not true).
19 Overlapping Quantifiers: When two quantifiers of the same sort appear adjacent to each other, the order does not make a difference. When different quantifiers appear adjacent to one another the order makes a difference, even when the statement function is nonrelational.
20 Using the Rules of Inference: The quantifier negation rule is applied in basically the same way as with single quantifiers. As the tilde operator is moved past a quantifier, the quantifier in question is switched to its correlative. 1. (Ǝx)(Ǝy)Pxy 2. (Ǝy) Pay 1, EI 3. Pab 2, EI 4. (Ǝx)Pxb 3, EG 5. (Ǝy)(Ǝx)Pxy 4, EG
21 The CQ rule is used in the same way. UI, EG and EI are all used in the same way. UG must not be used if the instantial variable y is free in any preceding line obtained by EI.
22 Translating and Negating Identity Relations: The only friend I have is Elizabeth. Elizabeth is not Nancy. Nancy is a Canadian. Therefore, there is a Canadian who is my friend. This relationship involves the following signs: = and Simple Identity Statements: Whoopi Goldberg is Caryn Johnson: w = c Whoopi Goldberg is not Roman Polanski: w r.
23 Only, The Only, and No Except : Only Nolan Ryan has struck out 5,000 batters: Sn (x) (Sx x = n) The only opera written by Beethoven is Fidelio. Of Bf (x) [Ox Bx) x = f] No nation except Australia is a continent: Na Ca (x) [(Nx Cx)] x = a] All Except All painters except Jackson Pollack make sense: Pj ~Mj (x) [(Px x j) Mx]
24 Superlatives: The largest planet is Jupiter: Pj (x)[(px x j) Ljx] Numerical Statements: There is at most one god: (x) (y) [Gx Gy) x = y] Definite Descriptions: The author of Middlemarch was a Victorian freethinker: (Ǝx) [Wxm (y)(wym y = x) Vx Fx]
25 Using the Rules of Inference: Special Rules, known collectively as Id govern the identity relation: Rule 1: a=a Rule 2: a=b is logically equivalent to b=a Rule 3: If a=b and b=c then a=c
SYMBOLIC LOGIC UNIT 10: SINGULAR SENTENCES
SYMBOLIC LOGIC UNIT 10: SINGULAR SENTENCES Singular Sentences name Paris is beautiful (monadic) predicate (monadic) predicate letter Bp individual constant Singular Sentences Bp These are our new simple
More information2. Use quantifiers to express the associative law for multiplication of real numbers.
1. Define statement function of one variable. When it will become a statement? Statement function is an expression containing symbols and an individual variable. It becomes a statement when the variable
More informationDenote John by j and Smith by s, is a bachelor by predicate letter B. The statements (1) and (2) may be written as B(j) and B(s).
PREDICATE CALCULUS Predicates Statement function Variables Free and bound variables Quantifiers Universe of discourse Logical equivalences and implications for quantified statements Theory of inference
More informationFormal 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 informationPropositional 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 informationPhilosophy 240 Symbolic Logic Russell Marcus Hamilton College Fall 2014
Philosophy 240 Symbolic Logic Russell Marcus Hamilton College Fall 2014 Class #23 - Translation into Predicate Logic II ( 3.2) Only as a Quantifier P Only Ps are Qs is logically equivalent to all Qs are
More information3. The Logic of Quantified Statements Summary. Aaron Tan August 2017
3. The Logic of Quantified Statements Summary Aaron Tan 28 31 August 2017 1 3. The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I Predicate; domain; truth set Universal quantifier,
More information4 Quantifiers and Quantified Arguments 4.1 Quantifiers
4 Quantifiers and Quantified Arguments 4.1 Quantifiers Recall from Chapter 3 the definition of a predicate as an assertion containing one or more variables such that, if the variables are replaced by objects
More informationCOMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
More informationPredicate Logic. 1 Predicate Logic Symbolization
1 Predicate Logic Symbolization innovation of predicate logic: analysis of simple statements into two parts: the subject and the predicate. E.g. 1: John is a giant. subject = John predicate =... is a giant
More informationPredicate Logic. Example. Statements in Predicate Logic. Some statements cannot be expressed in propositional logic, such as: Predicate Logic
Predicate Logic Predicate Logic (Rosen, Chapter 1.4-1.6) TOPICS Predicate Logic Quantifiers Logical Equivalence Predicate Proofs Some statements cannot be expressed in propositional logic, such as: All
More informationProposi'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 informationRULES OF UNIVERSAL INSTANTIATION AND GENERALIZATION, EXISTENTIAL INSTANTIATION AND GENERALIZATION, AND RULES OF QUANTIFIER EQUIVALENCE
1 UNIT 2 RULES OF UNIVERSAL INSTANTIATION AND GENERALIZATION, EXISTENTIAL INSTANTIATION AND GENERALIZATION, AND RULES OF QUANTIFIER EQUIVALENCE Contents 2.0 Objectives 2.1 Introduction 2.2 Rules of Quantification
More informationPredicate 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 informationSection 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 informationSymbolising Quantified Arguments
Symbolising Quantified Arguments 1. (i) Symbolise the following argument, given the universe of discourse is U = set of all animals. Animals are either male or female. Not all Cats are male, Therefore,
More informationRules 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 informationPhil 2B03 (McMaster University Final Examination) Page 1 of 4
Phil 2B03 (McMaster University Final Examination) Page 1 of 4 1. [SL proof] (a) Prove the formal validity of the following sequent: (F & G), F H, H G H (1) (F & G) Prem (2) F H Prem (3) H G Prem (4) H
More informationLecture Predicates and Quantifiers 1.5 Nested Quantifiers
Lecture 4 1.4 Predicates and Quantifiers 1.5 Nested Quantifiers Predicates The statement "x is greater than 3" has two parts. The first part, "x", is the subject of the statement. The second part, "is
More informationPredicate Logic. Andreas Klappenecker
Predicate Logic Andreas Klappenecker Predicates A function P from a set D to the set Prop of propositions is called a predicate. The set D is called the domain of P. Example Let D=Z be the set of integers.
More informationSection 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 informationChapter 3. The Logic of Quantified Statements
Chapter 3. The Logic of Quantified Statements 3.1. Predicates and Quantified Statements I Predicate in grammar Predicate refers to the part of a sentence that gives information about the subject. Example:
More informationPredicate Logic Quantifier Rules
Predicate Logic Quantifier Rules CS251 at CCUT, Spring 2017 David Lu May 8 th, 2017 Contents 1. Universal Instantiation (UI) 2. Existential Generalization (EG) 3. Universal Generalization (UG) 4. Existential
More informationReview. Propositions, propositional operators, truth tables. Logical Equivalences. Tautologies & contradictions
Review Propositions, propositional operators, truth tables Logical Equivalences. Tautologies & contradictions Some common logical equivalences Predicates & quantifiers Some logical equivalences involving
More informationIntroduction to Sets and Logic (MATH 1190)
Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition
More information3/29/2017. Logic. Propositions and logical operations. Main concepts: propositions truth values propositional variables logical operations
Logic Propositions and logical operations Main concepts: propositions truth values propositional variables logical operations 1 Propositions and logical operations A proposition is the most basic element
More informationPredicate 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 informationSection 1.1 Propositions
Set Theory & Logic Section 1.1 Propositions Fall, 2009 Section 1.1 Propositions In Chapter 1, our main goals are to prove sentences about numbers, equations or functions and to write the proofs. Definition.
More informationFormal (Natural) Deduction for Predicate Calculus
Formal (Natural) Deduction for Predicate Calculus Lila Kari University of Waterloo Formal (Natural) Deduction for Predicate Calculus CS245, Logic and Computation 1 / 42 Formal deducibility for predicate
More informationTest 1 Solutions(COT3100) (1) Prove that the following Absorption Law is correct. I.e, prove this is a tautology:
Test 1 Solutions(COT3100) Sitharam (1) Prove that the following Absorption Law is correct. I.e, prove this is a tautology: ( q (p q) (r p)) r Solution. This is Modus Tollens applied twice, with transitivity
More information(Refer Slide Time: 02:20)
Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 5 Logical Inference In the last class we saw about
More informationSection Summary. Section 1.5 9/9/2014
Section 1.5 Section Summary Nested Quantifiers Order of Quantifiers Translating from Nested Quantifiers into English Translating Mathematical Statements into Statements involving Nested Quantifiers Translated
More informationDERIVATIONS IN SYMBOLIC LOGIC I
DERIVATIONS IN SYMBOLIC LOGIC I Tomoya Sato Department of Philosophy University of California, San Diego Phil120: Symbolic Logic Summer 2014 TOMOYA SATO LECTURE 5: DERIVATIONS IN SYMBOLIC LOGIC I 1 / 40
More informationSection Summary. Predicates Variables Quantifiers. Negating Quantifiers. Translating English to Logic Logic Programming (optional)
Predicate Logic 1 Section Summary Predicates Variables Quantifiers Universal Quantifier Existential Quantifier Negating Quantifiers De Morgan s Laws for Quantifiers Translating English to Logic Logic Programming
More informationICS141: Discrete Mathematics for Computer Science I
ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross
More information2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic
CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares
More informationCS0441 Discrete Structures Recitation 3. Xiang Xiao
CS0441 Discrete Structures Recitation 3 Xiang Xiao Section 1.5 Q10 Let F(x, y) be the statement x can fool y, where the domain consists of all people in the world. Use quantifiers to express each of these
More informationTransparencies to accompany Rosen, Discrete Mathematics and Its Applications Section 1.3. Section 1.3 Predicates and Quantifiers
Section 1.3 Predicates and Quantifiers A generalization of propositions - propositional functions or predicates.: propositions which contain variables Predicates become propositions once every variable
More informationQuantifiers Here is a (true) statement about real numbers: Every real number is either rational or irrational.
Quantifiers 1-17-2008 Here is a (true) statement about real numbers: Every real number is either rational or irrational. I could try to translate the statement as follows: Let P = x is a real number Q
More informationLogic 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! 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 informationPREDICATE LOGIC. Schaum's outline chapter 4 Rosen chapter 1. September 11, ioc.pdf
PREDICATE LOGIC Schaum's outline chapter 4 Rosen chapter 1 September 11, 2018 margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, 2018 1 / 25 Contents 1 Predicates and quantiers 2 Logical equivalences
More informationPredicate Logic Thursday, January 17, 2013 Chittu Tripathy Lecture 04
Predicate Logic Today s Menu Predicate Logic Quantifiers: Universal and Existential Nesting of Quantifiers Applications Limitations of Propositional Logic Suppose we have: All human beings are mortal.
More informationDiscrete Mathematics and Its Applications
Discrete Mathematics and Its Applications Lecture 1: The Foundations: Logic and Proofs (1.3-1.5) MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 19, 2017 Outline 1 Logical
More informationLogic. Lesson 3: Tables and Models. Flavio Zelazek. October 22, 2014
Logic Lesson 3: Tables and Models Flavio Zelazek flavio.zelazek@gmail.com October 22, 2014 Flavio Zelazek (flavio.zelazek@gmail.com) Logic (Lesson 3: Tables and Models) October 22, 2014 1 / 29 Previous
More informationDiscrete 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 informationSection 1.7 Proof Methods and Strategy. Existence Proofs. xp(x). Constructive existence proof:
Section 1.7 Proof Methods and Strategy Existence Proofs We wish to establish the truth of xp(x). Constructive existence proof: - Establish P(c) is true for some c in the universe. - Then xp(x) is true
More informationDiscrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques
Discrete Mathematics & Mathematical Reasoning Predicates, Quantifiers and Proof Techniques Colin Stirling Informatics Some slides based on ones by Myrto Arapinis Colin Stirling (Informatics) Discrete Mathematics
More informationMat 243 Exam 1 Review
OBJECTIVES (Review problems: on next page) 1.1 Distinguish between propositions and non-propositions. Know the truth tables (i.e., the definitions) of the logical operators,,,, and Write truth tables for
More informationDiscrete Mathematics Basic Proof Methods
Discrete Mathematics 1-2. Basic Proof Methods Nature & Importance of Proofs In mathematics, a proof is: a correct(well-reasoned, logically valid) and complete (clear, detailed) argument that rigorously
More informationLecture 15: Validity and Predicate Logic
Lecture 15: Validity and Predicate Logic 1 Goals Today Learn the definition of valid and invalid arguments in terms of the semantics of predicate logic, and look at several examples. Learn how to get equivalents
More information1.3 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 informationINSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW PREDICATE LOGIC
1 CHAPTER 7. PREDICATE LOGIC 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW PREDICATE LOGIC 1 Predicate Logic 1.1 Introduction There are various arguments which cannot be dealt
More informationCS 214 Introduction to Discrete Structures. Chapter 1 Formal Logic. Mikel D. Petty, Ph.D.
CS 214 Introduction to Discrete Structures Chapter 1 Formal Logic Mikel D. Petty, Ph.D. Center for Modeling, Simulation, and Analysis CS 214 Formal Logic 1.2 Chapter sections and objectives 1.1 Statements,
More informationCSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI-2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 February 5, 2015 2 Announcements Homework 1 is due now. Homework 2 will be posted on the web site today. It is due Thursday, Feb. 12 at 10am in class.
More informationIntroduction to Predicate Logic Part 1. Professor Anita Wasilewska Lecture Notes (1)
Introduction to Predicate Logic Part 1 Professor Anita Wasilewska Lecture Notes (1) Introduction Lecture Notes (1) and (2) provide an OVERVIEW of a standard intuitive formalization and introduction to
More informationPredicate Logic: Sematics Part 1
Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with
More informationAll psychiatrists are doctors All doctors are college graduates All psychiatrists are college graduates
Predicate Logic In what we ve discussed thus far, we haven t addressed other kinds of valid inferences: those involving quantification and predication. For example: All philosophers are wise Socrates is
More information1 Predicates and Quantifiers
1 Predicates and Quantifiers We have seen how to represent properties of objects. For example, B(x) may represent that x is a student at Bryn Mawr College. Here B stands for is a student at Bryn Mawr College
More informationReview. Propositional Logic. Propositions atomic and compound. Operators: negation, and, or, xor, implies, biconditional.
Review Propositional Logic Propositions atomic and compound Operators: negation, and, or, xor, implies, biconditional Truth tables A closer look at implies Translating from/ to English Converse, inverse,
More informationDiscrete Mathematics Logics and Proofs. Liangfeng Zhang School of Information Science and Technology ShanghaiTech University
Discrete Mathematics Logics and Proofs Liangfeng Zhang School of Information Science and Technology ShanghaiTech University Resolution Theorem: p q p r (q r) p q p r q r p q r p q p p r q r T T T T F T
More information2-4: The Use of Quantifiers
2-4: The Use of Quantifiers The number x + 2 is an even integer is not a statement. When x is replaced by 1, 3 or 5 the resulting statement is false. However, when x is replaced by 2, 4 or 6 the resulting
More informationCSCE 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 informationTHE 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 informationA 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 informationPredicate Calculus. Lila Kari. University of Waterloo. Predicate Calculus CS245, Logic and Computation 1 / 59
Predicate Calculus Lila Kari University of Waterloo Predicate Calculus CS245, Logic and Computation 1 / 59 Predicate Calculus Alternative names: predicate logic, first order logic, elementary logic, restricted
More informationPredicate Logic: Introduction and Translations
Predicate Logic: Introduction and Translations Alice Gao Lecture 10 Based on work by J. Buss, L. Kari, A. Lubiw, B. Bonakdarpour, D. Maftuleac, C. Roberts, R. Trefler, and P. Van Beek 1/29 Outline Predicate
More informationMathematical Logic. Reasoning in First Order Logic. Chiara Ghidini. FBK-IRST, Trento, Italy
Reasoning in First Order Logic FBK-IRST, Trento, Italy April 12, 2013 Reasoning tasks in FOL Model checking Question: Is φ true in the interpretation I with the assignment a? Answer: Yes if I = φ[a]. No
More informationClass 29 - November 3 Semantics for Predicate Logic
Philosophy 240: Symbolic Logic Fall 2010 Mondays, Wednesdays, Fridays: 9am - 9:50am Hamilton College Russell Marcus rmarcus1@hamilton.edu Class 29 - November 3 Semantics for Predicate Logic I. Proof Theory
More informationFull file at
Transparencies to accompany Rosen, Discrete Mathematics and Its Applications Introduction Chapter 1 - Introduction Applications of discrete mathematics: Formal Languages (computer languages) Compiler Design
More informationUNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) I Semester Core Course. FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc (MATHEMATICS) (2011 Admission Onwards) I Semester Core Course FOUNDATIONS OF MATHEMATICS (MODULE I & ii) QUESTION BANK 1) If A and B are two sets
More informationNatural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms)
More informationIntroduction. Predicates and Quantifiers. Discrete Mathematics Andrei Bulatov
Introduction Predicates and Quantifiers Discrete Mathematics Andrei Bulatov Discrete Mathematics Predicates and Quantifiers 7-2 What Propositional Logic Cannot Do We saw that some declarative sentences
More informationProof. 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 informationLogic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
More informationDeduction by Daniel Bonevac. Chapter 8 Identity and Functions
Deduction by Daniel Bonevac Chapter 8 Identity and Functions Introduction 1 This chapter introduces two important extensions of Q that make it quite a bit more powerful. The first is the mathematical relation
More informationPredicate Calculus - Syntax
Predicate Calculus - Syntax Lila Kari University of Waterloo Predicate Calculus - Syntax CS245, Logic and Computation 1 / 26 The language L pred of Predicate Calculus - Syntax L pred, the formal language
More informationQuantifiers. P. Danziger
- 2 Quantifiers P. Danziger 1 Elementary Quantifiers (2.1) We wish to be able to use variables, such as x or n in logical statements. We do this by using the two quantifiers: 1. - There Exists 2. - For
More informationAnnouncements CompSci 102 Discrete Math for Computer Science
Announcements CompSci 102 Discrete Math for Computer Science Read for next time Chap. 1.4-1.6 Recitation 1 is tomorrow Homework will be posted by Friday January 19, 2012 Today more logic Prof. Rodger Most
More informationSolutions to Exercises (Sections )
s to Exercises (Sections 1.1-1.10) Section 1.1 Exercise 1.1.1: Identifying propositions (a) Have a nice day. : Command, not a proposition. (b) The soup is cold. : Proposition. Negation: The soup is not
More informationFirst-Order Logic (FOL)
First-Order Logic (FOL) Also called Predicate Logic or Predicate Calculus 2. First-Order Logic (FOL) FOL Syntax variables x, y, z, constants a, b, c, functions f, g, h, terms variables, constants or n-ary
More informationFirst 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 informationManual of Logical Style
Manual of Logical Style Dr. Holmes January 9, 2015 Contents 1 Introduction 2 2 Conjunction 3 2.1 Proving a conjunction...................... 3 2.2 Using a conjunction........................ 3 3 Implication
More informationA 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 informationLogic and Proofs. (A brief summary)
Logic and Proofs (A brief summary) Why Study Logic: To learn to prove claims/statements rigorously To be able to judge better the soundness and consistency of (others ) arguments To gain the foundations
More informationSymbolic Logic 3. For an inference to be deductively valid it is impossible for the conclusion to be false if the premises are true.
Symbolic Logic 3 Testing deductive validity with truth tables For an inference to be deductively valid it is impossible for the conclusion to be false if the premises are true. So, given that truth tables
More informationPacket #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics
CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus
More informationIt rains now. (true) The followings are not propositions.
Chapter 8 Fuzzy Logic Formal language is a language in which the syntax is precisely given and thus is different from informal language like English and French. The study of the formal languages is the
More informationPHIL 50 INTRODUCTION TO LOGIC 1 FREE AND BOUND VARIABLES MARCELLO DI BELLO STANFORD UNIVERSITY DERIVATIONS IN PREDICATE LOGIC WEEK #8
PHIL 50 INTRODUCTION TO LOGIC MARCELLO DI BELLO STANFORD UNIVERSITY DERIVATIONS IN PREDICATE LOGIC WEEK #8 1 FREE AND BOUND VARIABLES Before discussing the derivation rules for predicate logic, we should
More informationToday s Lecture. ICS 6B Boolean Algebra & Logic. Predicates. Chapter 1: Section 1.3. Propositions. For Example. Socrates is Mortal
ICS 6B Boolean Algebra & Logic Today s Lecture Chapter 1 Sections 1.3 & 1.4 Predicates & Quantifiers 1.3 Nested Quantifiers 1.4 Lecture Notes for Summer Quarter, 2008 Michele Rousseau Set 2 Ch. 1.3, 1.4
More informationUniversity of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura. March 1, :00 pm Duration: 1:15 hs
University of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura March 1, 2012 1:00 pm Duration: 1:15 hs Closed book, no calculators THIS MIDTERM AND ITS SOLUTION IS SUBJECT TO COPYRIGHT; NO PARTS OF
More informationChapter 1 Elementary Logic
2017-2018 Chapter 1 Elementary Logic The study of logic is the study of the principles and methods used in distinguishing valid arguments from those that are not valid. The aim of this chapter is to help
More informationSec$on Summary. Valid Arguments Inference Rules for Propositional Logic. Inference Rules for Quantified Statements. Building Arguments
Section 1.6 Sec$on Summary Valid Arguments Inference Rules for Propositional Logic Building Arguments Inference Rules for Quantified Statements Building Arguments 2 Revisi$ng the Socrates Example We have
More informationSection 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 informationi.e. The conclusion to the following argument says If you had an A, then you d have a ~(B v Z).
7.5 Conditional Proof (CP): Conditional Proof is a different way to do proofs. Using CP will always get you a horseshoe statement, so the best time to use it is when your conclusion is either a horseshoe
More informationG52DOA - Derivation of Algorithms Predicate Logic
G52DOA - Derivation of Algorithms Predicate Logic Venanzio Capretta Predicate Logic So far, we studied propositional logic, in which we started with unspecified propositional variables A, B, C, and combined
More informationCSE Discrete Structures
CSE 2315 - Discrete Structures Homework 1- Fall 2010 Due Date: Sept. 16 2010, 3:30 pm Statements, Truth Values, and Tautologies 1. Which of the following are statements? a) 3 + 7 = 10 b) All cars are blue.
More informationMAT2345 Discrete Math
Fall 2013 General Syllabus Schedule (note exam dates) Homework, Worksheets, Quizzes, and possibly Programs & Reports Academic Integrity Do Your Own Work Course Web Site: www.eiu.edu/~mathcs Course Overview
More informationCS 220: Discrete Structures and their Applications. Predicate Logic Section in zybooks
CS 220: Discrete Structures and their Applications Predicate Logic Section 1.6-1.10 in zybooks From propositional to predicate logic Let s consider the statement x is an odd number Its truth value depends
More informationMAT 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