Homework. Turn in Homework #4. The fifth homework assignment has been posted on the course website. It is due on Monday, March 2.

Size: px
Start display at page:

Download "Homework. Turn in Homework #4. The fifth homework assignment has been posted on the course website. It is due on Monday, March 2."

Transcription

1 Homework Turn in Homework #4. The fifth homework assignment has been posted on the course website. It is due on Monday, March 2.

2 First-Order Logic Formation Rules and Translations

3 First Order Logic

4 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,...

5 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,...

6 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Constants

7 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Constants

8 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Constants Variables

9 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,...

10 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Predicates: G, H, L,...

11 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Predicates: G, H, L,... Quantifiers: and

12 Well-Formed Formulas We add two formation rules to the rules we already had for zeroth-order logic. As before, the grammatically correct strings are called well-formed formulas.

13 Well-Formed Formulas New Formation Rules: 3. If Φ is a predicate and α 1, α 2,, α n are terms, then Φα 1 α 2 α n is a well-formed formula. 4. Let ϕ be a well-formed formula, and let β be a variable. Then ( β)ϕ and ( β)ϕ are both well-formed formulas.

14 Well-Formed Formulas Positive Examples: Ga (Ga Hbc) (~Lcxy ᴧ Gb) ( x)gx ( x)( y)hxy (( x)gx P)

15 Well-Formed Formulas Positive Examples: Ga (Ga Hbc) (~Lcxy ᴧ Gb) ( x)gx ( x)( y)hxy (( x)gx P) Negative Examples: GaH alx ( a) (( x) ᴧ Gx) ( B)Bc ( P)(P Ga)

16 Translations Predicates let us represent sentences like: Kermit is green. Kermit is a Muppet. Kermit plays the banjo.

17 First-Order Logic If P is a predicate symbol and a is a term, then Pa is a well-formed formula. The basic idea is to think of a simple sentence as composed of a logical subject a constant term, like Kermit and a logical predicate like is green.

18 First-Order Logic Predicates may be interpreted in many different ways. A predicate is a kind of truth function. assigns a property to an individual. indicates membership in a kind.

19 First-Order Logic Kermit is green. Constant Predicate

20 First-Order Logic Kermit is green. TRUE Constant Predicate

21 First-Order Logic Gonzo is green. Constant Predicate

22 First-Order Logic Gonzo is green. FALSE Constant Predicate

23 Quantifiers Add quantifiers and we can also represent sentences like: All Muppets are silly. Some Muppets wear hats.

24 Quantifiers Suppose ( β)ϕ and ( β)ϕ are well-formed formulas. Then we know that β is a variable term. Occurrences of β in ϕ are bound by the quantifiers. The variable β is also bound in the quantifier expressions ( β) and ( β).

25 Quantifiers If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence.

26 Quantifiers If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence. P (Q v R) Mxy Ga ( x)gx ( x)(mxy ᴧ Gy)

27 Quantifiers If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence. P (Q v R) Mxy Ga ( x)gx ( x)(mxy ᴧ Gy) Which of these are sentences?

28 Quantifiers Previously, we had no need to distinguish between well-formed formulas and sentences because every well-formed formula stood for a sentence in ordinary language. But a well-formed formula that contains a free variable does not stand for any sentence in ordinary language.

29 Quantifiers What do the quantifiers do though? Quantifiers let us talk about all of the constant terms without needing to actually list them all. That is convenient, since there are infinitely many constant terms.

30 Quantifiers What do the quantifiers do though? Quantifiers let us talk about all of the constant terms without needing to actually list them all. That is convenient, since there are infinitely many constant terms. We would get tired trying to write them all down!

31 Quantifiers Quantified formulas tell us something about all of the constant terms. The formula ( β)ϕ says that for every constant c, if you substitute c for β everywhere β appears in ϕ, then ϕ will come out true.

32 Quantifiers Let G be the predicate is green. Then the formula ( x)gx says that everything is green. The formula ( x)gx says that for every constant c, if you substitute c for x in Gx, then Gx will come out true.

33 Quantifiers Let G be the predicate is green. Then the formula ( x)gx says that everything is green. In other words, ( x)gx says that for every constant c, Gc is true. The universal quantifier works like an infinite conjunction.

34 Quantifiers Again, quantified formulas tell us something about all of the constant terms. The formula ( β)ϕ says that there is at least one constant c such that if you substitute c for β everywhere it appears in ϕ, then ϕ will be true.

35 Quantifiers Let H be the predicate is happy. Then the formula ( x)hx says that something is happy. The formula ( x)hx says that there is a constant c such that Hc true.

36 Quantifiers Let H be the predicate is happy. Then the formula ( x)hx says that something is happy. The formula ( x)hx says that there is a constant c such that Hc true. The existential quantifier works like an infinite disjunction.

37 Categoricals We ve seen that if G is the predicate is green, then the formula ( x)gx says that everything is green. What if we want to say that every green thing is colorful?

38 Categoricals We ve seen that if G is the predicate is green, then the formula ( x)gx says that everything is green. What if we want to say that every green thing is colorful? That would be a categorical sentence!

39 Categoricals There are four kinds of categorical sentence: All Ss are Ps. Some Ss are Ps. No Ss are Ps. Some Ss are not Ps. All Muppets are silly. Some Muppets wear hats. No Muppets wear hats. Some Muppets are not silly.

40 Categoricals There are four kinds of categorical sentence: All Ss are Ps. Some Ss are Ps. No Ss are Ps. Some Ss are not Ps. All Muppets are silly. Some Muppets wear hats. No Muppets wear hats. Some Muppets are not silly.

41 Categoricals There are four kinds of categorical sentence: All Ss are Ps. Some Ss are Ps. No Ss are Ps. Some Ss are not Ps. All Muppets are silly. Some Muppets wear hats. No Muppets wear hats. Some Muppets are not silly.

42 Categoricals Let s translate the categorical sentence, All Muppets are silly. Start with the predicates: M = is a Muppet S = is silly

43 Categoricals Suppose b stands for Beaker.

44 Categoricals Suppose b stands for Beaker.

45 Categoricals Suppose b stands for Beaker. Then

46 Categoricals Suppose b stands for Beaker. Then Mb = Beaker is a Muppet. Sb = Beaker is silly.

47 Categoricals Suppose b stands for Beaker. Then Mb = Beaker is a Muppet. Sb = Beaker is silly. (Mb ᴧ Sb) = Beaker is a Muppet and Beaker is silly.

48 Categoricals But we want to say that every Muppet is silly.

49 Categoricals But we want to say that every Muppet is silly.

50 Categoricals But we want to say that every Muppet is silly.

51 Categoricals But we want to say that every Muppet is silly.

52 Categoricals But we want to say that every Muppet is silly.

53 Categoricals But we want to say that every Muppet is silly. How do we make sure to cover all of the Muppets?

54 Categoricals We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb).

55 Categoricals We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb). (Mh ᴧ Sh)

56 Categoricals We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb). (Mh ᴧ Sh) (Mf ᴧ Sf)

57 Categoricals We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb). (Mh ᴧ Sh) (Mf ᴧ Sf) (Mk ᴧ Sk)

58 Categoricals Could we use the formula ( x)(mx ᴧ Sx) to translate the categorical All Muppets are silly?

59 Categoricals Could we use the formula ( x)(mx ᴧ Sx) to translate the categorical All Muppets are silly? NO

60 Categoricals Could we use the formula ( x)(mx ᴧ Sx) to translate the categorical All Muppets are silly? NO Why not?

61 Categoricals The formula ( x)(mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists.

62 Categoricals The formula ( x)(mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists. But lots of things aren t Muppets.

63 Categoricals The formula ( x)(mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists. And some things aren t silly either.

64 Categoricals The formula ( x)(mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists. And some things aren t silly either.

65 Categoricals So what do we mean when we say that all Muppets are silly?

66 Categoricals So what do we mean when we say that all Muppets are silly? For any object you want to pick, if the object is a Muppet, then it is silly.

67 Categoricals All Muppets are silly. ( x)(mx Sx)

68 Categoricals All Muppets are silly. ( x)(mx Sx) For any x you want

69 Categoricals All Muppets are silly. ( x)(mx Sx) For any x you want x is a Muppet

70 Categoricals All Muppets are silly. ( x)(mx Sx) For any x you want x is a Muppet only if

71 Categoricals All Muppets are silly. ( x)(mx Sx) x is silly For any x you want x is a Muppet only if

72 Categoricals All Muppets are silly. ( x)(mx Sx) x is silly For any x you want x is a Muppet only if

73 Categoricals All Muppets are silly. (Ma Sa) ᴧ (Mb Sb) ᴧ (Mc Sc) ᴧ : One way to think of a universal quantifier is as a big conjunction.

74 Next Time We will talk about the class of particular categorical sentences, and then we will talk about relations.

First-Order Natural Deduction. Part 1: Universal Introduction and Elimination

First-Order Natural Deduction. Part 1: Universal Introduction and Elimination First-Order Natural Deduction Part 1: Universal Introduction and Elimination Happy Birthday to Talan Two years old! Business Homework #7 will be posted later today. It is due on Monday, October 21. The

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

Topic DPL: Answers. Exercise 1.1a Explain why the rule &E for MPL is a sound rule.

Topic DPL: Answers. Exercise 1.1a Explain why the rule &E for MPL is a sound rule. Topic DPL: Answers Exercise 1.1a Explain why the rule &E for MPL is a sound rule. In MPL, if (φ&ψ) is true under some interpretation then φ and ψ are true under that interpretation too. Thus, if (φ&ψ)

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

SYMBOLIC LOGIC UNIT 10: SINGULAR SENTENCES

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 information

Mathematical Preliminaries. Sipser pages 1-28

Mathematical Preliminaries. Sipser pages 1-28 Mathematical Preliminaries Sipser pages 1-28 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation

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

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

Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)

Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010) http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product

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

Topic 1: Propositional logic

Topic 1: Propositional logic Topic 1: Propositional logic Guy McCusker 1 1 University of Bath Logic! This lecture is about the simplest kind of mathematical logic: propositional calculus. We discuss propositions, which are statements

More information

2. Use quantifiers to express the associative law for multiplication of real numbers.

2. 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 information

Section 1.1: Logical Form and Logical Equivalence

Section 1.1: Logical Form and Logical Equivalence Section 1.1: Logical Form and Logical Equivalence An argument is a sequence of statements aimed at demonstrating the truth of an assertion. The assertion at the end of an argument is called the conclusion,

More information

3. The Logic of Quantified Statements Summary. Aaron Tan August 2017

3. 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 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

Gödel s Incompleteness Theorems

Gödel s Incompleteness Theorems Seminar Report Gödel s Incompleteness Theorems Ahmet Aspir Mark Nardi 28.02.2018 Supervisor: Dr. Georg Moser Abstract Gödel s incompleteness theorems are very fundamental for mathematics and computational

More information

CS 2800: Logic and Computation Fall 2010 (Lecture 13)

CS 2800: Logic and Computation Fall 2010 (Lecture 13) CS 2800: Logic and Computation Fall 2010 (Lecture 13) 13 October 2010 1 An Introduction to First-order Logic In Propositional(Boolean) Logic, we used large portions of mathematical language, namely those

More information

MATH 22 INFERENCE & QUANTIFICATION. Lecture F: 9/18/2003

MATH 22 INFERENCE & QUANTIFICATION. Lecture F: 9/18/2003 MATH 22 Lecture F: 9/18/2003 INFERENCE & QUANTIFICATION Sixty men can do a piece of work sixty times as quickly as one man. One man can dig a post-hole in sixty seconds. Therefore, sixty men can dig a

More information

Proseminar on Semantic Theory Fall 2013 Ling 720 First Order (Predicate) Logic: Syntax and Natural Deduction 1

Proseminar on Semantic Theory Fall 2013 Ling 720 First Order (Predicate) Logic: Syntax and Natural Deduction 1 First Order (Predicate) Logic: Syntax and Natural Deduction 1 A Reminder of Our Plot I wish to provide some historical and intellectual context to the formal tools that logicians developed to study the

More information

Quantifiers Here is a (true) statement about real numbers: Every real number is either rational or irrational.

Quantifiers 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 information

In this episode of The Verification Corner, Rustan Leino talks about Loop Invariants. He gives a brief summary of the theoretical foundations and

In this episode of The Verification Corner, Rustan Leino talks about Loop Invariants. He gives a brief summary of the theoretical foundations and In this episode of The Verification Corner, Rustan Leino talks about Loop Invariants. He gives a brief summary of the theoretical foundations and shows how a program can sometimes be systematically constructed

More information

Section 2.3: Statements Containing Multiple Quantifiers

Section 2.3: Statements Containing Multiple Quantifiers Section 2.3: Statements Containing Multiple Quantifiers In this section, we consider statements such as there is a person in this company who is in charge of all the paperwork where more than one quantifier

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

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

CMPSCI 601: Tarski s Truth Definition Lecture 15. where

CMPSCI 601: Tarski s Truth Definition Lecture 15. where @ CMPSCI 601: Tarski s Truth Definition Lecture 15! "$#&%(') *+,-!".#/%0'!12 43 5 6 7 8:9 4; 9 9 < = 9 = or 5 6?>A@B!9 2 D for all C @B 9 CFE where ) CGE @B-HI LJKK MKK )HG if H ; C if H @ 1 > > > Fitch

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

a. (6.25 points) where we let b. (6.25 points) c. (6.25 points) d. (6.25 points) B 3 =(x)[ H(x) F (x)], and (11)

a. (6.25 points) where we let b. (6.25 points) c. (6.25 points) d. (6.25 points) B 3 =(x)[ H(x) F (x)], and (11) A1 Logic (25 points) For each of the following either prove it step-by-step using resolution or another sound and complete technique of your stated choice or disprove it by exhibiting in detail a relevant

More information

Denote 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).

Denote 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 information

Logic and Mathematics:

Logic and Mathematics: Logic and Mathematics: Mathematicians in Schools Program Lashi Bandara Mathematical Sciences Institute, Australian National University April 21, 2011 Contents 1 Russell s Paradox 1 2 Propositional Logic

More information

Announcements CompSci 102 Discrete Math for Computer Science

Announcements 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 information

THE LOGIC OF QUANTIFIED STATEMENTS

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

More information

Interpretations of PL (Model Theory)

Interpretations of PL (Model Theory) Interpretations of PL (Model Theory) 1. Once again, observe that I ve presented topics in a slightly different order from how I presented them in sentential logic. With sentential logic I discussed syntax

More information

Predicate Logic. Predicates. Math 173 February 9, 2010

Predicate Logic. Predicates. Math 173 February 9, 2010 Math 173 February 9, 2010 Predicate Logic We have now seen two ways to translate English sentences into mathematical symbols. We can capture the logical form of a sentence using propositional logic: variables

More information

Peano Arithmetic. by replacing the schematic letter R with a formula, then prefixing universal quantifiers to bind

Peano Arithmetic. by replacing the schematic letter R with a formula, then prefixing universal quantifiers to bind Peano Arithmetic Peano Arithmetic 1 or PA is the system we get from Robinson s Arithmetic by adding the induction axiom schema: ((R(0) v (œx)(r(x) 6 R(sx))) 6 (œx)r(x)). What this means is that any sentence

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 1

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 1 EECS 70 Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 1 Getting Started In order to be fluent in mathematical statements, you need to understand the basic framework of the language

More information

Before you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here.

Before you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here. Chapter 2 Mathematics and Logic Before you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here. 2.1 A Taste of Number Theory In this section, we will

More information

Math 10850, fall 2017, University of Notre Dame

Math 10850, fall 2017, University of Notre Dame Math 10850, fall 2017, University of Notre Dame Notes on first exam September 22, 2017 The key facts The first midterm will be on Thursday, September 28, 6.15pm-7.45pm in Hayes-Healy 127. What you need

More information

Logics - Introduction

Logics - Introduction Logics 1 Logics - Introduction So far we have seen a variety of operational formalisms based on some some notion of state and event/transition model the evolution of the system Now instead we will analyze

More information

Logic, Sets, and Proofs

Logic, Sets, and Proofs Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.

More information

Mathematical Logic Part One

Mathematical Logic Part One Mathematical Logic Part One Question: How do we formalize the definitions and reasoning we use in our proofs? Where We're Going Propositional Logic (oday) Basic logical connectives. ruth tables. Logical

More information

1 Predicates and Quantifiers

1 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 information

Math 31 Lesson Plan. Day 2: Sets; Binary Operations. Elizabeth Gillaspy. September 23, 2011

Math 31 Lesson Plan. Day 2: Sets; Binary Operations. Elizabeth Gillaspy. September 23, 2011 Math 31 Lesson Plan Day 2: Sets; Binary Operations Elizabeth Gillaspy September 23, 2011 Supplies needed: 30 worksheets. Scratch paper? Sign in sheet Goals for myself: Tell them what you re going to tell

More information

Recall that the expression x > 3 is not a proposition. Why?

Recall that the expression x > 3 is not a proposition. Why? Predicates and Quantifiers Predicates and Quantifiers 1 Recall that the expression x > 3 is not a proposition. Why? Notation: We will use the propositional function notation to denote the expression "

More information

Choosing Logical Connectives

Choosing Logical Connectives Choosing Logical Connectives 1. Too Few Connectives?: We have chosen to use only 5 logical connectives in our constructed language of logic, L1 (they are:,,,, and ). But, we might ask, are these enough?

More information

Intermediate Logic. First-Order Logic

Intermediate Logic. First-Order Logic Intermediate Logic Lecture Four First-Order Logic Rob Trueman rob.trueman@york.ac.uk University of York Introducing First-Order Logic First-Order Logic Introducing First-Order Logic Names Predicates Quantifiers

More information

Chapter 3. The Logic of Quantified Statements

Chapter 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 information

CM10196 Topic 2: Sets, Predicates, Boolean algebras

CM10196 Topic 2: Sets, Predicates, Boolean algebras CM10196 Topic 2: Sets, Predicates, oolean algebras Guy McCusker 1W2.1 Sets Most of the things mathematicians talk about are built out of sets. The idea of a set is a simple one: a set is just a collection

More information

Examples: P: it is not the case that P. P Q: P or Q P Q: P implies Q (if P then Q) Typical formula:

Examples: P: it is not the case that P. P Q: P or Q P Q: P implies Q (if P then Q) Typical formula: Logic: The Big Picture Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and

More information

Today s Lecture. ICS 6B Boolean Algebra & Logic. Predicates. Chapter 1: Section 1.3. Propositions. For Example. Socrates is Mortal

Today 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 information

Section Summary. Section 1.5 9/9/2014

Section 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 information

It rains now. (true) The followings are not propositions.

It 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 information

Deduction by Daniel Bonevac. Chapter 3 Truth Trees

Deduction by Daniel Bonevac. Chapter 3 Truth Trees Deduction by Daniel Bonevac Chapter 3 Truth Trees Truth trees Truth trees provide an alternate decision procedure for assessing validity, logical equivalence, satisfiability and other logical properties

More information

PHIL12A Section answers, 16 February 2011

PHIL12A Section answers, 16 February 2011 PHIL12A Section answers, 16 February 2011 Julian Jonker 1 How much do you know? 1. Show that the following sentences are equivalent. (a) (Ex 4.16) A B A and A B A B (A B) A A B T T T T T T T T T T T F

More information

MAT137 - Term 2, Week 4

MAT137 - Term 2, Week 4 MAT137 - Term 2, Week 4 Reminders: Your Problem Set 6 is due tomorrow at 3pm. Test 3 is next Friday, February 3, at 4pm. See the course website for details. Today we will: Talk more about substitution.

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: 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 information

Chapter 1 Review of Equations and Inequalities

Chapter 1 Review of Equations and Inequalities Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve

More information

First-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms

First-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO

More information

Mathematical Logic Part Two

Mathematical Logic Part Two Mathematical Logic Part Two Outline for Today Recap from Last Time The Contrapositive Using Propositional Logic First-Order Logic First-Order Translations Recap from Last Time Recap So Far A propositional

More information

Logic. Propositional Logic: Syntax. Wffs

Logic. Propositional Logic: Syntax. Wffs 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

1.1 Language and Logic

1.1 Language and Logic c Oksana Shatalov, Fall 2017 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,

More information

Question 1 (a). Explain different logical connectives with the help of examples. Ans: Logical Connectivity:-

Question 1 (a). Explain different logical connectives with the help of examples. Ans: Logical Connectivity:- Course Code : MCS-013 Course Title : Discrete Mathematics Assignment Number : MCA(1)/013/Assignment/17-18 Maximum Marks : 100 Weightage : 25% Last Dates for Submission : 15th October, 2017 (For July 2017

More information

Discrete Mathematics and Probability Theory Fall 2012 Vazirani Note 1

Discrete Mathematics and Probability Theory Fall 2012 Vazirani Note 1 CS 70 Discrete Mathematics and Probability Theory Fall 2012 Vazirani Note 1 Course Outline CS70 is a course on "Discrete Mathematics and Probability for Computer Scientists." The purpose of the course

More information

Proving logical equivalencies (1.3)

Proving logical equivalencies (1.3) EECS 203 Spring 2016 Lecture 2 Page 1 of 6 Proving logical equivalencies (1.3) One thing we d like to do is prove that two logical statements are the same, or prove that they aren t. Vocabulary time In

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

Introducing Proof 1. hsn.uk.net. Contents

Introducing Proof 1. hsn.uk.net. Contents Contents 1 1 Introduction 1 What is proof? 1 Statements, Definitions and Euler Diagrams 1 Statements 1 Definitions Our first proof Euler diagrams 4 3 Logical Connectives 5 Negation 6 Conjunction 7 Disjunction

More information

22c:145 Artificial Intelligence. First-Order Logic. Readings: Chapter 8 of Russell & Norvig.

22c:145 Artificial Intelligence. First-Order Logic. Readings: Chapter 8 of Russell & Norvig. 22c:145 Artificial Intelligence First-Order Logic Readings: Chapter 8 of Russell & Norvig. Einstein s Puzzle in Logic We used propositinal variables to specify everything: x 1 = house #1 is red ; x 2 =

More information

Final Exam Theory Quiz Answer Page

Final Exam Theory Quiz Answer Page Philosophy 120 Introduction to Logic Final Exam Theory Quiz Answer Page 1. (a) is a wff (and a sentence); its outer parentheses have been omitted, which is permissible. (b) is also a wff; the variable

More information

There are infinitely many set variables, X 0, X 1,..., each of which is

There are infinitely many set variables, X 0, X 1,..., each of which is 4. Second Order Arithmetic and Reverse Mathematics 4.1. The Language of Second Order Arithmetic. We ve mentioned that Peano arithmetic is sufficient to carry out large portions of ordinary mathematics,

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

CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC

CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called,

More information

Solutions to Sample Problems for Midterm

Solutions to Sample Problems for Midterm Solutions to Sample Problems for Midterm Problem 1. The dual of a proposition is defined for which contains only,,. It is For a compound proposition that only uses,, as operators, we obtained the dual

More information

All psychiatrists are doctors All doctors are college graduates All psychiatrists are college graduates

All 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 information

Philosophy 240 Symbolic Logic Russell Marcus Hamilton College Fall 2014

Philosophy 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 information

Line Integrals and Path Independence

Line Integrals and Path Independence Line Integrals and Path Independence We get to talk about integrals that are the areas under a line in three (or more) dimensional space. These are called, strangely enough, line integrals. Figure 11.1

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

AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic

AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic Propositional logic Logical connectives Rules for wffs Truth tables for the connectives Using Truth Tables to evaluate

More information

Essential facts about NP-completeness:

Essential facts about NP-completeness: CMPSCI611: NP Completeness Lecture 17 Essential facts about NP-completeness: Any NP-complete problem can be solved by a simple, but exponentially slow algorithm. We don t have polynomial-time solutions

More information

Completeness in the Monadic Predicate Calculus. We have a system of eight rules of proof. Let's list them:

Completeness in the Monadic Predicate Calculus. We have a system of eight rules of proof. Let's list them: Completeness in the Monadic Predicate Calculus We have a system of eight rules of proof. Let's list them: PI At any stage of a derivation, you may write down a sentence φ with {φ} as its premiss set. TC

More information

Mathematical Logic Part Three

Mathematical Logic Part Three Mathematical Logic Part Three Recap from Last Time What is First-Order Logic? First-order logic is a logical system for reasoning about properties of objects. Augments the logical connectives from propositional

More information

Introduction to Predicate Logic Part 1. Professor Anita Wasilewska Lecture Notes (1)

Introduction 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 information

INTRODUCTION TO LOGIC. Propositional Logic. Examples of syntactic claims

INTRODUCTION TO LOGIC. Propositional Logic. Examples of syntactic claims Introduction INTRODUCTION TO LOGIC 2 Syntax and Semantics of Propositional Logic Volker Halbach In what follows I look at some formal languages that are much simpler than English and define validity of

More information

Quantification in the predicate calculus

Quantification in the predicate calculus Quantification in the predicate calculus PHIL 43916 eptember 5, 2012 1. The problem posed by quantified sentences... 1 2. yntax of PC... 2 3. Bound and free iables... 3 4. Models and assignments... 4 5.

More information

Formal (Natural) Deduction for Predicate Calculus

Formal (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 information

Mathematical Logic Part Three

Mathematical Logic Part Three Mathematical Logic Part Three The Aristotelian Forms All As are Bs x. (A(x B(x Some As are Bs x. (A(x B(x No As are Bs x. (A(x B(x Some As aren t Bs x. (A(x B(x It It is is worth worth committing committing

More information

Comp487/587 - Boolean Formulas

Comp487/587 - Boolean Formulas Comp487/587 - Boolean Formulas 1 Logic and SAT 1.1 What is a Boolean Formula Logic is a way through which we can analyze and reason about simple or complicated events. In particular, we are interested

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

Lecture 3 : Predicates and Sets DRAFT

Lecture 3 : Predicates and Sets DRAFT CS/Math 240: Introduction to Discrete Mathematics 1/25/2010 Lecture 3 : Predicates and Sets Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last time we discussed propositions, which are

More information

One-to-one functions and onto functions

One-to-one functions and onto functions MA 3362 Lecture 7 - One-to-one and Onto Wednesday, October 22, 2008. Objectives: Formalize definitions of one-to-one and onto One-to-one functions and onto functions At the level of set theory, there are

More information

1 FUNDAMENTALS OF LOGIC NO.10 HERBRAND THEOREM Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth table Tautology

More information

Introduction to first-order logic:

Introduction to first-order logic: Introduction to first-order logic: First-order structures and languages. Terms and formulae in first-order logic. Interpretations, truth, validity, and satisfaction. Valentin Goranko DTU Informatics September

More information

PROBLEM SET 3: PROOF TECHNIQUES

PROBLEM SET 3: PROOF TECHNIQUES PROBLEM SET 3: PROOF TECHNIQUES CS 198-087: INTRODUCTION TO MATHEMATICAL THINKING UC BERKELEY EECS FALL 2018 This homework is due on Monday, September 24th, at 6:30PM, on Gradescope. As usual, this homework

More information

Discrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland

Discrete Mathematics. W. Ethan Duckworth. Fall 2017, Loyola University Maryland Discrete Mathematics W. Ethan Duckworth Fall 2017, Loyola University Maryland Contents 1 Introduction 4 1.1 Statements......................................... 4 1.2 Constructing Direct Proofs................................

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

THE LOGIC OF COMPOUND STATEMENTS CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.1 Logical Form and Logical Equivalence Copyright Cengage Learning. All rights reserved. Logical Form

More information

Ling 130 Notes: Syntax and Semantics of Propositional Logic

Ling 130 Notes: Syntax and Semantics of Propositional Logic Ling 130 Notes: Syntax and Semantics of Propositional Logic Sophia A. Malamud January 21, 2011 1 Preliminaries. Goals: Motivate propositional logic syntax and inferencing. Feel comfortable manipulating

More information

Mat 243 Exam 1 Review

Mat 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 information

Predicate Logic: Sematics Part 1

Predicate 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 information

First Order Logic: Syntax and Semantics

First Order Logic: Syntax and Semantics irst Order Logic: Syntax and Semantics COMP30412 Sean Bechhofer sean.bechhofer@manchester.ac.uk Logic Recap You should already know the basics of irst Order Logic (OL) It s a prerequisite of this course!

More information

Theorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1)

Theorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1) Week 1: Logic Lecture 1, 8/1 (Sections 1.1 and 1.3) Examples of theorems and proofs Theorem (Pythagoras). Let ABC be a right triangle, with legs of lengths a and b, and hypotenuse of length c. Then a +

More information

13. APPENDIX 1: THE SYNTAX OF PREDICATE LOGIC

13. APPENDIX 1: THE SYNTAX OF PREDICATE LOGIC 394 Hardegree, Symbolic Logic 13. APPENDIX 1: THE SYNTAX OF PREDICATE LOGIC In this appendix, we review the syntactic features of predicate logic that are crucial to understanding derivations in predicate

More information