Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies
|
|
- Kelley Peters
- 6 years ago
- Views:
Transcription
1 Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies Valentin Stockholm University September 2016
2 Propositions Proposition: statement which can be assigned a (unique) truth value: true or false. Some examples of propositions: The Sun is hot. The Moon is made of cheese. Two plus two equals twenty two. The 1001st decimal digit of the number π is 7. Some non-examples: Are you bored? Please, don t go away! She loves me. x is an integer. This sentence is false.
3 Propositional logical connectives not, called negation, denoted by ; and, called conjunction, denoted by (or, sometimes by &); or, called disjunction, denoted by ; if... then..., called implication, or conditional, denoted by ;... if and only if..., called equivalence, or biconditional, denoted by.
4 Using the propositions Building composite propositions Two plus two equals five and The sun is hot and the propositional connectives, we can produce: It is not the case that two plus two equals five Two plus two equals five and the sun is hot Two plus two equals five or the sun is hot If two plus two equals five then the sun is hot Two plus two equals five if and only if the sun is hot Likewise, using the propositions Logic is fun, Logic is boring, Logic is easy we can produce the composite proposition: Logic is not fun or, if Logic is easy then Logic is fun and Logic is not boring.
5 The propositional connectives as truth value functions Each propositional connective acts on the truth values of the component propositions in a precise way: A is true if and only if A is false. A B is true if and only if both A and B are true. A B is true if and only if either of A or B (possibly both) is true. A B is true if and only if the truth of A implies the truth of B, i.e., iff A is false or B is true. A B is true if and only if A and B have the same truth-values.
6 Truth tables These rules can be summarized in the following truth tables, where T stands for true and stands for false : p p T F F T p q p q p q p q p q T T T T T T T F F T F F F T F T T F F F F F T T
7 Applying the truth tables Given that Two plus two equals five is false and The Sun is hot is true, using the truth-tables we obtain that: It is not the case that two plus two equals five is true; Two plus two equals five and the Sun is hot is false; Two plus two equals five or the Sun is hot is true; If two plus two equals five, then the Sun is hot is true. If two plus two equals five if and only if the Sun is hot is true.
8 Some comments on the truth tables Conjunction is commutative: A B is equally true as B A. But, these do not always mean the same! Compare Bill pulled the trigger and Jack dropped dead with Jack dropped dead and Bill pulled the trigger. Disjunction is non-exclusive: I shall win or I shall die is still true if I both win and die. Compare also with I shall win unless I die. Implication is the most confusing of all. Why should False imply true and False imply false be true? Because there is no good reason to declare them false! The implication A B claims the truth of B only if A is true, else no claim is made. Thus, the implication A B is false in only one case: when A is true and yet B is false.
9 Some equivalent forms of the implication The implication is the most important and versatile propositional connective. It can appear in variety of phrases: If A then B A implies B A only if B B whenever A B if A A is sufficient for B B is necessary for A
10 Suppose that Computing the truth value of a proposition Logic is fun. : T; Logic is boring. : F; Logic is easy. : T; To compute the truth value of the composite proposition: Logic is not fun or, if Logic is easy then Logic is fun and Logic is not boring. we first write it in a symbolic form. For that, we first analyse and clarify it syntax, by placing auxiliary parentheses to indicate the order of applying the logical connectives: (Logic is not fun) or (if Logic is easy then (Logic is fun and Logic is not boring)).
11 Computing the truth value of a proposition, cont d We now introduce symbolic names for the atomic propositions occurring in it, e.g: A : Logic is fun. B : Logic is boring. C : Logic is easy. Then, the proposition can be written symbolically as: ( A) (C (A B)) Now, we compute its truth value step by step, applying the truth-tables of the respective logical connectives: ( T) (T (T F)) = F (T (T T)) = F (T T) = F T = T.
12 Propositional formulae Propositional constants: which represents a true proposition, and which represents a false proposition. Propositional variables: variables that range over propositions. Usually denoted by p, q, r, possibly with indices. Inductive definition of propositional formulae: 1. Every propositional constant and every propositional variable is a propositional formula. 2. If A is a propositional formula then A is a propositional formula. 3. If A, B are propositional formulae then (A B), (A B), (A B), (A B) are propositional formulae. Examples:,, p, p, p, q, (p q), (p 1 (p 2 p 1 )),... Outermost pairs of parentheses will often be omitted.
13 Construction trees, subformulae, main connectives Construction tree: a tree with nodes labelled with propositional constants, variables, and propositional connectives, such that: 1. Every leaf is labelled by a propositional constant or variable. 2. Propositional constants and variables label only leaves. 3. Every node labelled with has exactly one successor node. 4. Every node labelled with any of,,, has exactly two successor nodes - left and right successor. Every construction tree defines a formula C, built starting from the leaves and going towards the root, by applying at every node the formula construction rule corresponding to the label at that node.
14 Construction tree of a formula Leaves: propositional variables or constants unary connective ( ) binary connective (,,, ) the main connective The formulae constructed in the process are the subformulae of C. The connective labelling the root of the construction tree of a formula C is the main connective of C.
15 Construction tree: example Formula: Construction tree (p (q r)) r r q r p V
16 Parsing tree of a formula The parsing tree of a formula looks the same as the construction tree, but is produced in inverse order, starting from the main connective (of any), drawing edges to all main components, and then recursively producing the parsing trees for each of them. Leaves: propositional variables or constants unary connective ( ) binary connective (,,, ) the main connective
17 Formula: Parsing tree Parsing tree: example (p (q r)) r r q r p V
18 Truth tables of propositional formulae Example: (p (q r)) r p q r r r q r (q r) p (q r) (p (q r)) r T T T F T F T T T T T F T F T F T F T F T F T F T T T T F F T F F T T F F T T F T F T T T F T F T F T F F T F F T F F F
19 Simplified truth tables p q r (p (q r)) r T T T T T T T F F T T T F T T T F T T F T T T F F F T F T F T T T T F F F T T T F T T F F T T T F F T F F F T F F T T F T T T F F T T T F T F T F F F F T T T F T F T F F F T F F F
20 Tautologies Tautology (or, propositionally valid formula): a formula that obtains truth value T for every assignment of truth values to the occurring variables. Notation: = A. Examples: = p p, = (p p), = ((p (p q)) q) Testing tautologies with truth-tables: p q p q p (p q) (p (p q)) q T T T T T T F F F T F T T F T F F T F T
21 Contradictions, satisfiable formulae Contradiction is a formula that always takes truth value F. Examples: p p, ((p q) p) Thus, the negation of a tautology is a contradiction and the negation of a contradiction is a tautology. A formula is satisfiable if it is not a contradiction. Example: p, p q, etc.
CSCE 222 Discrete Structures for Computing. Propositional Logic. Dr. Hyunyoung Lee. !!!!!! Based on slides by Andreas Klappenecker
CSCE 222 Discrete Structures for Computing Propositional Logic Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Propositions A proposition is a declarative sentence that is either true or false
More informationCSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
More informationCompound Propositions
Discrete Structures Compound Propositions Producing new propositions from existing propositions. Logical Operators or Connectives 1. Not 2. And 3. Or 4. Exclusive or 5. Implication 6. Biconditional Truth
More informationLecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
More informationProofs. Joe Patten August 10, 2018
Proofs Joe Patten August 10, 2018 1 Statements and Open Sentences 1.1 Statements A statement is a declarative sentence or assertion that is either true or false. They are often labelled with a capital
More informationDISCRETE STRUCTURES WEEK5 LECTURE1
DISCRETE STRUCTURES WEEK5 LECTURE1 Let s get started with... Logic! Spring 2010 CPCS 222 - Discrete Structures 2 Logic Crucial for mathematical reasoning Important for program design Used for designing
More informationPROPOSITIONAL CALCULUS
PROPOSITIONAL CALCULUS A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both. These are not propositions! Connectives and
More informationChapter 4: Classical Propositional Semantics
Chapter 4: Classical Propositional Semantics Language : L {,,, }. Classical Semantics assumptions: TWO VALUES: there are only two logical values: truth (T) and false (F), and EXTENSIONALITY: the logical
More informationChapter Summary. Propositional Logic. Predicate Logic. Proofs. The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.
Chapter 1 Chapter Summary Propositional Logic The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.3) Predicate Logic The Language of Quantifiers (1.4) Logical Equivalences (1.4)
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 informationLecture 7. Logic. Section1: Statement Logic.
Ling 726: Mathematical Linguistics, Logic, Section : Statement Logic V. Borschev and B. Partee, October 5, 26 p. Lecture 7. Logic. Section: Statement Logic.. Statement Logic..... Goals..... Syntax of Statement
More informationA statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial
More informationLearning Goals of CS245 Logic and Computation
Learning Goals of CS245 Logic and Computation Alice Gao April 27, 2018 Contents 1 Propositional Logic 2 2 Predicate Logic 4 3 Program Verification 6 4 Undecidability 7 1 1 Propositional Logic Introduction
More informationChapter 1, Section 1.1 Propositional Logic
Discrete Structures Chapter 1, Section 1.1 Propositional Logic These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 6 th ed., by Kenneth H. Rosen, published
More informationSection 1.1 Propositional Logic. proposition : true = T (or 1) or false = F (or 0) (binary logic) the moon is made of green cheese
Section 1.1 Propositional Logic proposition : true = T (or 1) or false = F (or 0) (binary logic) the moon is made of green cheese go to town! X - imperative What time is it? X - interrogative propositional
More informationPropositions. Frequently, we will use the word statement instead of proposition.
Propositional Logic Propositions A proposition is a declaration of fact that is either true or false, but not both. Examples and non-examples: One plus two equals four (proposition) Mozart is the greatest
More informationWhat is Logic? Introduction to Logic. Simple Statements. Which one is statement?
What is Logic? Introduction to Logic Peter Lo Logic is the study of reasoning It is specifically concerned with whether reasoning is correct Logic is also known as Propositional Calculus CS218 Peter Lo
More informationPropositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.
Propositional Logic Winter 2012 Propositional Logic: Section 1.1 Proposition A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?
More information2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
More informationTruth-Functional Logic
Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence
More informationPropositional Logic. Spring Propositional Logic Spring / 32
Propositional Logic Spring 2016 Propositional Logic Spring 2016 1 / 32 Introduction Learning Outcomes for this Presentation Learning Outcomes... At the conclusion of this session, we will Define the elements
More informationNatural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
More informationCSE 20: Discrete Mathematics
Spring 2018 Summary Last time: Today: Logical connectives: not, and, or, implies Using Turth Tables to define logical connectives Logical equivalences, tautologies Some applications Proofs in propositional
More informationComp487/587 - 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 informationChapter 1, Part I: Propositional Logic. With Question/Answer Animations
Chapter 1, Part I: Propositional Logic With Question/Answer Animations Chapter Summary Propositional Logic The Language of Propositions Applications Logical Equivalences Predicate Logic The Language of
More informationDiscrete Mathematics and Its Applications
Discrete Mathematics and Its Applications Lecture 1: Proposition logic MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 12, 2017 Outline 1 Propositions 2 Connectives
More informationAI 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 informationLogic. Readings: Coppock and Champollion textbook draft, Ch
Logic Readings: Coppock and Champollion textbook draft, Ch. 3.1 3 1. Propositional logic Propositional logic (a.k.a propositional calculus) is concerned with complex propositions built from simple propositions
More information1.1 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 informationSec$on Summary. Propositions Connectives. Truth Tables. Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional
Section 1.1 Sec$on Summary Propositions Connectives Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional ruth ables 2 Proposi$ons A proposition is a declarative
More informationArtificial Intelligence. Propositional logic
Artificial Intelligence Propositional logic Propositional Logic: Syntax Syntax of propositional logic defines allowable sentences Atomic sentences consists of a single proposition symbol Each symbol stands
More informationDiscrete Structures of Computer Science Propositional Logic I
Discrete Structures of Computer Science Propositional Logic I Gazihan Alankuş (Based on original slides by Brahim Hnich) July 26, 2012 1 Use of Logic 2 Statements 3 Logic Connectives 4 Truth Tables Use
More informationPropositional Logic: Part II - Syntax & Proofs 0-0
Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems
More informationThe Importance of Being Formal. Martin Henz. February 5, Propositional Logic
The Importance of Being Formal Martin Henz February 5, 2014 Propositional Logic 1 Motivation In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets
More informationDefinition 2. Conjunction of p and q
Proposition Propositional Logic CPSC 2070 Discrete Structures Rosen (6 th Ed.) 1.1, 1.2 A proposition is a statement that is either true or false, but not both. Clemson will defeat Georgia in football
More informationDiscrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
More informationSection 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 informationIntroduction 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 information2.2: Logical Equivalence: The Laws of Logic
Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q
More informationOverview. 1. Introduction to Propositional Logic. 2. Operations on Propositions. 3. Truth Tables. 4. Translating Sentences into Logical Expressions
Note 01 Propositional Logic 1 / 10-1 Overview 1. Introduction to Propositional Logic 2. Operations on Propositions 3. Truth Tables 4. Translating Sentences into Logical Expressions 5. Preview: Propositional
More informationExample. Logic. Logical Statements. Outline of logic topics. Logical Connectives. Logical Connectives
Logic Logic is study of abstract reasoning, specifically, concerned with whether reasoning is correct. Logic focuses on relationship among statements as opposed to the content of any particular statement.
More informationChapter 1: Formal Logic
Chapter 1: Formal Logic Dr. Fang (Daisy) Tang ftang@cpp.edu www.cpp.edu/~ftang/ CS 130 Discrete Structures Logic: The Foundation of Reasoning Definition: the foundation for the organized, careful method
More informationMath 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 informationPacket #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 informationWe last time we began introducing equivalency laws.
Monday, January 14 MAD2104 Discrete Math 1 Course website: www/mathfsuedu/~wooland/mad2104 Today we will continue in Course Notes Chapter 22 We last time we began introducing equivalency laws Today we
More informationPropositional Logic: Semantics
Propositional Logic: Semantics Alice Gao Lecture 4, September 19, 2017 Semantics 1/56 Announcements Semantics 2/56 The roadmap of propositional logic Semantics 3/56 FCC spectrum auction an application
More informationLING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, Propositional logic
LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, 2016 Propositional logic 1 Vocabulary of propositional logic Vocabulary (1) a. Propositional letters: p, q, r, s, t, p 1, q 1,..., p 2, q 2,...
More informationPropositional Logic. Yimei Xiang 11 February format strictly follow the laws and never skip any step.
Propositional Logic Yimei Xiang yxiang@fas.harvard.edu 11 February 2014 1 Review Recursive definition Set up the basis Generate new members with rules Exclude the rest Subsets vs. proper subsets Sets of
More informationSupplementary exercises in propositional logic
Supplementary exercises in propositional logic The purpose of these exercises is to train your ability to manipulate and analyze logical formulas. Familiarize yourself with chapter 7.3-7.5 in the course
More informationCS1021. Why logic? Logic about inference or argument. Start from assumptions or axioms. Make deductions according to rules of reasoning.
3: Logic Why logic? Logic about inference or argument Start from assumptions or axioms Make deductions according to rules of reasoning Logic 3-1 Why logic? (continued) If I don t buy a lottery ticket on
More informationTHE 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 informationDiscrete Mathematics and Applications COT3100
Discrete Mathematics and Applications CO3100 Dr. Ungor Sources: Slides are based on Dr. G. Bebis material. uesday, January 7, 2014 oundations of Logic: Overview Propositional logic: (Sections 1.1-1.3)
More informationIntroduction to Decision Sciences Lecture 2
Introduction to Decision Sciences Lecture 2 Andrew Nobel August 24, 2017 Compound Proposition A compound proposition is a combination of propositions using the basic operations. For example (p q) ( p)
More informationTecniche di Verifica. Introduction to Propositional Logic
Tecniche di Verifica Introduction to Propositional Logic 1 Logic A formal logic is defined by its syntax and semantics. Syntax An alphabet is a set of symbols. A finite sequence of these symbols is called
More informationPropositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
More informationPropositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
More informationLogic and Discrete Mathematics. Section 3.5 Propositional logical equivalence Negation of propositional formulae
Logic and Discrete Mathematics Section 3.5 Propositional logical equivalence Negation of propositional formulae Slides version: January 2015 Logical equivalence of propositional formulae Propositional
More informationPropositional and Predicate Logic - II
Propositional and Predicate Logic - II Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - II WS 2016/2017 1 / 16 Basic syntax Language Propositional logic
More information3 Propositional Logic
3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists
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 informationHW1 graded review form? HW2 released CSE 20 DISCRETE MATH. Fall
CSE 20 HW1 graded review form? HW2 released DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Translate sentences from English to propositional logic using appropriate
More informationPropositional logic (revision) & semantic entailment. p. 1/34
Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)
More informationDiscrete Mathematics
Discrete Mathematics Discrete mathematics is devoted to the study of discrete or distinct unconnected objects. Classical mathematics deals with functions on real numbers. Real numbers form a continuous
More information1.1 Language and Logic
c Oksana Shatalov, Spring 2018 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 informationCSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter p. 1/33
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter 4.1-4.8 p. 1/33 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer
More information~ p is always false. Based on the basic truth table for disjunction, if q is true then p ~
MAT 101 Solutions Exam 2 (Logic, Part I) Multiple-Choice Questions 1. D Because this sentence contains exactly ten words, it is stating that it is false. But if it is taken to be false, then it has to
More informationA brief introduction to Logic. (slides from
A brief introduction to Logic (slides from http://www.decision-procedures.org/) 1 A Brief Introduction to Logic - Outline Propositional Logic :Syntax Propositional Logic :Semantics Satisfiability and validity
More informationPropositional Logic and Semantics
Propositional Logic and Semantics English is naturally ambiguous. For example, consider the following employee (non)recommendations and their ambiguity in the English language: I can assure you that no
More information1 Propositional Logic
1 Propositional Logic Required reading: Foundations of Computation. Sections 1.1 and 1.2. 1. Introduction to Logic a. Logical consequences. If you know all humans are mortal, and you know that you are
More informationLogic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.
Math 0413 Appendix A.0 Logic Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another. This type of logic is called propositional.
More information7 LOGICAL AGENTS. OHJ-2556 Artificial Intelligence, Spring OHJ-2556 Artificial Intelligence, Spring
109 7 LOGICAL AGENS We now turn to knowledge-based agents that have a knowledge base KB at their disposal With the help of the KB the agent aims at maintaining knowledge of its partially-observable environment
More informationThe Calculus of Computation: Decision Procedures with Applications to Verification. Part I: FOUNDATIONS. by Aaron Bradley Zohar Manna
The Calculus of Computation: Decision Procedures with Applications to Verification Part I: FOUNDATIONS by Aaron Bradley Zohar Manna 1. Propositional Logic(PL) Springer 2007 1-1 1-2 Propositional Logic(PL)
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 information2/13/2012. Logic: Truth Tables. CS160 Rosen Chapter 1. Logic?
Logic: Truth Tables CS160 Rosen Chapter 1 Logic? 1 What is logic? Logic is a truth-preserving system of inference Truth-preserving: If the initial statements are true, the inferred statements will be true
More informationIntroduction. Applications of discrete mathematics:
Introduction Applications of discrete mathematics: Formal Languages (computer languages) Compiler Design Data Structures Computability Automata Theory Algorithm Design Relational Database Theory Complexity
More informationSyntax of propositional logic. Syntax tree of a formula. Semantics of propositional logic (I) Subformulas
Syntax of propositional logic Syntax tree of a formula An atomic formula has the form A i where i =, 2, 3,.... Formulas are defined by the following inductive process: Every formula can be represented
More information10/5/2012. Logic? What is logic? Propositional Logic. Propositional Logic (Rosen, Chapter ) Logic is a truth-preserving system of inference
Logic? Propositional Logic (Rosen, Chapter 1.1 1.3) TOPICS Propositional Logic Truth Tables Implication Logical Proofs 10/1/12 CS160 Fall Semester 2012 2 What is logic? Logic is a truth-preserving system
More informationPropositional Logic: Equivalence
Propositional Logic: Equivalence Alice Gao Lecture 5 Based on work by J. Buss, L. Kari, A. Lubiw, B. Bonakdarpour, D. Maftuleac, C. Roberts, R. Trefler, and P. Van Beek 1/42 Outline Propositional Logic:
More informationPropositional Language - Semantics
Propositional Language - Semantics Lila Kari University of Waterloo Propositional Language - Semantics CS245, Logic and Computation 1 / 41 Syntax and semantics Syntax Semantics analyzes Form analyzes Meaning
More informationEquivalence and Implication
Equivalence and Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February 7 8, 2018 Alice E. Fischer Laws of Logic... 1/33 1 Logical Equivalence Contradictions and Tautologies 2 3 4 Necessary
More informationDiscrete Structures & Algorithms. Propositional Logic EECE 320 // UBC
Discrete Structures & Algorithms Propositional Logic EECE 320 // UBC 1 Review of last lecture Pancake sorting A problem with many applications Bracketing (bounding a function) Proving bounds for pancake
More informationThe following techniques for methods of proofs are discussed in our text: - Vacuous proof - Trivial proof
Ch. 1.6 Introduction to Proofs The following techniques for methods of proofs are discussed in our text - Vacuous proof - Trivial proof - Direct proof - Indirect proof (our book calls this by contraposition)
More informationLogic 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 informationPHIL12A 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 informationINTRODUCTION 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 informationPropositional Calculus: Formula Simplification, Essential Laws, Normal Forms
P Formula Simplification, Essential Laws, Normal Forms Lila Kari University of Waterloo P Formula Simplification, Essential Laws, Normal CS245, Forms Logic and Computation 1 / 26 Propositional calculus
More informationDigital Logic Design: a rigorous approach c
Digital Logic Design: a rigorous approach c Chapter 6: Propositional Logic Guy Even Moti Medina School of Electrical Engineering Tel-Aviv Univ. November 4, 2015 Book Homepage: http://www.eng.tau.ac.il/~guy/even-medina
More informationWhat is the decimal (base 10) representation of the binary number ? Show your work and place your final answer in the box.
Question 1. [10 marks] Part (a) [2 marks] What is the decimal (base 10) representation of the binary number 110101? Show your work and place your final answer in the box. 2 0 + 2 2 + 2 4 + 2 5 = 1 + 4
More informationAnnouncements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before
More informationCS100: 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 informationMore Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences. Computability and Logic
More Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences Computability and Logic Equivalences Involving Conditionals Some Important Equivalences Involving Conditionals
More informationUNIT-I: Propositional Logic
1. Introduction to Logic: UNIT-I: Propositional Logic Logic: logic comprises a (formal) language for making statements about objects and reasoning about properties of these objects. Statements in a logical
More informationPropositional Logic. Fall () Propositional Logic Fall / 30
Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically
More information15414/614 Optional Lecture 1: Propositional Logic
15414/614 Optional Lecture 1: Propositional Logic Qinsi Wang Logic is the study of information encoded in the form of logical sentences. We use the language of Logic to state observations, to define concepts,
More informationPropositional Logic. CS 3234: Logic and Formal Systems. Martin Henz and Aquinas Hobor. August 26, Generated on Tuesday 31 August, 2010, 16:54
Propositional Logic CS 3234: Logic and Formal Systems Martin Henz and Aquinas Hobor August 26, 2010 Generated on Tuesday 31 August, 2010, 16:54 1 Motivation In traditional logic, terms represent sets,
More informationIntroduction Propositional Logic. Discrete Mathematics Andrei Bulatov
Introduction Propositional Logic Discrete Mathematics Andrei Bulatov Discrete Mathematics Propositional Logic 2-2 What is Logic? Computer science is a mere continuation of logic by other means Georg Gottlob
More informationChapter 4, Logic using Propositional Calculus Handout
ECS 20 Chapter 4, Logic using Propositional Calculus Handout 0. Introduction to Discrete Mathematics. 0.1. Discrete = Individually separate and distinct as opposed to continuous and capable of infinitesimal
More informationKnowledge base (KB) = set of sentences in a formal language Declarative approach to building an agent (or other system):
Logic Knowledge-based agents Inference engine Knowledge base Domain-independent algorithms Domain-specific content Knowledge base (KB) = set of sentences in a formal language Declarative approach to building
More informationARTIFICIAL INTELLIGENCE
ARTIFICIAL INTELLIGENCE LECTURE # 03 Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 1 Review of Last Lecture Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 2 Today s
More informationcis32-ai lecture # 18 mon-3-apr-2006
cis32-ai lecture # 18 mon-3-apr-2006 today s topics: propositional logic cis32-spring2006-sklar-lec18 1 Introduction Weak (search-based) problem-solving does not scale to real problems. To succeed, problem
More information