Publication Data. Reading Options. Licence and permissions ISBN Mark Jago, 2007
|
|
- Leonard Newton
- 5 years ago
- Views:
Transcription
1 Running Head The World is all that is the case http// Philosophy Insights General Editor: Mark ddis Formal Logic Mark Jago What makes an argument valid? For advice on use of this ebook please scroll to page 2
2 Publication Data Mark Jago, 2007 The uthor has asserted his right to be identified as the author of this Work in accordance with the Copyright, Designs and Patents ct Published by Humanities-Ebooks.co.uk Tirril Hall, Tirril, Penrith C10 2JE Reading Options * Before continuing, please use the command View > fit to page and then progress by using the next page arrows at the top or bottom right of the Viewer screen. * To navigate through the contents use the Bookmarks at the left of the screen. * To search, click on the search symbol in the toolbar and select show all results. * For ease of reading, use <CTRL+L> to enlarge the page to full screen * Use <CTRL+L> to return to the full menu, with its bookmarks and search tool. * Hyperlinks (if any) appear in Blue Underlined Text. Licence and permissions This book is licensed for a particular computer or computers. The file itself may be copied, but the copy will not open until the new user obtains a licence from the Humanities-Ebooks website in the usual manner. The original purchaser may license the same work for a second computer by applying to support@humanities-ebooks.co.uk with proof of purchase. Permissions: it is permissible to print one (watermarked) copy of the book for your own use, but not to copy and paste text. ISBN
3 Formal Logic Mark Jago Bibliographical Entry: Jago, Mark. Formal Logic. Philosophy Insights. Tirril: Humanities-Ebooks, 2007
4 Note on the uthor Mark Jago is a lecturer in the Department of Philosophy at the University of Nottingham, UK and a Junior Research ssociate in the Research Group on the Philosophy of Information at the University of Oxford. He wrote the Wittgenstein guide in the Philosophy Insights series and has published articles on truth, belief, logic, fiction and information. Personal website:
5 Philosophy Insights: Formal Logic 5 Contents Introduction 7 1 Logical Reasoning Preliminaries Valid rguments Valid Forms of Inference Exercises Propositional Logic Introduction Logical Connectives The Logical Language Construction Trees Truth Tables Valuations Exercises Entailment and Equivalence Logical Entailment Equivalence Equivalence Schemes Exercises Proof Trees Proofs in Logic The Proof Tree Method Examples of Proof Trees Decidability Valuations From Open Finished Trees Soundness and Completeness Exercises
6 Philosophy Insights: Formal Logic 6 5 First Order Logic More Valid rguments Constants, Predicates and Relations Existence and Generality Formation Rules Binary Relations Semantics for First-Order Logic Satisfaction Exercises Identity The Puzzle of Identity Identity in First-Order Logic Expressing t Least, t Most and Exactly Definite Descriptions Leibniz s Law and Second Order Logic Exercises Proof Trees for First Order Logic Rules for Quantifiers Rules for Identity Undecidability Constructing Models from Open Branches Soundness and Completeness Exercises ppendix. Basic Set Theory 91 ppendix B. Infinity 92 References and Further Reading 94 nswers to Exercises 95
7 Introduction Logical reasoning is vital to philosophy. Descartes for one recognized this in his Rules for the Direction of the Mind (1628), where he writes: RULE 4: There is need of a method for investigating the truth about things. RULE 5:... we shall be observing this method exactly if we reduce complex and obscure propositions step by step to simpler ones, and then, by retracing our steps, try to rise from intuition of all of the simplest ones to knowledge of all the rest. Descartes aim was first to find principles whose truth he could be certain of and then to deduce further truths from these. This raises the question, just what counts as reasoning correctly from one proposition to another? This is what we hope to understand through studying logic. There are different views as to what studying logic should achieve, including the following: We should aim to discover logical truths, i.e. sentences that could not possibly be false and which we can discover to be true a priori. We should aim to discover valid forms of reasoning to use in our arguments. We should aim to discover the principles of logical entailment, so that we can ascertain the facts that are entailed by what we know to be the case. Fortunately for us, these approaches to logic all turn out to be interchangeable, at least in the form of logic that we will study here, known as classical logic. Strange as it sounds at first, there is not one body of doctrine or method that can be labelled logic. There are disagreements over what principles apply to the notion of logical entailment and over what counts as a valid argument. These disagreements constitute the philosophy of logic, which I will not go into in this book. s a rule of thumb, whenever someone speaks of logic, unqualified as this or that style of logic, they will mean classical first order logic. This is certainly true in most philosophy classes (at least, those not dealing with technical subjects such as the philosophy of logic or mathematics). Classical logic is also adopted as the logic of choice in mathematics and electronics, although not always in computer science. 1 1 Classical logic is focused on truth, whereas computer scientists are often focused on the kinds of tasks that computers can do and in particular, what computers can prove. Focusing on proof rather than truth is the province of constructive logics.
8 7. Proof Trees for First Order Logic Recap: Rules for the Logical Connectives The basics of proof trees in first order logic are just the same as the basics of proof trees for propositional logic (section 4.2). ll of the rules that we encountered there are still usable. B B B B B B B B B ( B) ( B) ( B) ( B) B B B B B Here, and B can be any first order sentences, atomic or complex. These rules do not allow us to deal with quantified sentences or identity, which we will look at below. 7.1 Rules for Quantifiers Rules for x and x The rules for negated quantified sentences x and x are obvious, given how x and x are interrelated: x x x x
9 Philosophy Insights: Formal Logic 91 ppendix. Basic Set Theory set is just a collection of objects. The collection is abstract: objects do not have to be arranged in space or time in any particular way form a set. For example, there is a set consisting of Tony Blair, the Emperor ugustus and the number 17. If a, b and c are objects, we write the set that contains them (and no further objects) as {a,b,c} Here, a, b and c are called the elements or members of the set. Sets themselves are considered to be objects, so that a set can be an element of another set, for example, the set {a,b,{a,c}} What makes a set the set it is, rather than some other set, is just its members, so that there cannot be two distinct sets that have exactly the same members. Thus, {a,c,b}, {a,b,c} and {a,b,b,c} are different ways of writing the very same set. But note that {a,b,{c}} is not the same set as {a,b,c}, for the former contains the set {c} whereas the latter does not. We count the collection of no objects at all as a set. This is called the empty set and is written (some authors use {} ). 1 I will use capital roman letters X, Y, Z to name sets, the symbol to abbreviate is a member of and / to abbreviate is not a member of. The union of two sets X and Y, X Y, is the set that contains all the members of X and all the members of Y (and no more). n object a X Y, therefore, if and only if either a X or a Y. For example, if X = {a,b,c} and Y = {b,c,d} then X Y = {a,b,c,d}. The intersection of X and Y, X Y, is the set that contains all of the objects that are both members of X and members of Y. n object a X Y if and only if a X and a Y. If X and Y are as above, then their intersection X Y = {b,c}. X is a subset of Y, written X Y, if and only if every member of X is also a member of Y. {a} is a subset of {a,b}, as is {b}. Note that X X and X for every X. The set of all subsets of X, including and X itself, is the power set of X, written (X). 2 1 There can only be the one empty set. If there were two distinct empty sets, then at least one object must belong to one but not the other. But since no objects belong to the empty set, this cannot be the case. 2 The power set of of X can also be written 2 X.
Propositional Logic. Part I
Part I Propositional Logic 1 Classical Logic and the Material Conditional 1.1 Introduction 1.1.1 The first purpose of this chapter is to review classical propositional logic, including semantic tableaux.
More informationAlgebra of Sets. Aditya Ghosh. April 6, 2018 It is recommended that while reading it, sit with a pen and a paper.
Algebra of Sets Aditya Ghosh April 6, 2018 It is recommended that while reading it, sit with a pen and a paper. 1 The Basics This article is only about the algebra of sets, and does not deal with the foundations
More informationCSC 501 Semantics of Programming Languages
CSC 501 Semantics of Programming Languages Subtitle: An Introduction to Formal Methods. Instructor: Dr. Lutz Hamel Email: hamel@cs.uri.edu Office: Tyler, Rm 251 Books There are no required books in this
More information[Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics
400 lecture note #4 [Ch 6] Set Theory 1. Basic Concepts and Definitions 1) Basics Element: ; A is a set consisting of elements x which is in a/another set S such that P(x) is true. Empty set: notated {
More informationDiscrete Mathematics Lecture 4. Harper Langston New York University
Discrete Mathematics Lecture 4 Harper Langston New York University Sequences Sequence is a set of (usually infinite number of) ordered elements: a 1, a 2,, a n, Each individual element a k is called a
More informationSets. Sets. Subset, universe. Specifying sets, membership. Examples: Specifying a set using a predicate. Examples
Sets 2/36 We will not give a precise definition of what is a set, but we will say precisely what you can do with it. Sets Lectures 7 and 8 (hapter 16) (Think of a set as a collection of things of which
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Final exam The final exam is Saturday December 16 11:30am-2:30pm. Lecture A will take the exam in Lecture B will take the exam
More informationChapter 3. Set Theory. 3.1 What is a Set?
Chapter 3 Set Theory 3.1 What is a Set? A set is a well-defined collection of objects called elements or members of the set. Here, well-defined means accurately and unambiguously stated or described. Any
More informationHow invariants help writing loops Author: Sander Kooijmans Document version: 1.0
How invariants help writing loops Author: Sander Kooijmans Document version: 1.0 Why this document? Did you ever feel frustrated because of a nasty bug in your code? Did you spend hours looking at the
More informationBootcamp. Christoph Thiele. Summer An example of a primitive universe
Bootcamp Christoph Thiele Summer 2012 0.1 An example of a primitive universe A primitive universe consists of primitive objects and primitive sets. This allows to form primitive statements as to which
More informationSemantics via Syntax. f (4) = if define f (x) =2 x + 55.
1 Semantics via Syntax The specification of a programming language starts with its syntax. As every programmer knows, the syntax of a language comes in the shape of a variant of a BNF (Backus-Naur Form)
More informationBOOLEAN ALGEBRA AND CIRCUITS
UNIT 3 Structure BOOLEAN ALGEBRA AND CIRCUITS Boolean Algebra and 3. Introduction 3. Objectives 3.2 Boolean Algebras 3.3 Logic 3.4 Boolean Functions 3.5 Summary 3.6 Solutions/ Answers 3. INTRODUCTION This
More informationCS 512, Spring 2017: Take-Home End-of-Term Examination
CS 512, Spring 2017: Take-Home End-of-Term Examination Out: Tuesday, 9 May 2017, 12:00 noon Due: Wednesday, 10 May 2017, by 11:59 am Turn in your solutions electronically, as a single PDF file, by placing
More informationCS Bootcamp Boolean Logic Autumn 2015 A B A B T T T T F F F T F F F F T T T T F T F T T F F F
1 Logical Operations 1.1 And The and operator is a binary operator, denoted as, &,, or sometimes by just concatenating symbols, is true only if both parameters are true. A B A B F T F F F F The expression
More informationCSC Discrete Math I, Spring Sets
CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its
More informationKnowledge Representation
Knowledge Representation References Rich and Knight, Artificial Intelligence, 2nd ed. McGraw-Hill, 1991 Russell and Norvig, Artificial Intelligence: A modern approach, 2nd ed. Prentice Hall, 2003 Outline
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Final exam The final exam is Saturday March 18 8am-11am. Lecture A will take the exam in GH 242 Lecture B will take the exam
More informationHoare Logic. COMP2600 Formal Methods for Software Engineering. Rajeev Goré
Hoare Logic COMP2600 Formal Methods for Software Engineering Rajeev Goré Australian National University Semester 2, 2016 (Slides courtesy of Ranald Clouston) COMP 2600 Hoare Logic 1 Australian Capital
More informationPropositional Logic. Andreas Klappenecker
Propositional Logic Andreas Klappenecker Propositions A proposition is a declarative sentence that is either true or false (but not both). Examples: College Station is the capital of the USA. There are
More informationSets and set operations. Lecture 5 ICOM 4075
Sets and set operations Lecture 5 ICOM 4075 Reviewing sets s defined in a previous lecture, a setis a collection of objects that constitute the elementsof the set We say that a set containsits elements,
More informationReflection in the Chomsky Hierarchy
Reflection in the Chomsky Hierarchy Henk Barendregt Venanzio Capretta Dexter Kozen 1 Introduction We investigate which classes of formal languages in the Chomsky hierarchy are reflexive, that is, contain
More informationReasoning About Programs Panagiotis Manolios
Reasoning About Programs Panagiotis Manolios Northeastern University March 22, 2012 Version: 58 Copyright c 2012 by Panagiotis Manolios All rights reserved. We hereby grant permission for this publication
More informationHandout 9: Imperative Programs and State
06-02552 Princ. of Progr. Languages (and Extended ) The University of Birmingham Spring Semester 2016-17 School of Computer Science c Uday Reddy2016-17 Handout 9: Imperative Programs and State Imperative
More informationLECTURE 2 An Introduction to Boolean Algebra
IST 210: Boot Camp Ritendra Datta LECTURE 2 An Introduction to Boolean Algebra 2.1. Outline of Lecture Fundamentals Negation, Conjunction, and Disjunction Laws of Boolean Algebra Constructing Truth Tables
More informationLecture 6,
Lecture 6, 4.16.2009 Today: Review: Basic Set Operation: Recall the basic set operator,!. From this operator come other set quantifiers and operations:!,!,!,! \ Set difference (sometimes denoted, a minus
More informationSOFTWARE ENGINEERING DESIGN I
2 SOFTWARE ENGINEERING DESIGN I 3. Schemas and Theories The aim of this course is to learn how to write formal specifications of computer systems, using classical logic. The key descriptional technique
More information(QiuXin Hui) 7.2 Given the following, can you prove that the unicorn is mythical? How about magical? Horned? Decide what you think the right answer
(QiuXin Hui) 7.2 Given the following, can you prove that the unicorn is mythical? How about magical? Horned? Decide what you think the right answer is yourself, then show how to get the answer using both
More informationAutomata Theory for Reasoning about Actions
Automata Theory for Reasoning about Actions Eugenia Ternovskaia Department of Computer Science, University of Toronto Toronto, ON, Canada, M5S 3G4 eugenia@cs.toronto.edu Abstract In this paper, we show
More informationTopic 3: Propositions as types
Topic 3: Propositions as types May 18, 2014 Propositions as types We have seen that the main mathematical objects in a type theory are types. But remember that in conventional foundations, as based on
More informationTo prove something about all Boolean expressions, we will need the following induction principle: Axiom 7.1 (Induction over Boolean expressions):
CS 70 Discrete Mathematics for CS Fall 2003 Wagner Lecture 7 This lecture returns to the topic of propositional logic. Whereas in Lecture 1 we studied this topic as a way of understanding proper reasoning
More informationLECTURE 8: SETS. Software Engineering Mike Wooldridge
LECTURE 8: SETS Mike Wooldridge 1 What is a Set? The concept of a set is used throughout mathematics; its formal definition matches closely our intuitive understanding of the word. Definition: A set is
More informationModule 8. Other representation formalisms. Version 2 CSE IIT, Kharagpur
Module 8 Other representation formalisms 8.1 Instructional Objective The students should understand the syntax and semantic of semantic networks Students should learn about different constructs and relations
More informationSection 2.4: Arguments with Quantified Statements
Section 2.4: Arguments with Quantified Statements In this section, we shall generalize the ideas we developed in Section 1.3 to arguments which involve quantified statements. Most of the concepts we shall
More information2. BOOLEAN ALGEBRA 2.1 INTRODUCTION
2. BOOLEAN ALGEBRA 2.1 INTRODUCTION In the previous chapter, we introduced binary numbers and binary arithmetic. As you saw in binary arithmetic and in the handling of floating-point numbers, there is
More information1 Introduction CHAPTER ONE: SETS
1 Introduction CHAPTER ONE: SETS Scientific theories usually do not directly describe the natural phenomena under investigation, but rather a mathematical idealization of them that abstracts away from
More information2.1 Sets 2.2 Set Operations
CSC2510 Theoretical Foundations of Computer Science 2.1 Sets 2.2 Set Operations Introduction to Set Theory A set is a structure, representing an unordered collection (group, plurality) of zero or more
More informationTo prove something about all Boolean expressions, we will need the following induction principle: Axiom 7.1 (Induction over Boolean expressions):
CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 7 This lecture returns to the topic of propositional logic. Whereas in Lecture Notes 1 we studied this topic as a way of understanding
More informationTaibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103. Chapter 2. Sets
Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 2 Sets Slides are adopted from Discrete Mathematics and It's Applications Kenneth H.
More informationSTABILITY AND PARADOX IN ALGORITHMIC LOGIC
STABILITY AND PARADOX IN ALGORITHMIC LOGIC WAYNE AITKEN, JEFFREY A. BARRETT Abstract. Algorithmic logic is the logic of basic statements concerning algorithms and the algorithmic rules of deduction between
More informationChapter 2 & 3: Representations & Reasoning Systems (2.2)
Chapter 2 & 3: A Representation & Reasoning System & Using Definite Knowledge Representations & Reasoning Systems (RRS) (2.2) Simplifying Assumptions of the Initial RRS (2.3) Datalog (2.4) Semantics (2.5)
More informationChapter 3: Propositional Languages
Chapter 3: Propositional Languages We define here a general notion of a propositional language. We show how to obtain, as specific cases, various languages for propositional classical logic and some non-classical
More informationThis is already grossly inconvenient in present formalisms. Why do we want to make this convenient? GENERAL GOALS
1 THE FORMALIZATION OF MATHEMATICS by Harvey M. Friedman Ohio State University Department of Mathematics friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ May 21, 1997 Can mathematics be
More informationGraph Theory Questions from Past Papers
Graph Theory Questions from Past Papers Bilkent University, Laurence Barker, 19 October 2017 Do not forget to justify your answers in terms which could be understood by people who know the background theory
More information2 Review of Set Theory
2 Review of Set Theory Example 2.1. Let Ω = {1, 2, 3, 4, 5, 6} 2.2. Venn diagram is very useful in set theory. It is often used to portray relationships between sets. Many identities can be read out simply
More informationGraph Theory. 1 Introduction to Graphs. Martin Stynes Department of Mathematics, UCC January 26, 2011
Graph Theory Martin Stynes Department of Mathematics, UCC email: m.stynes@ucc.ie January 26, 2011 1 Introduction to Graphs 1 A graph G = (V, E) is a non-empty set of nodes or vertices V and a (possibly
More informationThe Further Mathematics Support Programme
Degree Topics in Mathematics Groups A group is a mathematical structure that satisfies certain rules, which are known as axioms. Before we look at the axioms, we will consider some terminology. Elements
More information3.4 Deduction and Evaluation: Tools Conditional-Equational Logic
3.4 Deduction and Evaluation: Tools 3.4.1 Conditional-Equational Logic The general definition of a formal specification from above was based on the existence of a precisely defined semantics for the syntax
More information"Relations for Relationships"
M359 An explanation from Hugh Darwen "Relations for Relationships" This note might help those who have struggled with M359's so-called "relation for relationship" method of representing, in a relational
More informationMath 302 Introduction to Proofs via Number Theory. Robert Jewett (with small modifications by B. Ćurgus)
Math 30 Introduction to Proofs via Number Theory Robert Jewett (with small modifications by B. Ćurgus) March 30, 009 Contents 1 The Integers 3 1.1 Axioms of Z...................................... 3 1.
More informationCSE101: Design and Analysis of Algorithms. Ragesh Jaiswal, CSE, UCSD
Recap. Growth rates: Arrange the following functions in ascending order of growth rate: n 2 log n n log n 2 log n n/ log n n n Introduction Algorithm: A step-by-step way of solving a problem. Design of
More informationThis Lecture. We will first introduce some basic set theory before we do counting. Basic Definitions. Operations on Sets.
Sets A B C This Lecture We will first introduce some basic set theory before we do counting. Basic Definitions Operations on Sets Set Identities Defining Sets Definition: A set is an unordered collection
More information9/19/12. Why Study Discrete Math? What is discrete? Sets (Rosen, Chapter 2) can be described by discrete math TOPICS
What is discrete? Sets (Rosen, Chapter 2) TOPICS Discrete math Set Definition Set Operations Tuples Consisting of distinct or unconnected elements, not continuous (calculus) Helps us in Computer Science
More informationarxiv: v2 [cs.ai] 18 Sep 2013
Lambda Dependency-Based Compositional Semantics Percy Liang September 19, 2013 arxiv:1309.4408v2 [cs.ai] 18 Sep 2013 Abstract This short note presents a new formal language, lambda dependency-based compositional
More informationSWEN-220 Mathematical Models of Software
SWEN-220 Mathematical Models of Software Introduction to Alloy Signatures, Fields, Facts, Predicates, and Assertions 2017 - Thomas Reichlmayr & Michael Lutz 1 Topics What is Alloy? Atoms, Sets, Relations
More informationFirst-Order Logic PREDICATE LOGIC. Syntax. Terms
First-Order Logic PREDICATE LOGIC Aim of this lecture: to introduce first-order predicate logic. More expressive than propositional logic. Consider the following argument: all monitors are ready; X12 is
More informationMathematics for Computer Scientists 2 (G52MC2)
Mathematics for Computer Scientists 2 (G52MC2) L07 : Operations on sets School of Computer Science University of Nottingham October 29, 2009 Enumerations We construct finite sets by enumerating a list
More informationA set with only one member is called a SINGLETON. A set with no members is called the EMPTY SET or 2 N
Mathematical Preliminaries Read pages 529-540 1. Set Theory 1.1 What is a set? A set is a collection of entities of any kind. It can be finite or infinite. A = {a, b, c} N = {1, 2, 3, } An entity is an
More informationFoundations of AI. 9. Predicate Logic. Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution
Foundations of AI 9. Predicate Logic Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution Wolfram Burgard, Andreas Karwath, Bernhard Nebel, and Martin Riedmiller 09/1 Contents Motivation
More information1. true / false By a compiler we mean a program that translates to code that will run natively on some machine.
1. true / false By a compiler we mean a program that translates to code that will run natively on some machine. 2. true / false ML can be compiled. 3. true / false FORTRAN can reasonably be considered
More informationPropositional Logic Formal Syntax and Semantics. Computability and Logic
Propositional Logic Formal Syntax and Semantics Computability and Logic Syntax and Semantics Syntax: The study of how expressions are structured (think: grammar) Semantics: The study of the relationship
More informationBinary Decision Diagrams
Logic and roof Hilary 2016 James Worrell Binary Decision Diagrams A propositional formula is determined up to logical equivalence by its truth table. If the formula has n variables then its truth table
More informationIntroduction to Sets and Logic (MATH 1190)
Introduction to Sets and Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University Dec 4, 2014 Outline 1 2 3 4 Definition A relation R from a set A to a set
More informationLecture 4. First order logic is a formal notation for mathematics which involves:
0368.4435 Automatic Software Verification April 14, 2015 Lecture 4 Lecturer: Mooly Sagiv Scribe: Nimrod Busany, Yotam Frank Lesson Plan 1. First order logic recap. 2. The SMT decision problem. 3. Basic
More informationCSE 215: Foundations of Computer Science Recitation Exercises Set #9 Stony Brook University. Name: ID#: Section #: Score: / 4
CSE 215: Foundations of Computer Science Recitation Exercises Set #9 Stony Brook University Name: ID#: Section #: Score: / 4 Unit 14: Set Theory: Definitions and Properties 1. Let C = {n Z n = 6r 5 for
More informationConsider a description of arithmetic. It includes two equations that define the structural types of digit and operator:
Syntax A programming language consists of syntax, semantics, and pragmatics. We formalize syntax first, because only syntactically correct programs have semantics. A syntax definition of a language lists
More informationDefinition: A context-free grammar (CFG) is a 4- tuple. variables = nonterminals, terminals, rules = productions,,
CMPSCI 601: Recall From Last Time Lecture 5 Definition: A context-free grammar (CFG) is a 4- tuple, variables = nonterminals, terminals, rules = productions,,, are all finite. 1 ( ) $ Pumping Lemma for
More informationLecture 5. Logic I. Statement Logic
Ling 726: Mathematical Linguistics, Logic. Statement Logic V. Borschev and B. Partee, September 27, 2 p. Lecture 5. Logic I. Statement Logic. Statement Logic...... Goals..... Syntax of Statement Logic....2.
More informationLTCS Report. Concept Descriptions with Set Constraints and Cardinality Constraints. Franz Baader. LTCS-Report 17-02
Technische Universität Dresden Institute for Theoretical Computer Science Chair for Automata Theory LTCS Report Concept Descriptions with Set Constraints and Cardinality Constraints Franz Baader LTCS-Report
More informationLogic: TD as search, Datalog (variables)
Logic: TD as search, Datalog (variables) Computer Science cpsc322, Lecture 23 (Textbook Chpt 5.2 & some basic concepts from Chpt 12) June, 8, 2017 CPSC 322, Lecture 23 Slide 1 Lecture Overview Recap Top
More informationSets. {1, 2, 3, Calvin}.
ets 2-24-2007 Roughly speaking, a set is a collection of objects. he objects are called the members or the elements of the set. et theory is the basis for mathematics, and there are a number of axiom systems
More informationMaterial from Recitation 1
Material from Recitation 1 Darcey Riley Frank Ferraro January 18, 2011 1 Introduction In CSC 280 we will be formalizing computation, i.e. we will be creating precise mathematical models for describing
More informationCMPSCI 250: Introduction to Computation. Lecture #7: Quantifiers and Languages 6 February 2012
CMPSCI 250: Introduction to Computation Lecture #7: Quantifiers and Languages 6 February 2012 Quantifiers and Languages Quantifier Definitions Translating Quantifiers Types and the Universe of Discourse
More informationComputer Science Technical Report
Computer Science Technical Report Feasibility of Stepwise Addition of Multitolerance to High Atomicity Programs Ali Ebnenasir and Sandeep S. Kulkarni Michigan Technological University Computer Science
More informationLecture 1: Conjunctive Queries
CS 784: Foundations of Data Management Spring 2017 Instructor: Paris Koutris Lecture 1: Conjunctive Queries A database schema R is a set of relations: we will typically use the symbols R, S, T,... to denote
More informationLecture Notes 15 Number systems and logic CSS Data Structures and Object-Oriented Programming Professor Clark F. Olson
Lecture Notes 15 Number systems and logic CSS 501 - Data Structures and Object-Oriented Programming Professor Clark F. Olson Number systems The use of alternative number systems is important in computer
More informationFinite Automata. Dr. Nadeem Akhtar. Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur
Finite Automata Dr. Nadeem Akhtar Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur PhD Laboratory IRISA-UBS University of South Brittany European University
More informationOperational Semantics 1 / 13
Operational Semantics 1 / 13 Outline What is semantics? Operational Semantics What is semantics? 2 / 13 What is the meaning of a program? Recall: aspects of a language syntax: the structure of its programs
More informationCore Membership Computation for Succinct Representations of Coalitional Games
Core Membership Computation for Succinct Representations of Coalitional Games Xi Alice Gao May 11, 2009 Abstract In this paper, I compare and contrast two formal results on the computational complexity
More informationTeacher Activity: page 1/9 Mathematical Expressions in Microsoft Word
Teacher Activity: page 1/9 Mathematical Expressions in Microsoft Word These instructions assume that you are familiar with using MS Word for ordinary word processing *. If you are not comfortable entering
More informationTh(N, +) is decidable
Theorem 6.12 Th(N, +) is decidable Presented by: Brian Lee Two Domains 1. We can give an algorithm to decide truth 2. A problem is undecidable First Order Logic Also known as First order predicate calculus
More informationReview of Sets. Review. Philippe B. Laval. Current Semester. Kennesaw State University. Philippe B. Laval (KSU) Sets Current Semester 1 / 16
Review of Sets Review Philippe B. Laval Kennesaw State University Current Semester Philippe B. Laval (KSU) Sets Current Semester 1 / 16 Outline 1 Introduction 2 Definitions, Notations and Examples 3 Special
More informationCS3110 Spring 2017 Lecture 12: DRAFT: Constructive Real Numbers continued
CS3110 Spring 2017 Lecture 12: DRAFT: Constructive Real Numbers continued Robert Constable Reminder: Prelim in class next Tuesday. It will not cover the real numbers beyond lecture 11 and comments on lecture
More information3.1 Constructions with sets
3 Interlude on sets Sets and functions are ubiquitous in mathematics. You might have the impression that they are most strongly connected with the pure end of the subject, but this is an illusion: think
More informationUNIVERSITY OF EDINBURGH COLLEGE OF SCIENCE AND ENGINEERING SCHOOL OF INFORMATICS INFR08008 INFORMATICS 2A: PROCESSING FORMAL AND NATURAL LANGUAGES
UNIVERSITY OF EDINBURGH COLLEGE OF SCIENCE AND ENGINEERING SCHOOL OF INFORMATICS INFR08008 INFORMATICS 2A: PROCESSING FORMAL AND NATURAL LANGUAGES Saturday 10 th December 2016 09:30 to 11:30 INSTRUCTIONS
More informationA computer implemented philosophy of mathematics
A computer implemented philosophy of mathematics M. Randall Holmes May 14, 2018 This paper presents a philosophical view of the basic foundations of mathematics, which is implemented in actual computer
More informationPropositional Calculus. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson
Propositional Calculus CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Objective: To provide students with the concepts and techniques from propositional calculus
More informationCOMP Logic for Computer Scientists. Lecture 17
COMP 1002 Logic for Computer Scientists Lecture 17 5 2 J Puzzle: the barber In a certain village, there is a (male) barber who shaves all and only those men of the village who do not shave themselves.
More informationLING 510, Lab 3 September 23, 2013
LING 510, Lab 3 September 23, 2013 Agenda: Go over Homework 1 Go over JYW, if there are questions Go over function application (what we ended with on Thursday) 1. Frequently missed questions on Homework
More informationCS 275 Automata and Formal Language Theory. First Problem of URMs. (a) Definition of the Turing Machine. III.3 (a) Definition of the Turing Machine
CS 275 Automata and Formal Language Theory Course Notes Part III: Limits of Computation Chapt. III.3: Turing Machines Anton Setzer http://www.cs.swan.ac.uk/ csetzer/lectures/ automataformallanguage/13/index.html
More informationCMPSCI 250: Introduction to Computation. Lecture #1: Things, Sets and Strings David Mix Barrington 22 January 2014
CMPSCI 250: Introduction to Computation Lecture #1: Things, Sets and Strings David Mix Barrington 22 January 2014 Things, Sets, and Strings The Mathematical Method Administrative Stuff The Objects of Mathematics
More informationCantor s Diagonal Argument for Different Levels of Infinity
JANUARY 2015 1 Cantor s Diagonal Argument for Different Levels of Infinity Michael J. Neely University of Southern California http://www-bcf.usc.edu/ mjneely Abstract These notes develop the classic Cantor
More informationAn Evolution of Mathematical Tools
An Evolution of Mathematical Tools From Conceptualization to Formalization Here's what we do when we build a formal model (or do a computation): 0. Identify a collection of objects/events in the real world.
More informationBoolean Algebra & Digital Logic
Boolean Algebra & Digital Logic Boolean algebra was developed by the Englishman George Boole, who published the basic principles in the 1854 treatise An Investigation of the Laws of Thought on Which to
More informationCHAPTER 10 GRAPHS AND TREES. Alessandro Artale UniBZ - artale/z
CHAPTER 10 GRAPHS AND TREES Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/z SECTION 10.1 Graphs: Definitions and Basic Properties Copyright Cengage Learning. All rights reserved. Graphs: Definitions
More informationUncertain Data Models
Uncertain Data Models Christoph Koch EPFL Dan Olteanu University of Oxford SYNOMYMS data models for incomplete information, probabilistic data models, representation systems DEFINITION An uncertain data
More informationConstructive Coherent Translation of Propositional Logic
Constructive Coherent Translation of Propositional Logic JRFisher@cpp.edu (started: 2009, latest: January 18, 2016) Abstract Propositional theories are translated to coherent logic rules using what are
More informationComplexity Theory. Compiled By : Hari Prasad Pokhrel Page 1 of 20. ioenotes.edu.np
Chapter 1: Introduction Introduction Purpose of the Theory of Computation: Develop formal mathematical models of computation that reflect real-world computers. Nowadays, the Theory of Computation can be
More information6. Relational Algebra (Part II)
6. Relational Algebra (Part II) 6.1. Introduction In the previous chapter, we introduced relational algebra as a fundamental model of relational database manipulation. In particular, we defined and discussed
More informationInduction and Semantics in Dafny
15-414 Lecture 11 1 Instructor: Matt Fredrikson Induction and Semantics in Dafny TA: Ryan Wagner Encoding the syntax of Imp Recall the abstract syntax of Imp: a AExp ::= n Z x Var a 1 + a 2 b BExp ::=
More informationIntroductory logic and sets for Computer scientists
Introductory logic and sets for Computer scientists Nimal Nissanke University of Reading ADDISON WESLEY LONGMAN Harlow, England II Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario
More information