King s College London
|
|
- Derrick Gardner
- 5 years ago
- Views:
Transcription
1 King s College London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board. MA EXAMINATION MODAL LOGIC Examination Period 2 (May/June 2016) TIME ALLOWED: TWO HOURS ANSWER TWO QUESTIONS, ONE FROM EACH SECTION NB Candidates should avoid overlap in their answers ANSWER EACH QUESTION ON A NEW PAGE OF YOUR ANSWER BOOK AND WRITE ITS NUMBER IN THE SPACE PROVIDED. DO NOT REMOVE THIS PAPER FROM THE EXAMINATION ROOM TURN OVER WHEN INSTRUCTED 2016 King s College London
2 SECTION A 1. (a) Is the following valid in T: (p q) ( p q)? Justify your answer. (b) Show that the characteristic thesis of the Brouwerian system, p P, is not valid in S4. (c) i. Provide a model to show that ( p p) is not valid in T. ii. Provide a model that shows that ( p p) is not valid in S4. (d) Provide an S5 model in which xp x x P x is false at some world. (e) Explain why, if domains are allowed to vary freely, the converse of the Barcan Formula is not valid even in T. 2. Let three frames (G 1, R 1 ), (G 2, R 2 ), (G 3, R 3 ), be defined by G 1 = {a}, R 1 = { a, a } G 2 = {a, b}, R 2 = { a, b, b, a, b, b } G 3 = {a, b}, R 3 = { a, a, a, b, b, b } ( x, y in R i means xr i y). (a) For each of the three frames determine which of the following formulas are valid on the frame. i. P P ii. P P iii. P P 2
3 iv. P P v. P P Moreover, if one of the formulas φ is not valid on frame (G i, R i ), give a world x in G i and a forcing relation between G i and {P } such that x φ. (b) Show that a frame (G, R) is reflexive if and only if every formula of the form P P is valid in it. 3. Use the propositional tableau proof systems to prove the following three formulas (a) (p p) in the T system, (b) ( p q) ( p q) in the system S4, (c) (( ( p p) ( p p)) in the system S5. Use the constant domain tableau system to determine whether (d) is valid on all K models with constant domain (d) ( ( x)a(x) ( x)b(x)) ( x) (A(x) B(x)). Use the variable domain tableau system to determine whether (e) is valid on all K models with varying domain (e) ( ( x)a(x) ( x)b(x)) ( x)(a(x) B(x)). 3
4 SECTION B 4. Given the following facts a closed tableau is not satisfiable, applying tableau extension rules to a satisfiable tableau results in a satisfiable tableau, an open tableau, in which all rules that can be applied have been applied, is satisfiable, if X has no L-proof, then there is a saturated tableau for X with an open branch, An open branch of a saturated tableau for X can be identified with a model with a world that makes X true. Show the following (a) The Tableau Method for Propositional Modal Logic is sound and complete. (b) If a tableau for P is satisfiable, then so is the tableau that results from applying the tableau extension rule for. (Of course, you may not now assume the second bullet point above). 5. In the quantified modal logic of Fitting & Mendelsohn: Some constants don t designate; some constants are rigid designators; some rigid designators designate existents and some don t; some constants are nonrigid; some of the constants that are nonrigid designate non existents; some of the constants that are nonrigid designate in other worlds but not in this one. Give examples of this variety and explain how the semantics treats the constants in each case. 4
5 6. (a) Describe Predicate Abstraction and explain what problem in quantified modal logic (without abstraction) it is meant to solve. (b) Formulate De Re and De Dicto readings of the following sentences in the language of quantified modal logic with predicate abstraction and indicate which of the readings is the more likely one: i. The mayor of London will be a Labour member ii. The number of planets is necessarily odd. (c) Give a formalisation of the premisses and conclusion in the language of quantified modal logic with predicate abstraction such that the following argument is valid: Ken knows that the Morning Star is the Evening Star. Ken doesn t know that the Morning Star is Venus. Ken doesn t know that the Evening Star is Venus. 5 FINAL PAGE
This paper is also taken by Combined Studies Students. Optional Subject (i): Set Theory and Further Logic
UNIVERSITY OF LONDON BA EXAMINATION for Internal Students This paper is also taken by Combined Studies Students PHILOSOPHY Optional Subject (i): Set Theory and Further Logic Answer THREE questions, at
More information(c) Establish the following claims by means of counterexamples.
sample paper I N T R O D U C T I O N T O P H I L O S O P H Y S E C T I O N A : L O G I C Volker Halbach Michaelmas 2008 I thank the author of the 2008 Trinity Term paper, Andrew Bacon, and Kentaro Fujimoto
More informationList of errors in and suggested modifications for First-Order Modal Logic Melvin Fitting and Richard L. Mendelsohn August 11, 2013
List of errors in and suggested modifications for First-Order Modal Logic Melvin Fitting and Richard L. Mendelsohn August 11, 2013 James W. Garson has answered a question we raised, in a paper that is
More informationSemantics and Pragmatics of NLP
Semantics and Pragmatics of NLP Alex Ewan School of Informatics University of Edinburgh 28 January 2008 1 2 3 Taking Stock We have: Introduced syntax and semantics for FOL plus lambdas. Represented FOL
More informationLogical Structures in Natural Language: First order Logic (FoL)
Logical Structures in Natural Language: First order Logic (FoL) Raffaella Bernardi Università degli Studi di Trento e-mail: bernardi@disi.unitn.it Contents 1 How far can we go with PL?................................
More information1 Completeness Theorem for First Order Logic
1 Completeness Theorem for First Order Logic There are many proofs of the Completeness Theorem for First Order Logic. We follow here a version of Henkin s proof, as presented in the Handbook of Mathematical
More informationFirst Order Logic: Syntax and Semantics
CS1081 First Order Logic: Syntax and Semantics COMP30412 Sean Bechhofer sean.bechhofer@manchester.ac.uk Problems Propositional logic isn t very expressive As an example, consider p = Scotland won on Saturday
More informationModal Logic & Kripke Semantics
Modal Logic & Kripke Semantics Modal logic allows one to model possible truths reasoning what is possible and what is simply not possible when we don t have complete knowledge. For example: it is possible
More informationThe Logical Contingency of Identity Hanoch Ben-Yami
The Logical Contingency of Identity Hanoch Ben-Yami ABSTRACT. I show that intuitive and logical considerations do not justify introducing Leibniz s Law of the Indiscernibility of Identicals in more than
More informationLogic: First Order Logic
Logic: First Order Logic Raffaella Bernardi bernardi@inf.unibz.it P.zza Domenicani 3, Room 2.28 Faculty of Computer Science, Free University of Bolzano-Bozen http://www.inf.unibz.it/~bernardi/courses/logic06
More informationWilliamson s Modal Logic as Metaphysics
Williamson s Modal Logic as Metaphysics Ted Sider Modality seminar 1. Methodology The title of this book may sound to some readers like Good as Evil, or perhaps Cabbages as Kings. If logic and metaphysics
More informationTHE LOGICAL CONTINGENCY OF IDENTITY. HANOCH BEN-YAMI Central European University ABSTRACT
EuJAP Vol. 14, No. 2, 2018 1 LEIBNIZ, G. W. 161.2 THE LOGICAL CONTINGENCY OF IDENTITY HANOCH BEN-YAMI Central European University Original scientific article Received: 23/03/2018 Accepted: 04/07/2018 ABSTRACT
More informationFormal Logic Lecture 11
Faculty of Philosophy Formal Logic Lecture 11 Peter Smith Peter Smith: Formal Logic, Lecture 11 1 Outline Where next? Introducing PL trees Branching trees Peter Smith: Formal Logic, Lecture 11 2 Where
More informationComputational Semantics Day 4: Extensionality and intensionality
Computational Semantics Day 4: Extensionality and intensionality Jan van Eijck 1 & Christina Unger 2 1 CWI, Amsterdam, and UiL-OTS, Utrecht, The Netherlands 2 CITEC, Bielefeld University, Germany ESSLLI
More informationIntroduction to Predicate Logic Part 1. Professor Anita Wasilewska Lecture Notes (1)
Introduction to Predicate Logic Part 1 Professor Anita Wasilewska Lecture Notes (1) Introduction Lecture Notes (1) and (2) provide an OVERVIEW of a standard intuitive formalization and introduction to
More informationIntroduction to Logic in Computer Science: Autumn 2006
Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Today s class will be an introduction
More informationUniversity of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura. February 9, :30 pm Duration: 1:50 hs. Closed book, no calculators
University of Ottawa CSI 2101 Midterm Test Instructor: Lucia Moura February 9, 2010 11:30 pm Duration: 1:50 hs Closed book, no calculators Last name: First name: Student number: There are 5 questions and
More informationLecture 13: Soundness, Completeness and Compactness
Discrete Mathematics (II) Spring 2017 Lecture 13: Soundness, Completeness and Compactness Lecturer: Yi Li 1 Overview In this lecture, we will prvoe the soundness and completeness of tableau proof system,
More informationSystems of modal logic
499 Modal and Temporal Logic Systems of modal logic Marek Sergot Department of Computing Imperial College, London utumn 2008 Further reading: B.F. Chellas, Modal logic: an introduction. Cambridge University
More informationKE/Tableaux. What is it for?
CS3UR: utomated Reasoning 2002 The term Tableaux refers to a family of deduction methods for different logics. We start by introducing one of them: non-free-variable KE for classical FOL What is it for?
More informationINTRODUCTION TO LOGIC 8 Identity and Definite Descriptions
8.1 Qualitative and Numerical Identity INTRODUCTION TO LOGIC 8 Identity and Definite Descriptions Volker Halbach Keith and Volker have the same car. Keith and Volker have identical cars. Keith and Volker
More informationNon-normal Worlds. Daniel Bonevac. February 5, 2012
Non-normal Worlds Daniel Bonevac February 5, 2012 Lewis and Langford (1932) devised five basic systems of modal logic, S1 - S5. S4 and S5, as we have seen, are normal systems, equivalent to K ρτ and K
More informationLecture 15: Validity and Predicate Logic
Lecture 15: Validity and Predicate Logic 1 Goals Today Learn the definition of valid and invalid arguments in terms of the semantics of predicate logic, and look at several examples. Learn how to get equivalents
More informationYork University. Faculty of Science and Engineering MATH 1090, Section M Final Examination, April NAME (print, in ink): Instructions, remarks:
York University Faculty of Science and Engineering MATH 1090, Section M ination, NAME (print, in ink): (Family name) (Given name) Instructions, remarks: 1. In general, carefully read all instructions in
More informationEXERCISE 10 SOLUTIONS
CSE541 EXERCISE 10 SOLUTIONS Covers Chapters 10, 11, 12 Read and learn all examples and exercises in the chapters as well! QUESTION 1 Let GL be the Gentzen style proof system for classical logic defined
More informationINTRODUCTION TO LOGIC 1 Sets, Relations, and Arguments. Why logic? Arguments
The Logic Manual INTRODUCTION TO LOGIC 1 Sets, Relations, and Arguments Volker Halbach Pure logic is the ruin of the spirit. Antoine de Saint-Exupéry The Logic Manual web page for the book: http://logicmanual.philosophy.ox.ac.uk/
More informationSection 2.1: Introduction to the Logic of Quantified Statements
Section 2.1: Introduction to the Logic of Quantified Statements In the previous chapter, we studied a branch of logic called propositional logic or propositional calculus. Loosely speaking, propositional
More informationStanford University CS103: Math for Computer Science Handout LN9 Luca Trevisan April 25, 2014
Stanford University CS103: Math for Computer Science Handout LN9 Luca Trevisan April 25, 2014 Notes for Lecture 9 Mathematical logic is the rigorous study of the way in which we prove the validity of mathematical
More information1 Completeness Theorem for Classical Predicate
1 Completeness Theorem for Classical Predicate Logic The relationship between the first order models defined in terms of structures M = [M, I] and valuations s : V AR M and propositional models defined
More informationCHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS
CHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS 1 Language There are several propositional languages that are routinely called classical propositional logic languages. It is due to the functional dependency
More informationCHAPTER 10. Predicate Automated Proof Systems
CHAPTER 10 ch10 Predicate Automated Proof Systems We define and discuss here a Rasiowa and Sikorski Gentzen style proof system QRS for classical predicate logic. The propositional version of it, the RS
More information240 Metaphysics. Frege s Puzzle. Chapter 26
240 Metaphysics Frege s Puzzle Frege s Puzzle 241 Frege s Puzzle In his 1879 Begriffsschrift (or Concept-Writing ), Gottlob Frege developed a propositional calculus to determine the truth values of propositions
More informationI thank the author of the examination paper on which sample paper is based. VH
I thank the author of the examination paper on which sample paper is based. VH 1. (a) Which of the following expressions is a sentence of L 1 or an abbreviation of one? If an expression is neither a sentence
More informationMarie Duží
Marie Duží marie.duzi@vsb.cz 1 Formal systems, Proof calculi A proof calculus (of a theory) is given by: 1. a language 2. a set of axioms 3. a set of deduction rules ad 1. The definition of a language
More informationSection 3.1: Direct Proof and Counterexample 1
Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion
More informationPhilosophy 244: #14 Existence and Identity
Philosophy 244: #14 Existence and Identity Existence Predicates The problem we ve been having is that (a) we want to allow models that invalidate the CBF ( xα x α), (b) these will have to be models in
More informationLogic I - Session 22. Meta-theory for predicate logic
Logic I - Session 22 Meta-theory for predicate logic 1 The course so far Syntax and semantics of SL English / SL translations TT tests for semantic properties of SL sentences Derivations in SD Meta-theory:
More informationCS280, Spring 2004: Final
CS280, Spring 2004: Final 1. [4 points] Which of the following relations on {0, 1, 2, 3} is an equivalence relation. (If it is, explain why. If it isn t, explain why not.) Just saying Yes or No with no
More informationMeaning and Reference INTENSIONAL AND MODAL LOGIC. Intensional Logic. Frege: Predicators (general terms) have
INTENSIONAL AND MODAL LOGIC Meaning and Reference Why do we consider extensions to the standard logical language(s)? Requirements of knowledge representation / domain modelling Intensional expressions:
More informationGenerality & Existence II
Generality & Existence II Modality & Quantifiers Greg Restall arché, st andrews 2 december 2015 My Aim To analyse the quantifiers Greg Restall Generality & Existence II 2 of 60 My Aim To analyse the quantifiers
More informationLogic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
More information5. And. 5.1 The conjunction
5. And 5.1 The conjunction To make our logical language more easy and intuitive to use, we can now add to it elements that make it able to express the equivalents of other sentences from a natural language
More informationPredicate Logic. CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo
Predicate Logic CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 22 Outline 1 From Proposition to Predicate
More informationDiscrete Mathematics
Slides for Part IA CST 2015/16 Discrete Mathematics Prof Marcelo Fiore Marcelo.Fiore@cl.cam.ac.uk What are we up to? Learn to read and write, and also work with,
More informationCOMP2411 Lecture 10: Propositional Logic Programming. Note: This material is not covered in the book. Resolution Applied to Horn Clauses
COMP2411 Lecture 10: Propositional Logic Programming Note: This material is not covered in the book Consider two Horn clauses Resolution Applied to Horn Clauses p p 1... p n and q q 1... q m Suppose these
More informationKripke on Frege on Sense and Reference. David Chalmers
Kripke on Frege on Sense and Reference David Chalmers Kripke s Frege Kripke s Frege Theory of Sense and Reference: Some Exegetical Notes Focuses on Frege on the hierarchy of senses and on the senses of
More informationTR : Binding Modalities
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2012 TR-2012011: Binding Modalities Sergei N. Artemov Tatiana Yavorskaya (Sidon) Follow this and
More informationNotation for Logical Operators:
Notation for Logical Operators: always true always false... and...... or... if... then...... if-and-only-if... x:x p(x) x:x p(x) for all x of type X, p(x) there exists an x of type X, s.t. p(x) = is equal
More informationPredicate Logic - Deductive Systems
CS402, Spring 2018 G for Predicate Logic Let s remind ourselves of semantic tableaux. Consider xp(x) xq(x) x(p(x) q(x)). ( xp(x) xq(x) x(p(x) q(x))) xp(x) xq(x), x(p(x) q(x)) xp(x), x(p(x) q(x)) xq(x),
More informationClassical Possibilism and Fictional Objects
Classical Possibilism and Fictional Objects Erich Rast erich@snafu.de Institute for the Philosophy of Language (IFL) Universidade Nova de Lisboa 15. July 2009 Overview 1 Actualism versus Possibilism 2
More informationApproximations of Modal Logic K
WoLLIC 2005 Preliminary Version Approximations of Modal Logic K Guilherme de Souza Rabello 2 Department of Mathematics Institute of Mathematics and Statistics University of Sao Paulo, Brazil Marcelo Finger
More informationGS03/4023: Validation and Verification Predicate Logic Jonathan P. Bowen Anthony Hall
GS03/4023: Validation and Verification Predicate Logic Jonathan P. Bowen www.cs.ucl.ac.uk/staff/j.bowen/gs03 Anthony Hall GS03 W1 L3 Predicate Logic 12 January 2007 1 Overview The need for extra structure
More informationCS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati
CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati Important 1. No questions about the paper will be entertained during
More informationINTRODUCTION TO PREDICATE LOGIC HUTH AND RYAN 2.1, 2.2, 2.4
INTRODUCTION TO PREDICATE LOGIC HUTH AND RYAN 2.1, 2.2, 2.4 Neil D. Jones DIKU 2005 Some slides today new, some based on logic 2004 (Nils Andersen), some based on kernebegreber (NJ 2005) PREDICATE LOGIC:
More informationINTRODUCTION TO LOGIC 8 Identity and Definite Descriptions
INTRODUCTION TO LOGIC 8 Identity and Definite Descriptions Volker Halbach The analysis of the beginning would thus yield the notion of the unity of being and not-being or, in a more reflected form, the
More informationReasoning Under Uncertainty: Introduction to Probability
Reasoning Under Uncertainty: Introduction to Probability CPSC 322 Uncertainty 1 Textbook 6.1 Reasoning Under Uncertainty: Introduction to Probability CPSC 322 Uncertainty 1, Slide 1 Lecture Overview 1
More informationThe Skolemization of existential quantifiers in intuitionistic logic
The Skolemization of existential quantifiers in intuitionistic logic Matthias Baaz and Rosalie Iemhoff Institute for Discrete Mathematics and Geometry E104, Technical University Vienna, Wiedner Hauptstrasse
More informationRecall 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 informationA Tableau Calculus for Minimal Modal Model Generation
M4M 2011 A Tableau Calculus for Minimal Modal Model Generation Fabio Papacchini 1 and Renate A. Schmidt 2 School of Computer Science, University of Manchester Abstract Model generation and minimal model
More informationPřednáška 12. Důkazové kalkuly Kalkul Hilbertova typu. 11/29/2006 Hilbertův kalkul 1
Přednáška 12 Důkazové kalkuly Kalkul Hilbertova typu 11/29/2006 Hilbertův kalkul 1 Formal systems, Proof calculi A proof calculus (of a theory) is given by: A. a language B. a set of axioms C. a set of
More informationMathematical Logic. Reasoning in First Order Logic. Chiara Ghidini. FBK-IRST, Trento, Italy
Reasoning in First Order Logic FBK-IRST, Trento, Italy April 12, 2013 Reasoning tasks in FOL Model checking Question: Is φ true in the interpretation I with the assignment a? Answer: Yes if I = φ[a]. No
More information2. Use quantifiers to express the associative law for multiplication of real numbers.
1. Define statement function of one variable. When it will become a statement? Statement function is an expression containing symbols and an individual variable. It becomes a statement when the variable
More informationModal Logics. Most applications of modal logic require a refined version of basic modal logic.
Modal Logics Most applications of modal logic require a refined version of basic modal logic. Definition. A set L of formulas of basic modal logic is called a (normal) modal logic if the following closure
More informationMATH 501: Discrete Mathematics
January 5, 016 : Final exam Model Solutions Instructions. Please read carefully before proceeding. (a) The duration of this exam is 180 minutes. (b) Non-programmable calculators are allowed. (c) No books
More informationReferences A CONSTRUCTIVE INTRODUCTION TO FIRST ORDER LOGIC. The Starting Point. Goals of foundational programmes for logic:
A CONSTRUCTIVE INTRODUCTION TO FIRST ORDER LOGIC Goals of foundational programmes for logic: Supply an operational semantic basis for extant logic calculi (ex post) Rational reconstruction of the practice
More informationLing 98a: The Meaning of Negation (Week 5)
Yimei Xiang yxiang@fas.harvard.edu 15 October 2013 1 Review Negation in propositional logic, oppositions, term logic of Aristotle Presuppositions Projection and accommodation Three-valued logic External/internal
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 informationFirst Order Logic Implication (4A) Young W. Lim 4/6/17
First Order Logic (4A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationBertrand Russell, Herbrand s Theorem, and the Assignment Statement
Bertrand Russell, Herbrand s Theorem, and the Assignment Statement Melvin Fitting Dept. Mathematics and Computer Science Lehman College (CUNY), Bronx, NY 10468 fitting@alpha.lehman.cuny.edu http://math240.lehman.cuny.edu/fitting
More informationRelevant Logic. Daniel Bonevac. March 20, 2013
March 20, 2013 The earliest attempts to devise a relevance logic that avoided the problem of explosion centered on the conditional. FDE, however, has no conditional operator, or a very weak one. If we
More informationFirst-Degree Entailment
March 5, 2013 Relevance Logics Relevance logics are non-classical logics that try to avoid the paradoxes of material and strict implication: p (q p) p (p q) (p q) (q r) (p p) q p (q q) p (q q) Counterintuitive?
More informationReasoning Under Uncertainty: Introduction to Probability
Reasoning Under Uncertainty: Introduction to Probability CPSC 322 Lecture 23 March 12, 2007 Textbook 9 Reasoning Under Uncertainty: Introduction to Probability CPSC 322 Lecture 23, Slide 1 Lecture Overview
More informationSymbolising Quantified Arguments
Symbolising Quantified Arguments 1. (i) Symbolise the following argument, given the universe of discourse is U = set of all animals. Animals are either male or female. Not all Cats are male, Therefore,
More information1 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 informationQuantification and Modality
Quantification and Modality Terry Langendoen Professor Emeritus of Linguistics University of Arizona Linguistics Colloquium University of Arizona 13 Mar 2009 Appearance The modal operators are like disguised
More informationPredicate Calculus - Semantic Tableau (2/2) Moonzoo Kim CS Division of EECS Dept. KAIST
Predicate Calculus - Semantic Tableau (2/2) Moonzoo Kim CS Division of EECS Dept. KAIST moonzoo@cs.kaist.ac.kr http://pswlab.kaist.ac.kr/courses/cs402-07 1 Formal construction is explained in two steps
More informationModal Epistemic Logic with Subjunctive Markers and Knowability
Modal Epistemic Logic with Subjunctive Markers and Knowability Dr. Helge Rückert University of Mannheim Germany rueckert@rumms.uni-mannheim.de http://www.phil.uni-mannheim.de/fakul/phil2/rueckert/index.html
More informationCPPE TURN OVER
[4] [1] [4] [16] The solutions are highly incomplete and only intended to give a rough idea. 1. (a) Which of the following expressions is an abbreviation of a sentence of L 1? If an expression is an abbreviation
More informationHandout: Proof of the completeness theorem
MATH 457 Introduction to Mathematical Logic Spring 2016 Dr. Jason Rute Handout: Proof of the completeness theorem Gödel s Compactness Theorem 1930. For a set Γ of wffs and a wff ϕ, we have the following.
More information1 FUNDAMENTALS OF LOGIC NO.8 SEMANTICS OF PREDICATE LOGIC Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth
More informationRecitation 4: Quantifiers and basic proofs
Math 299 Recitation 4: Quantifiers and basic proofs 1. Quantifiers in sentences are one of the linguistic constructs that are hard for computers to handle in general. Here is a nice pair of example dialogues:
More informationPrinciples of Knowledge Representation and Reasoning
Principles of Knowledge Representation and Reasoning Modal Logics Bernhard Nebel, Malte Helmert and Stefan Wölfl Albert-Ludwigs-Universität Freiburg May 2 & 6, 2008 Nebel, Helmert, Wölfl (Uni Freiburg)
More informationChapter 9. Modal Language, Syntax, and Semantics
Chapter 9 Modal Language, Syntax, and Semantics In chapter 6 we saw that PL is not expressive enough to represent valid arguments and semantic relationships that employ quantified expressions some and
More informationLogic: First Order Logic
Logic: First Order Logic Raffaella Bernardi bernardi@inf.unibz.it P.zza Domenicani 3, Room 2.28 Faculty of Computer Science, Free University of Bolzano-Bozen http://www.inf.unibz.it/~bernardi/courses/logic06
More informationarxiv: v4 [math.lo] 6 Apr 2018
Complexity of the interpretability logic IL arxiv:1710.05599v4 [math.lo] 6 Apr 2018 Luka Mikec luka.mikec@math.hr Fedor Pakhomov pakhfn@mi.ras.ru Monday 2 nd April, 2018 Abstract Mladen Vuković vukovic@math.hr
More informationRealization Using the Model Existence Theorem
Realization Using the Model Existence Theorem Melvin Fitting e-mail: melvin.fitting@lehman.cuny.edu web page: comet.lehman.cuny.edu/fitting May 15, 2013 Abstract Justification logics refine modal logics
More information5. And. 5.1 The conjunction
5. And 5.1 The conjunction To make our logical language more easy and intuitive to use, we can now add to it elements that make it able to express the equivalents of other sentences from a natural language
More informationPropositional Logic Arguments (5A) Young W. Lim 11/30/16
Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationCSE 1400 Applied Discrete Mathematics Definitions
CSE 1400 Applied Discrete Mathematics Definitions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Arithmetic 1 Alphabets, Strings, Languages, & Words 2 Number Systems 3 Machine
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 informationDoc112: Hardware. Department of Computing, Imperial College London. Doc112: Hardware Lecture 1 Slide 1
Doc112: Hardware Department of Computing, Imperial College London Doc112: Hardware Lecture 1 Slide 1 First Year Computer Hardware Course Lecturers Duncan Gillies Bjoern Schuller Doc112: Hardware Lecture
More informationS4LP and Local Realizability
S4LP and Local Realizability Melvin Fitting Lehman College CUNY 250 Bedford Park Boulevard West Bronx, NY 10548, USA melvin.fitting@lehman.cuny.edu Abstract. The logic S4LP combines the modal logic S4
More informationSection 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 informationExercises for the Logic Course
Exercises for the Logic Course First Order Logic Course Web Page http://www.inf.unibz.it/~artale/dml/dml.htm Computer Science Free University of Bozen-Bolzano December 22, 2017 1 Exercises 1.1 Formalisation
More informationPredicate 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 informationPropositional Logic Arguments (5A) Young W. Lim 11/29/16
Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationPredicate Logic: Sematics Part 1
Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with
More informationINTRODUCTION TO LOGIC
INTRODUCTION TO LOGIC 6 Natural Deduction Volker Halbach There s nothing you can t prove if your outlook is only sufficiently limited. Dorothy L. Sayers http://www.philosophy.ox.ac.uk/lectures/ undergraduate_questionnaire
More informationLogic in Computer Science (COMP118) Tutorial Problems 1
Logic in Computer Science (COMP118) Tutorial Problems 1 1. Let p 1 denote the proposition: Tom s house is red; p 2 denote the proposition: Jim s house is red; p 3 denote the proposition: Mary s house is
More informationCOMP219: Artificial Intelligence. Lecture 19: Logic for KR
COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof
More information