AI Principles, Semester 2, Week 2, Lecture 4 Introduction to Logic Thinking, reasoning and deductive logic Validity of arguments, Soundness of

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "AI Principles, Semester 2, Week 2, Lecture 4 Introduction to Logic Thinking, reasoning and deductive logic Validity of arguments, Soundness of"

Transcription

1 AI Principles, Semester 2, Week 2, Lecture 4 Introduction to Logic Thinking, reasoning and deductive logic Validity of arguments, Soundness of arguments Formal systems Axioms, Inference, and Proof Propositional Logic and Predicate Logic are formal systems Semantics 1

2 Review of logic from last semester What did you learn about logic in last semesters lectures, (or know from elsewhere)? 2

3 Different kinds of movement of thought (from Guttenplan 1986, chapter 1) Three descriptions of an episode of thinking. One could characterise them as movements of thought. The subject in each case has one, and then other thoughts, which together form a series through which there is a kind of motion. Logic could be seen as containing laws of this sort of motion. Not all movements of thought of thought are the concern of logic 3

4 How do thoughts differ? (from Guttenplan 1986, chapter 1) Example 1: Brown is sitting at his desk gazing out of the window. He notices that the buds are just beginning to open on the trees. This reminds him of the unusually warm weather which was experienced last year at this time. That thought prompts the further thought that he must have his central heating boiler seen to as soon as possible. 4

5 How do thoughts differ? (from Guttenplan 1986, chapter 1) Example 2: Smith finds that her car won t start. She remembers that when Jones s car failed to start, it was because the distributor was wet. She also recalls reading that distributor problems are common in the sort of car she has. She is aware of how damp it is today. She concludes, therefore, that a wet distributor is the cause of the trouble. 5

6 How do thoughts differ? (from Guttenplan 1986, chapter 1) Example 3: Green is planning his summer holiday. He knows that he can go by aeroplane or car. If he goes by aeroplane, he will get there faster, but will be unable to take much luggage. If he goes by car, he can take much more. He recognises that the success of his holiday depends on his having the right sort of clothing for the unpredictable weather. He could not take the needed clothing on the aeroplane. He concludes that if the holiday is to be successful, he will have to go by car. 6

7 Questions to ponder about the three descriptions of thought What is the difference between Brown s thoughts, and the other two kinds of thoughts? Which thoughts might be described as making arguments? How do Smith and Green s thoughts differ? Which of these three kinds of thought shows reasoning? Which of these kinds of thought shows deductive reasoning? 7

8 Describing the three styles of thought Brown s thoughts. We might describe Brown s thoughts as associative thinking or thought by spreading activation. Patterns in his perceptual stimuli gives rise to a chain of thoughts which activate further thoughts. This kind of pattern recognition is not usually studied as part of logic reasoning, but may be studied with other AI representations, such as Semantic Networks or Artificial Neural Networks. Smith s thoughts. We might describe Smith s thoughts as a form of inductive reasoning. Smith relies upon past events and consideration of probabilities to inform her decisions. This kind of thought is termed inductive logic. Green s thoughts. Green s thoughts might be described as using deductive reasoning. In the description of Green s thought has Ifthen statements and facts, and makes conclusions after reasoning about these conditional statements and facts. 8

9 Inductive and deductive reasoning both share the concept of truth and falsity that may not be shared by other types of thinking. Inductive reasoning is going from true statements to true statements but it is a shaky process Deductive reasoning assumes the premises to true. Then if the premises are true the conclusions have to be true. There is nothing shaky about the transition. For deductive reasoning we can see that the process of going from premise to conclusions is independent of the truth of any assumptions. Arguments have premises and conclusions that may be true or false. When reasoning is deductive there is only one case that can be ruled out: that is when the premises for the argument are all true and the conclusion is false. 9

10 Valid and invalid deductive arguments Validity = An argument is VALID if and only if it is necessary that if all it premises are true, its conclusion is true The intuitive idea captured by this definition is this: If it is possible for the conclusion of an argument to be false when its premises are all true, then the argument is not reliable (that is, it is invalid) If true premises guarantee a true conclusion then the argument is valid. Alternatively, an argument is VALID if and only if it is impossible for all the premises to be true while the conclusion is false. When an argument is valid its premises ENTAIL its conclusion. 10

11 Examples of deductive arguments? The sun has shone every day for thousands of years, therefore the sun will shine tomorrow The sun is using up fuel which has a finite supply, therefore one day the sun will no longer shine If he is Napolean Bonaparte, then he is the Emperor of France He believes he is Napolean Bonaparte, therefore he believes he is Emperor of France (modal logic for reasoning about beliefs in 4 th logic lecture) 11

12 Sound arguments An argument is SOUND if and only if it is valid and all its premises are true. All sound arguments have true conclusions. An argument may be unsound in two ways: if it is invalid, or it has one or more false premises. 12

13 More on validity and soundness of arguments Why are the concepts of validity and soundness important in automated reasoning? If you have a reasoning system where all the arguments it produces are valid, how can you ensure that they are also sound? 13

14 Representing arguments in formal systems Logical deductive arguments have been represented in natural language since the time of Classical Greek philosophers. However, what drawbacks are there in using natural language to automate reasoning? 14

15 Representing arguments in formal systems Logical deductive arguments have been represented in natural language since the time of Classical Greek philosophers. However, what drawbacks are there in using natural language to automate reasoning? Natural language is ambigous, the automated processing of natural language is a difficult challenge in its own right. An answer is to represent argument and reasoning within formal systems 15

16 Formal Systems Formal systems in mathematics possess a number of elements: 1. A finite set of symbols 2. A grammar (syntax), that is a way of constructing well-formed formulae (wff) out of the symbols. (There should exist a decision procedure that can always tell whether a formulae is a wff) 3. A set of axioms 4. A set of inference rules 5. A set of theorems. This set includes all the axioms, plus all wffs which can be derived from previous-derived theorems by means of rules of inference. (There may not exist a decision procedure for deciding whether a wff is a theorem or not) *a formal system does not require any axioms to start with 16

17 The MU Puzzle In the MU puzzle, what were the: 1. set of symbols 2. grammar, 3. set of axioms 4. set of inference rules 5. set of theorems. 17

18 Definition of Proof within a formal system A sequence of wffs [ω 1, ω 2,...ω n is called a proof (or a deduction) of ω n from a set of wffs iff each ω i in the sequence is either in or can be inferred from a wff (or wffs) earlier in the sequence by using one of the rules of inference. If there is a proof of ω n from, we say that ω n is a theorem of the set ω n (ω n can be proved from ) R ω n (ω n can be proved from using the inference rules in R) (Nilsson page 221) 18

19 Definition of Proof within a formal system and linking inference with entailment A proof in a formal system is a sequence of well formed formulas leading from the initial set of well formed formulae to the new well formed formulae that is the subject of the proof 19

20 Propositional Logic and Predicate Logic are formal systems Both Propositional Logic and Predicate Logic are types of formal system, and we will spend the next few lectures learning how to form and use expressions in these logics. In the final lecture we will take a brief look at how to automate reasoning in Propositional Logic, and look at some other logics, which include Modal Logic, Temporal Logic, and Fuzzy Logic. What all these different logics possess in addition to syntax and rules of inference is semantics 20

21 Semantics Semantics has to do with associating elements of a logical language with aspects of the real world. In propositional logic, logical atoms are associated with propositions about the world. An association of atoms with propositions is called an interpretation. (Russel and Norvig page 203, Nilsson page 222) 21

22 Sentences Entails Sentences Semantics Representation World Semantics Aspect of the real world Follows Aspect of the real world 22

23 Conclusion Different kinds of thought and reasoning Whats it got to do with the MU puzzle Propositional Logic and Predicate Logic are formal systems Semantics 23

mywbut.com Propositional Logic inference rules

mywbut.com Propositional Logic inference rules Propositional Logic inference rules 1 5.5 Rules of Inference Here are some examples of sound rules of inference. Each can be shown to be sound once and for all using a truth table. The left column contains

More information

The Predicate Calculus in AI

The Predicate Calculus in AI Last time, we: The Predicate Calculus in AI Motivated the use of Logic as a representational language for AI (Can derive new facts syntactically - simply by pushing symbols around) Described propositional

More information

Propositional Logic and Methods of Inference SEEM

Propositional Logic and Methods of Inference SEEM Propositional Logic and Methods of Inference SEEM 5750 1 Logic Knowledge can also be represented by the symbols of logic, which is the study of the rules of exact reasoning. Logic is also of primary importance

More information

Predicate logic Proofs Artificial intelligence. Predicate logic. SET07106 Mathematics for Software Engineering

Predicate logic Proofs Artificial intelligence. Predicate logic. SET07106 Mathematics for Software Engineering Predicate logic SET07106 Mathematics for Software Engineering School of Computing Edinburgh Napier University Module Leader: Uta Priss 2010 Copyright Edinburgh Napier University Predicate logic Slide 1/24

More information

CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

More information

Lecture 13 of 41. More Propositional and Predicate Logic

Lecture 13 of 41. More Propositional and Predicate Logic Lecture 13 of 41 More Propositional and Predicate Logic Monday, 20 September 2004 William H. Hsu, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Sections 8.1-8.3, Russell and Norvig

More information

Discrete Mathematics

Discrete Mathematics Slides for Part IA CST 2014/15 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, mathematical

More information

Handout #1: Mathematical Reasoning

Handout #1: Mathematical Reasoning Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

More information

Artificial Intelligence Automated Reasoning

Artificial Intelligence Automated Reasoning Artificial Intelligence Automated Reasoning Andrea Torsello Automated Reasoning Very important area of AI research Reasoning usually means deductive reasoning New facts are deduced logically from old ones

More information

Deductive Systems. Marco Piastra. Artificial Intelligence. Artificial Intelligence - A.A Deductive Systems [1]

Deductive Systems. Marco Piastra. Artificial Intelligence. Artificial Intelligence - A.A Deductive Systems [1] Artificial Intelligence Deductive Systems Marco Piastra Artificial Intelligence - A.A. 2012- Deductive Systems 1] Symbolic calculus? A wff is entailed by a set of wff iff every model of is also model of

More information

CS510 Software Engineering

CS510 Software Engineering CS510 Software Engineering Propositional Logic Asst. Prof. Mathias Payer Department of Computer Science Purdue University TA: Scott A. Carr Slides inspired by Xiangyu Zhang http://nebelwelt.net/teaching/15-cs510-se

More information

Notes on Modal Logic

Notes on Modal Logic Notes on Modal Logic Notes for Philosophy 151 Eric Pacuit January 28, 2009 These short notes are intended to supplement the lectures and text ntroduce some of the basic concepts of Modal Logic. The primary

More information

MATHS 315 Mathematical Logic

MATHS 315 Mathematical Logic MATHS 315 Mathematical Logic Second Semester, 2006 Contents 2 Formal Statement Logic 1 2.1 Post production systems................................. 1 2.2 The system L.......................................

More information

Truth Conditional Meaning of Sentences. Ling324 Reading: Meaning and Grammar, pg

Truth Conditional Meaning of Sentences. Ling324 Reading: Meaning and Grammar, pg Truth Conditional Meaning of Sentences Ling324 Reading: Meaning and Grammar, pg. 69-87 Meaning of Sentences A sentence can be true or false in a given situation or circumstance. (1) The pope talked to

More information

2nd Computer Science and Engineering, 1st Computer Science and Engineering & Mathematics

2nd Computer Science and Engineering, 1st Computer Science and Engineering & Mathematics 1. COURSE TITLE Discrete Structures and Logic 1.1. Course number 17824 1.2. Course area Discrete Structures and Logic 1.3. Course type Core course 1.4. Course level Undergraduate 1.5. Year 2nd Computer

More information

Fundamentals of Mathematics Lecture 6: Propositional Logic

Fundamentals of Mathematics Lecture 6: Propositional Logic Fundamentals of Mathematics Lecture 6: Propositional Logic Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T F

More information

CSE 459/598: Logic for Computer Scientists (Spring 2012)

CSE 459/598: Logic for Computer Scientists (Spring 2012) CSE 459/598: Logic for Computer Scientists (Spring 2012) Time and Place: T Th 10:30-11:45 a.m., M1-09 Instructor: Joohyung Lee (joolee@asu.edu) Instructor s Office Hours: T Th 4:30-5:30 p.m. and by appointment

More information

DEDUCTIVE & INDUCTIVE REASONING

DEDUCTIVE & INDUCTIVE REASONING DEDUCTIVE & INDUCTIVE REASONING Expectations 1. Take notes on inductive and deductive reasoning. 2. This is an information based presentation -- I simply want you to be able to apply this information to

More information

Encoding Mathematics in First-Order Logic

Encoding Mathematics in First-Order Logic Applied Logic Lecture 19 CS 486 Spring 2005 Thursday, April 14, 2005 Encoding Mathematics in First-Order Logic Over the past few lectures, we have seen the syntax and the semantics of first-order logic,

More information

Rigorous Software Development CSCI-GA 3033-009

Rigorous Software Development CSCI-GA 3033-009 Rigorous Software Development CSCI-GA 3033-009 Instructor: Thomas Wies Spring 2013 Lecture 11 Semantics of Programming Languages Denotational Semantics Meaning of a program is defined as the mathematical

More information

(Refer Slide Time: 05:02)

(Refer Slide Time: 05:02) Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 1 Propositional Logic This course is about discrete

More information

The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs

The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments

More information

Course Outline Department of Computing Science Faculty of Science. COMP 3710-3 Applied Artificial Intelligence (3,1,0) Fall 2015

Course Outline Department of Computing Science Faculty of Science. COMP 3710-3 Applied Artificial Intelligence (3,1,0) Fall 2015 Course Outline Department of Computing Science Faculty of Science COMP 710 - Applied Artificial Intelligence (,1,0) Fall 2015 Instructor: Office: Phone/Voice Mail: E-Mail: Course Description : Students

More information

A Few Basics of Probability

A Few Basics of Probability A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study

More information

Foundations of Artificial Intelligence

Foundations of Artificial Intelligence Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard, Bernhard Nebel and Martin Riedmiller Albert-Ludwigs-Universität Freiburg Contents 1 Agents

More information

Lecture 3. Mathematical Induction

Lecture 3. Mathematical Induction Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

More information

CS 441 Discrete Mathematics for CS Lecture 2. Propositional logic. CS 441 Discrete mathematics for CS. Course administration

CS 441 Discrete Mathematics for CS Lecture 2. Propositional logic. CS 441 Discrete mathematics for CS. Course administration CS 441 Discrete Mathematics for CS Lecture 2 Propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Course administration Homework 1 First homework assignment is out today will be posted

More information

2. The Language of First-order Logic

2. The Language of First-order Logic 2. The Language of First-order Logic KR & R Brachman & Levesque 2005 17 Declarative language Before building system before there can be learning, reasoning, planning, explanation... need to be able to

More information

Chapter 1 LOGIC AND PROOF

Chapter 1 LOGIC AND PROOF Chapter 1 LOGIC AND PROOF To be able to understand mathematics and mathematical arguments, it is necessary to have a solid understanding of logic and the way in which known facts can be combined to prove

More information

Introduction to formal semantics -

Introduction to formal semantics - Introduction to formal semantics - Introduction to formal semantics 1 / 25 structure Motivation - Philosophy paradox antinomy division in object und Meta language Semiotics syntax semantics Pragmatics

More information

Beyond Propositional Logic Lukasiewicz s System

Beyond Propositional Logic Lukasiewicz s System Beyond Propositional Logic Lukasiewicz s System Consider the following set of truth tables: 1 0 0 1 # # 1 0 # 1 1 0 # 0 0 0 0 # # 0 # 1 0 # 1 1 1 1 0 1 0 # # 1 # # 1 0 # 1 1 0 # 0 1 1 1 # 1 # 1 Brandon

More information

Predicate Logic. PHI 201 Introductory Logic Spring 2011

Predicate Logic. PHI 201 Introductory Logic Spring 2011 Predicate Logic PHI 201 Introductory Logic Spring 2011 This is a summary of definitions in Predicate Logic from the text The Logic Book by Bergmann et al. 1 The Language PLE Vocabulary The vocabulary of

More information

R-Calculus: A Logical Framework for Scientific Discovery

R-Calculus: A Logical Framework for Scientific Discovery R-Calculus: A Logical Framework for Scientific Discovery Wei Li State Key Laboratory of Software Development Environment Beihang University August 9, 2013 Outline 1 Motivation 2 Key points of R-calculus

More information

Definition 10. A proposition is a statement either true or false, but not both.

Definition 10. A proposition is a statement either true or false, but not both. Chapter 2 Propositional Logic Contrariwise, continued Tweedledee, if it was so, it might be; and if it were so, it would be; but as it isn t, it ain t. That s logic. (Lewis Carroll, Alice s Adventures

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

More information

Correspondence analysis for strong three-valued logic

Correspondence analysis for strong three-valued logic Correspondence analysis for strong three-valued logic A. Tamminga abstract. I apply Kooi and Tamminga s (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (K 3 ).

More information

Likewise, we have contradictions: formulas that can only be false, e.g. (p p).

Likewise, we have contradictions: formulas that can only be false, e.g. (p p). CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formula

More information

Logic in general. Inference rules and theorem proving

Logic in general. Inference rules and theorem proving Logical Agents Knowledge-based agents Logic in general Propositional logic Inference rules and theorem proving First order logic Knowledge-based agents Inference engine Knowledge base Domain-independent

More information

CS532, Winter 2010 Lecture Notes: First-Order Logic: Syntax and Semantics

CS532, Winter 2010 Lecture Notes: First-Order Logic: Syntax and Semantics CS532, Winter 2010 Lecture Notes: First-Order Logic: Syntax and Semantics Dr Alan Fern, afern@csorstedu January 8, 2010 1 Limits of Propositional Logic Propositional logic assumes that the world or system

More information

Logic and Proofs. Chapter 1

Logic and Proofs. Chapter 1 Section 1.0 1.0.1 Chapter 1 Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs

More information

INDUCTIVE & DEDUCTIVE RESEARCH APPROACH

INDUCTIVE & DEDUCTIVE RESEARCH APPROACH INDUCTIVE & DEDUCTIVE RESEARCH APPROACH Meritorious Prof. Dr. S. M. Aqil Burney Director UBIT Chairman Department of Computer Science University of Karachi burney@computer.org www.drburney.net Designed

More information

Summary Last Lecture. Automated Reasoning. Outline of the Lecture. Definition sequent calculus. Theorem (Normalisation and Strong Normalisation)

Summary Last Lecture. Automated Reasoning. Outline of the Lecture. Definition sequent calculus. Theorem (Normalisation and Strong Normalisation) Summary Summary Last Lecture sequent calculus Automated Reasoning Georg Moser Institute of Computer Science @ UIBK Winter 013 (Normalisation and Strong Normalisation) let Π be a proof in minimal logic

More information

Consistency, completeness of undecidable preposition of Principia Mathematica. Tanmay Jaipurkar

Consistency, completeness of undecidable preposition of Principia Mathematica. Tanmay Jaipurkar Consistency, completeness of undecidable preposition of Principia Mathematica Tanmay Jaipurkar October 21, 2013 Abstract The fallowing paper discusses the inconsistency and undecidable preposition of Principia

More information

CS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers

CS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x)

More information

Subjects, Predicates and The Universal Quantifier

Subjects, Predicates and The Universal Quantifier Introduction to Logic Week Sixteen: 14 Jan., 2008 Subjects, Predicates and The Universal Quantifier 000. Office Hours this term: Tuesdays and Wednesdays 1-2, or by appointment; 5B-120. 00. Results from

More information

Mathematical Logic. Tableaux Reasoning for Propositional Logic. Chiara Ghidini. FBK-IRST, Trento, Italy

Mathematical Logic. Tableaux Reasoning for Propositional Logic. Chiara Ghidini. FBK-IRST, Trento, Italy Tableaux Reasoning for Propositional Logic FBK-IRST, Trento, Italy Outline of this lecture An introduction to Automated Reasoning with Analytic Tableaux; Today we will be looking into tableau methods for

More information

DISCRETE MATH: LECTURE 3

DISCRETE MATH: LECTURE 3 DISCRETE MATH: LECTURE 3 DR. DANIEL FREEMAN 1. Chapter 2.2 Conditional Statements If p and q are statement variables, the conditional of q by p is If p then q or p implies q and is denoted p q. It is false

More information

Artificial Intelligence (LISP)

Artificial Intelligence (LISP) Artificial Intelligence (LISP) Introduction Artificial Intelligence (AI) is a broad field, and means different things to different people. It is concerned with getting computers to do tasks that require

More information

Gödel s Theorem: An Incomplete Guide to Its Use and Abuse

Gödel s Theorem: An Incomplete Guide to Its Use and Abuse Gödel s Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzén Summary by Xavier Noria August 2006 fxn@hashref.com This unique exposition of Kurt Gödel s stunning incompleteness theorems for

More information

Cognitive Logic versus Mathematical Logic

Cognitive Logic versus Mathematical Logic Cognitive Logic versus Mathematical Logic Pei Wang Department of Computer and Information Sciences Temple University pei.wang@temple.edu http://www.cis.temple.edu/ pwang/ Abstract First-order predicate

More information

Automation Principles

Automation Principles Automation Principles 8/15/07 These principles concern finding efficient computational ways to perform human tasks. Tasks can be physical, such as running an assembly line, driving a car, controlling airplane

More information

LECTURE 10: FIRST-ORDER LOGIC. Software Engineering Mike Wooldridge

LECTURE 10: FIRST-ORDER LOGIC. Software Engineering Mike Wooldridge LECTURE 10: FIRST-ORDER LOGIC Mike Wooldridge 1 Why not Propositional Logic? Consider the following statements: all monitors are ready; X12 is a monitor. We saw in an earlier lecture that these statements

More information

First-Order Predicate Logic (2)

First-Order Predicate Logic (2) First-Order Predicate Logic (2) Predicate Logic (2) Understanding first-order predicate logic formulas. Satisfiability and undecidability of satisfiability. Tautology, logical consequence, and logical

More information

The epistemic structure of de Finetti s betting problem

The epistemic structure of de Finetti s betting problem The epistemic structure of de Finetti s betting problem Tommaso Flaminio 1 and Hykel Hosni 2 1 IIIA - CSIC Campus de la Univ. Autònoma de Barcelona s/n 08193 Bellaterra, Spain. Email: tommaso@iiia.csic.es

More information

Propositional Logic. 1. Semantics and Propositions. LX Semantics September 19, 2008

Propositional Logic. 1. Semantics and Propositions. LX Semantics September 19, 2008 Propositional Logic LX 502 - Semantics September 19, 2008 1. Semantics and Propositions Natural language is used to communicate information about the world, typically between a speaker and an addressee.

More information

Fixed-Point Logics and Computation

Fixed-Point Logics and Computation 1 Fixed-Point Logics and Computation Symposium on the Unusual Effectiveness of Logic in Computer Science University of Cambridge 2 Mathematical Logic Mathematical logic seeks to formalise the process of

More information

Satisfiability Checking

Satisfiability Checking Satisfiability Checking First-Order Logic Prof. Dr. Erika Ábrahám RWTH Aachen University Informatik 2 LuFG Theory of Hybrid Systems WS 14/15 Satisfiability Checking Prof. Dr. Erika Ábrahám (RWTH Aachen

More information

Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

More information

Vagueness at the interface between logic, philosophy, and linguistics

Vagueness at the interface between logic, philosophy, and linguistics LoMoReVI Cejkovice, September 17 Vagueness at the interface between logic, philosophy, and linguistics Pleasures and pitfalls of interdisciplinarity Chris Fermüller Vienna University of Technology Theory

More information

CS 2710 Foundations of AI. Lecture 12. Inference in FOL. CS 2710 Foundations of AI. Optimization competition winners.

CS 2710 Foundations of AI. Lecture 12. Inference in FOL. CS 2710 Foundations of AI. Optimization competition winners. Lecture 12 Inference in FOL Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Optimization competition winners CS2710 Homework 3 Simulated annealing 1. Valko Michal 2. Zhang Yue 3. Villamarin Ricardo

More information

Self-Referential Probabilities

Self-Referential Probabilities Self-Referential Probabilities And a Kripkean semantics Catrin Campbell-Moore Munich Center for Mathematical Philosophy Bridges 2 September 2015 Supported by Introduction Languages that can talk about

More information

Predicate Logic. M.A.Galán, TDBA64, VT-03

Predicate Logic. M.A.Galán, TDBA64, VT-03 Predicate Logic 1 Introduction There are certain arguments that seem to be perfectly logical, yet they cannot be specified by using propositional logic. All cats have tails. Tom is a cat. From these two

More information

Introduction to Logic: Argumentation and Interpretation. Vysoká škola mezinárodních a veřejných vztahů PhDr. Peter Jan Kosmály, Ph.D. 9. 3.

Introduction to Logic: Argumentation and Interpretation. Vysoká škola mezinárodních a veřejných vztahů PhDr. Peter Jan Kosmály, Ph.D. 9. 3. Introduction to Logic: Argumentation and Interpretation Vysoká škola mezinárodních a veřejných vztahů PhDr. Peter Jan Kosmály, Ph.D. 9. 3. 2016 tests. Introduction to Logic: Argumentation and Interpretation

More information

General Philosophy. Dr Peter Millican, Hertford College. Lecture 3: Induction

General Philosophy. Dr Peter Millican, Hertford College. Lecture 3: Induction General Philosophy Dr Peter Millican, Hertford College Lecture 3: Induction Hume s s Fork 2 Enquiry IV starts with a vital distinction between types of proposition: Relations of ideas can be known a priori

More information

SQL INJECTION ATTACKS By Zelinski Radu, Technical University of Moldova

SQL INJECTION ATTACKS By Zelinski Radu, Technical University of Moldova SQL INJECTION ATTACKS By Zelinski Radu, Technical University of Moldova Where someone is building a Web application, often he need to use databases to store information, or to manage user accounts. And

More information

COMP 250 Fall Mathematical induction Sept. 26, (n 1) + n = n + (n 1)

COMP 250 Fall Mathematical induction Sept. 26, (n 1) + n = n + (n 1) COMP 50 Fall 016 9 - Mathematical induction Sept 6, 016 You will see many examples in this course and upcoming courses of algorithms for solving various problems It many cases, it will be obvious that

More information

The psychological theory of persons

The psychological theory of persons The psychological theory of persons Last week were discussing dualist views of persons, according to which human beings are immaterial things distinct from their bodies. We closed by discussing some problems

More information

PHILOSOPHY 4360/5360 METAPHYSICS

PHILOSOPHY 4360/5360 METAPHYSICS PHILOSOPHY 4360/5360 METAPHYSICS Topic IV: The Nature of the Mind Arguments for (Persisting) Substance Dualism Argument 1: The Modal Argument from Personal Identity This first type of argument is advanced

More information

-123- A Three-Valued Interpretation for a Relevance Logic. In thi.s paper an entailment relation which holds between certain

-123- A Three-Valued Interpretation for a Relevance Logic. In thi.s paper an entailment relation which holds between certain Dr. Frederick A. Johnson Department of Philosophy Colorado State University Fort Collins, Colorado 80523-23- A Three-Valued Interpretation for a Relevance Logic In thi.s paper an entailment relation which

More information

2. Propositional Equivalences

2. Propositional Equivalences 2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p Definition

More information

Formal Logic Lecture 2

Formal Logic Lecture 2 Faculty of Philosophy Formal Logic Lecture 2 Peter Smith Peter Smith: Formal Logic, Lecture 2 1 Outline Validity again Systematicity and formality Modality and the invalidity principle The counterexample

More information

conditional statement conclusion Vocabulary Flash Cards Chapter 2 (p. 66) Chapter 2 (p. 69) Chapter 2 (p. 66) Chapter 2 (p. 76)

conditional statement conclusion Vocabulary Flash Cards Chapter 2 (p. 66) Chapter 2 (p. 69) Chapter 2 (p. 66) Chapter 2 (p. 76) biconditional statement conclusion Chapter 2 (p. 69) conditional statement conjecture Chapter 2 (p. 76) contrapositive converse Chapter 2 (p. 67) Chapter 2 (p. 67) counterexample deductive reasoning Chapter

More information

Lecture 15 Empiricism. David Hume Enquiry Concerning Human Understanding

Lecture 15 Empiricism. David Hume Enquiry Concerning Human Understanding Lecture 15 Empiricism David Hume Enquiry Concerning Human Understanding 1 Agenda 1. David Hume 2. Relations of Ideas and Matters of Fact 3. Cause and Effect 4. Can We Rationally Justify that the Future

More information

CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo

CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo Propositional Logic CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 37 Discrete Mathematics What is Discrete

More information

One natural response would be to cite evidence of past mornings, and give something like the following argument:

One natural response would be to cite evidence of past mornings, and give something like the following argument: Hume on induction Suppose you were asked to give your reasons for believing that the sun will come up tomorrow, in the form of an argument for the claim that the sun will come up tomorrow. One natural

More information

1.3 Induction and Other Proof Techniques

1.3 Induction and Other Proof Techniques 4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

More information

(Refer Slide Time: 1:41)

(Refer Slide Time: 1:41) Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

More information

Software Modeling and Verification

Software Modeling and Verification Software Modeling and Verification Alessandro Aldini DiSBeF - Sezione STI University of Urbino Carlo Bo Italy 3-4 February 2015 Algorithmic verification Correctness problem Is the software/hardware system

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

The Cosmological Argument

The Cosmological Argument A quick overview An a posteriori argument Everything that exists in the universe exists because it was caused by something else. That something was caused by something else It is necessary for something

More information

Mathematical Induction

Mathematical Induction Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How

More information

Computational Logic and Cognitive Science: An Overview

Computational Logic and Cognitive Science: An Overview Computational Logic and Cognitive Science: An Overview Session 1: Logical Foundations Technical University of Dresden 25th of August, 2008 University of Osnabrück Who we are Helmar Gust Interests: Analogical

More information

CSE373: Data Structures and Algorithms Lecture 2: Proof by Induction. Linda Shapiro Winter 2015

CSE373: Data Structures and Algorithms Lecture 2: Proof by Induction. Linda Shapiro Winter 2015 CSE373: Data Structures and Algorithms Lecture 2: Proof by Induction Linda Shapiro Winter 2015 Background on Induction Type of mathematical proof Typically used to establish a given statement for all natural

More information

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

More information

Machine Learning: Overview

Machine Learning: Overview Machine Learning: Overview Why Learning? Learning is a core of property of being intelligent. Hence Machine learning is a core subarea of Artificial Intelligence. There is a need for programs to behave

More information

FACULTY OF INFORMATION TECHNOLOGY

FACULTY OF INFORMATION TECHNOLOGY FACULTY OF INFORMATION TECHNOLOGY Course Specifications: (MATH 251) Month, Year: Fall 2008 University: Misr University for Science and Technology Faculty : Faculty of Information Technology Course Specifications

More information

Midterm Examination 1 with Solutions - Math 574, Frank Thorne Thursday, February 9, 2012

Midterm Examination 1 with Solutions - Math 574, Frank Thorne Thursday, February 9, 2012 Midterm Examination 1 with Solutions - Math 574, Frank Thorne (thorne@math.sc.edu) Thursday, February 9, 2012 1. (3 points each) For each sentence below, say whether it is logically equivalent to the sentence

More information

Predicate Logic Review

Predicate Logic Review Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginning

More information

Formal Verification and Linear-time Model Checking

Formal Verification and Linear-time Model Checking Formal Verification and Linear-time Model Checking Paul Jackson University of Edinburgh Automated Reasoning 21st and 24th October 2013 Why Automated Reasoning? Intellectually stimulating and challenging

More information

Mathematical Induction. Lecture 10-11

Mathematical Induction. Lecture 10-11 Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

More information

Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from

More information

Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both. M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

More information

Artificial Intelligence. 5. First-Order Logic

Artificial Intelligence. 5. First-Order Logic Artificial Intelligence Artificial Intelligence 5. First-Order Logic Lars Schmidt-Thieme Information Systems and Machine Learning Lab (ISMLL) Institute of Economics and Information Systems & Institute

More information

Descartes rationalism

Descartes rationalism Michael Lacewing Descartes rationalism Descartes Meditations provide an extended study in establishing knowledge through rational intuition and deduction. We focus in this handout on three central claims:

More information

SEARCHING AND KNOWLEDGE REPRESENTATION. Angel Garrido

SEARCHING AND KNOWLEDGE REPRESENTATION. Angel Garrido Acta Universitatis Apulensis ISSN: 1582-5329 No. 30/2012 pp. 147-152 SEARCHING AND KNOWLEDGE REPRESENTATION Angel Garrido ABSTRACT. The procedures of searching of solutions of problems, in Artificial Intelligence

More information

Davidson s Contribution to the Philosophy of Logic and Language

Davidson s Contribution to the Philosophy of Logic and Language Davidson s Contribution to the Philosophy of Logic and Language Gilbert Harman Princeton University Saturday, April 30, 2005 Finite Primitives believes-that-socrates-was-a-philosopher (Scheffler) multiply

More information

Perfect being theology and modal truth

Perfect being theology and modal truth Perfect being theology and modal truth Jeff Speaks February 9, 2016 Perfect being theology is the attempt to use the principle that God is the greatest possible being to derive claims about the divine

More information

Inference Rules and Proof Methods

Inference Rules and Proof Methods Inference Rules and Proof Methods Winter 2010 Introduction Rules of Inference and Formal Proofs Proofs in mathematics are valid arguments that establish the truth of mathematical statements. An argument

More information

The result of the bayesian analysis is the probability distribution of every possible hypothesis H, given one real data set D. This prestatistical approach to our problem was the standard approach of Laplace

More information

Th e ontological argument distinguishes itself from the cosmological

Th e ontological argument distinguishes itself from the cosmological Aporia vol. 18 no. 1 2008 Charles Hartshorne and the Ontological Argument Joshua Ernst Th e ontological argument distinguishes itself from the cosmological and teleological arguments for God s existence

More information