Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing February, 2018
|
|
|
- Holly Davidson
- 6 years ago
- Views:
Transcription
1 Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February, 2018 Alice E. Fischer... 1/28
2 1 Examples and Varieties Order of Quantifiers and Negations 2 3 Universal Existential 4 Universal Modus Ponens Universal Modus Tollens Universal Transitivity 5 Alice E. Fischer... 2/28
3 Examples and Varieties Order of Quantifiers and Negations Examples and Varieties Order of Quantifiers Negations of Multiply Quantified Statements Alice E. Fischer... 3/28
4 For All, There Exists Examples and Varieties Order of Quantifiers and Negations A statement can have multiple quantifiers: All married people have a spouse. m People, Married(m) s People, Spouse(m, s). All squares on a completed Sudoko board hold a digit s squares, d , d is written in s. All columns on a completed Sudoko board contain a square with the digit 1 c Sudoku columns, s squares, s c 1 s. Alice E. Fischer... 4/28
5 Examples and Varieties Order of Quantifiers and Negations Turning quantified statements into English. Let the predicate T (p, x) mean p teaches x, Example p Pros x Skaters, T (p, x). Wrong English: For all pros, there exists a skater such that all pros teach the skater. Right: you need to use the x and y in the English statement. For all pros p, there exists a skater x, such that p teaches x. If you focus on any particular pro, p, there exists a skater x, such that p teaches x. Alice E. Fischer... 5/28
6 Same Quantifier Twice Examples and Varieties Order of Quantifiers and Negations A statement can have two quantifiers that are alike. There are pairs of men and women who are not married. m men, w women, m is not married to w. For all digits d , and for all rows r a completed Sudoko board, d occurs once in r. d , and r completed Sudoku rows, Once(d, r). Alice E. Fischer... 6/28
7 Two or More Quantifiers Examples and Varieties Order of Quantifiers and Negations Given two consecutive quantifiers in a series of two or more: If they are the same quantifier, the order does not matter. ( y Student)( x Teacher)( z Course), Teaches(x, y, z) ( x Teacher)( z Course)( y Student), Teaches(x, y, z) If they are different quantifiers, the order matters greatly. ( x Z + )( y Z ), x + y = 0. (true) ( y Z )( x Z + ), x + y = 0. (false) Alice E. Fischer... 7/28
8 Examples and Varieties Order of Quantifiers and Negations Order of quantifiers: All-Exists vs. Exists-All x y is not at all the same as y x a) Every man is descended from a mother. ( m Men) ( f femalepeople), f is the mother of m. b) A woman is the ancestor of every man. ( f femalepeople)( m Men), f is the ancestor of m. To disprove a, find a man without a mother. To disprove b, find two men who do not have a common ancestor. (Remember Eve!) Alice E. Fischer... 8/28
9 Examples and Varieties Order of Quantifiers and Negations : Interpreting Quantified Statements CS Students Ann, Bob, Cal, Don, Eva, Flo belong to three clubs. Hacking team Robotics club Programming team Ann Flo Cal Don Eva Cal Bob Ann Ann Don Eva Given the diagram, say whether each statement is true or false. 1 z Students w Students, SameClub(z, w) 2 s Students c clubs, Member(s, c). 3 t Students d clubs, Member(t, d). 4 b clubs v Students, Member(v, b) 5 s Students z Students, SameClub(s, z). 6 x Students y Students, SameClub(x, y) Alice E. Fischer... 9/28
10 Examples and Varieties Order of Quantifiers and Negations Negating a Statement with Two or More Quantifiers To negate a multiply quantified statement, negate each quantifier in order, from left to right. Start with this true predicate that says there is no largest or smallest integer : ( y Z)( x Z)( z Z), x < y < z. To prove this, choose y=any integer, x=y-1 and z=y+1. Now let us negate it; the result will be a predicate that is false: ( y Z)( x Z)( z Z), x < y < z. ( y Z) ( x Z)( z Z), x < y < z. ( y Z)( x Z) ( z Z), x < y < z. ( y Z)( x Z)( z Z), (x < y < z.) ( y Z)( x Z)( z Z), x y z.) To disprove this, choose x=1 and z=10. Alice E. Fischer... 10/28
11 Universal Modus Ponens x, P(x) Q(x) P(k) for a particular k Q(k) for that k. Example: If an integer is even, then its square is even. The integer 10 is even. Therefore, 100 is even. Universal Modus Tollens x, P(x).Q(x) Q(k) for a particular k P(k) for that k Example: All cows have hoofs. I do not have hoofs. Therefore, I am not a cow. Alice E. Fischer... 11/28
12 Inverse Error and Converse Error Inverse Error x, P(x) Q(x) Q(k) for a particular k P(k) for that k. Invalid! Example: All dogs have four legs, My pet fluffy has four legs. Therefore Fluffy is a dog. Not true, Fluffy is a cat. Converse Error x, P(x).Q(x) P(k) for a particular k Q(k) for that k Invalid! Example: All lawyers are educated. I am not a lawyer. Therefore I am not educated. Not true! Alice E. Fischer... 12/28
13 Outline Universal Existential Making an argument using quantified statements uses the same rules of inference as for propositional calculus. But we need one extra concept. is the process of applying a general quantified statement to a particular element(s) from the domain for which it applies. Sometimes we want to answer general questions: do all birds fly? Other times we might be more particular: which bird doesn t fly? allows us to focus parts of an argument on particular individuals in the domain to answer a specific question. Alice E. Fischer... 13/28
14 Universal Universal Existential Universal instantiation states that a quantified statement that is true for every element of a domain is equally true for one specific element of the domain. Example - All men are nerds x Men, Nerd(x) If this statement is true, and Fred is a man, then Fred is a nerd, Nerd(Fred). This version of instantiation is used commonly in everyday speech. Alice E. Fischer... 14/28
15 Existential Universal Existential Existential instantiation states that if a quantified statement is true for at least one element of a domain, we can give such an element an arbitrary name. Example - Some men are nerds x Men, Nerd(x) Using existential instantiation, we can give a convenient name to one of the qualifying men, say, SmartGuy. This version of instantiation is used commonly in mathematics. Alice E. Fischer... 15/28
16 Universal Modus Ponens Universal Modus Tollens Universal Transitivity When we made arguments in propositional calculus, we presented many different rules of inference that could be used to generate new statements that eventually led to a conclusion. The same rules apply to predicate calculus, with the help of instantiation. We will look at three of the most useful rules: modus ponens, modus tollens, and transitivity. Alice E. Fischer... 16/28
17 Universal Modus Ponens Universal Modus Ponens Universal Modus Tollens Universal Transitivity Universal Modus Ponens states: x, P(x) Q(x) P(k) for a particular k Q(k) for that k. In other words, if the general premise of P(x) Q(x) is true for all x in the domain, and we know that the fact P(k) is true for a particular element, k, then we can conclude that the fact Q(k) is also true. Alice E. Fischer... 17/28
18 An Example Outline Universal Modus Ponens Universal Modus Tollens Universal Transitivity Here is a classic argument to which universal modus ponens applies: All men are mortal. Socrates is a man. Socrates is mortal. x People, Man(x) Mortal(x) Man(Socrates) Mortal(Socrates) This can be demonstrated using Venn diagrams: Alice E. Fischer... 18/28
19 Beware the Converse Error Universal Modus Ponens Universal Modus Tollens Universal Transitivity Consider this argument: All men are mortal. Socrates is mortal. Socrates is a man. x People, Man(x) Mortal(x) Mortal(Socrates) Man(Socrates) This invalid argument, suffers from the converse error: Alice E. Fischer... 19/28
20 Beware the Inverse Error Universal Modus Ponens Universal Modus Tollens Universal Transitivity Consider this argument: All men are mortal. Zeus is not a man. Zeus is not mortal. x People, Man(x) Mortal(x) Man(Zeus) Mortal(Zeus) This invalid argument, suffers from the inverse error: Alice E. Fischer... 20/28
21 Universal Modus Tollens Universal Modus Ponens Universal Modus Tollens Universal Transitivity Universal Modus Tollens states x, P(x) Q(x) Q(k) for a particular k P(k) for that k In other words, if the general premise of P(x) Q(x) is true for all x in the domain, and we know that the fact Q(k) is false for a particular element, k, then we can conclude that the fact P(k) is also false. This is based on the fact that the contrapositive version of a statement is equivalent to the statement. Alice E. Fischer... 21/28
22 An Example Outline Universal Modus Ponens Universal Modus Tollens Universal Transitivity Here is a valid argument to which universal modus tollens applies: All men are mortal. Zeus is not mortal. Zeus is not a man. x People, Man(x) Mortal(x) Mortal(Zeus) Man(Zeus) This can be demonstrated using Venn diagrams: Alice E. Fischer... 22/28
23 Transitivity Outline Universal Modus Ponens Universal Modus Tollens Universal Transitivity The Law of Universal Transitivity states: x, P(x) Q(x) x, Q(x) R(x) x, P(x) R(x) In other words, if the general premise of P(x) Q(x) is true for all x in the domain, and it is also true that Q(x) R(x), then we can conclude that P(x) R(x). This can be validated by specifying P(k) is true and using universal modus ponens to show that R(k) is true, for all k. Alice E. Fischer... 23/28
24 An Example Outline Universal Modus Ponens Universal Modus Tollens Universal Transitivity Here is a valid argument to which universal transitivity applies: All men are mortal. All mortals die. All men die. Fred will be a dead man. x People, Man(x) Mortal(x) x People, Mortal(x) Die(x) x People, Man(x) Die(x) Man(Fred) Die(Fred) This can be demonstrated using Venn diagrams: Alice E. Fischer... 24/28
25 Some example arguments Consider these facts: 1 Marcus was a man 2 Marcus was a Pompeian 3 All Pompeians were Romans 4 Caesar was a ruler 5 All Romans were either loyal to Caesar or hated him 6 Everyone is loyal to someone 7 Men only try to assassinate rulers they are not loyal to 8 Marcus tried to assassinate Caesar Alice E. Fischer... 25/28
26 Converting From English to Predicates 1 Marcus was a man 2 Marcus was a Pompeian 3 All Pompeians were Romans 4 Caesar was a ruler 5 All Romans were either loyal to Caesar or hated him 6 Everyone is loyal to someone 7 Men only try to assassinate rulers they are not loyal to 8 Marcus tried to assassinate Caesar 1 Man(Marcus) 2 Pompeian(Marcus) 3 x People, Pompeian(x) Roman(x) 4 Ruler(Caesar) 5 x People, Roman(x) (Loyalto(x, Caesar) Hate(x, Caesar)) (Loyalto(x, Caesar) Hate(x, Caesar)) 6 x People, y People, Loyalto(x, y) 7 x People, y People, Man(x) Ruler(y) Tryassassinate(x, y) Loyalto(x, y) 8 Tryassassinate(Marcus, Caesar) Alice E. Fischer... 26/28
27 Did Marcus hate Caesar? 9. Loyalto(Marcus, Caesar) Wrong: Tried instantiation with 6, picking x as Marcus and y as Caesar. It was ok to pick x as Marcus because of universal instantiation. It was not ok to pick Caesar as the model for y using existential instantiation, because Caesar has special properties that are not shared by all Romans. 9. Loyalto(Marcus, Caesar) Use universal modus ponens with 1, 4, 8 and 7. Alice E. Fischer... 27/28
28 Did Marcus hate Caesar? 10. Roman(Marcus) Use universal modus ponens with 2 and (Loyalto(Marcus, Caesar) Hate(Marcus, Caesar)) (Loyalto(Marcus, Caesar) Hate(Marcus, Caesar)) Use universal modus ponens with 10 and Hate(Marcus, Caesar) Use 9 and 11 and the definitions of And and Or Alice E. Fischer... 28/28
Announcements. CS311H: Discrete Mathematics. First Order Logic, Rules of Inference. Satisfiability, Validity in FOL. Example.
Announcements CS311H: Discrete Mathematics First Order Logic, Rules of Inference Instructor: Işıl Dillig Homework 1 is due now! Homework 2 is handed out today Homework 2 is due next Wednesday Instructor:
Announcements. CS243: Discrete Structures. First Order Logic, Rules of Inference. Review of Last Lecture. Translating English into First-Order Logic
Announcements CS243: Discrete Structures First Order Logic, Rules of Inference Işıl Dillig Homework 1 is due now Homework 2 is handed out today Homework 2 is due next Tuesday Işıl Dillig, CS243: Discrete
Revisiting the Socrates Example
Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments Rules of Inference for Quantified Statements Building Arguments for Quantified
Philosophy 1100: Ethics
Philosophy 1100: Ethics Topic 1 - Course Introduction: 1. What is Philosophy? 2. What is Ethics? 3. Logic a. Truth b. Arguments c. Validity d. Soundness What is Philosophy? The Three Fundamental Questions
MATH1061/MATH7861 Discrete Mathematics Semester 2, Lecture 5 Valid and Invalid Arguments. Learning Goals
MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Learning Goals 1. Understand the meaning of necessary and sufficient conditions (carried over from Wednesday). 2. Understand the difference between
Logic Appendix: More detailed instruction in deductive logic
Logic Appendix: More detailed instruction in deductive logic Standardizing and Diagramming In Reason and the Balance we have taken the approach of using a simple outline to standardize short arguments,
Logic: A Brief Introduction. Ronald L. Hall, Stetson University
Logic: A Brief Introduction Ronald L. Hall, Stetson University 2012 CONTENTS Part I Critical Thinking Chapter 1 Basic Training 1.1 Introduction 1.2 Logic, Propositions and Arguments 1.3 Deduction and Induction
Chapter 1. Introduction. 1.1 Deductive and Plausible Reasoning Strong Syllogism
Contents 1 Introduction 3 1.1 Deductive and Plausible Reasoning................... 3 1.1.1 Strong Syllogism......................... 3 1.1.2 Weak Syllogism.......................... 4 1.1.3 Transitivity
Section 3.5. Symbolic Arguments. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 3.5 Symbolic Arguments What You Will Learn Symbolic arguments Standard forms of arguments 3.5-2 Symbolic Arguments A symbolic argument consists of a set of premises and a conclusion. It is called
Part II: How to Evaluate Deductive Arguments
Part II: How to Evaluate Deductive Arguments Week 4: Propositional Logic and Truth Tables Lecture 4.1: Introduction to deductive logic Deductive arguments = presented as being valid, and successful only
Venn Diagrams and Categorical Syllogisms. Unit 5
Venn Diagrams and Categorical Syllogisms Unit 5 John Venn 1834 1923 English logician and philosopher noted for introducing the Venn diagram Used in set theory, probability, logic, statistics, and computer
16. Universal derivation
16. Universal derivation 16.1 An example: the Meno In one of Plato s dialogues, the Meno, Socrates uses questions and prompts to direct a young slave boy to see that if we want to make a square that has
Section 3.5. Symbolic Arguments. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 3.5 Symbolic Arguments INB able of Contents Date opic Page # July 28, 2014 Section 3.5 Examples 84 July 28, 2014 Section 3.5 Notes 85 July 28, 2014 Section 3.6 Examples 86 July 28, 2014 Section
Foundations of Non-Monotonic Reasoning
Foundations of Non-Monotonic Reasoning Notation S A - from a set of premisses S we can derive a conclusion A. Example S: All men are mortal Socrates is a man. A: Socrates is mortal. x.man(x) mortal(x)
Transition to Quantified Predicate Logic
Transition to Quantified Predicate Logic Predicates You may remember (but of course you do!) during the first class period, I introduced the notion of validity with an argument much like (with the same
(Refer Slide Time 03:00)
Artificial Intelligence Prof. Anupam Basu Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 15 Resolution in FOPL In the last lecture we had discussed about
Complications for Categorical Syllogisms. PHIL 121: Methods of Reasoning February 27, 2013 Instructor:Karin Howe Binghamton University
Complications for Categorical Syllogisms PHIL 121: Methods of Reasoning February 27, 2013 Instructor:Karin Howe Binghamton University Overall Plan First, I will present some problematic propositions and
Study Guides. Chapter 1 - Basic Training
Study Guides Chapter 1 - Basic Training Argument: A group of propositions is an argument when one or more of the propositions in the group is/are used to give evidence (or if you like, reasons, or grounds)
Part 2 Module 4: Categorical Syllogisms
Part 2 Module 4: Categorical Syllogisms Consider Argument 1 and Argument 2, and select the option that correctly identifies the valid argument(s), if any. Argument 1 All bears are omnivores. All omnivores
10.3 Universal and Existential Quantifiers
M10_COPI1396_13_SE_C10.QXD 10/22/07 8:42 AM Page 441 10.3 Universal and Existential Quantifiers 441 and Wx, and so on. We call these propositional functions simple predicates, to distinguish them from
Exercise Sets. KS Philosophical Logic: Modality, Conditionals Vagueness. Dirk Kindermann University of Graz July 2014
Exercise Sets KS Philosophical Logic: Modality, Conditionals Vagueness Dirk Kindermann University of Graz July 2014 1 Exercise Set 1 Propositional and Predicate Logic 1. Use Definition 1.1 (Handout I Propositional
Semantic Entailment and Natural Deduction
Semantic Entailment and Natural Deduction Alice Gao Lecture 6, September 26, 2017 Entailment 1/55 Learning goals Semantic entailment Define semantic entailment. Explain subtleties of semantic entailment.
7. Some recent rulings of the Supreme Court were politically motivated decisions that flouted the entire history of U.S. legal practice.
M05_COPI1396_13_SE_C05.QXD 10/12/07 9:00 PM Page 193 5.5 The Traditional Square of Opposition 193 EXERCISES Name the quality and quantity of each of the following propositions, and state whether their
9.1 Intro to Predicate Logic Practice with symbolizations. Today s Lecture 3/30/10
9.1 Intro to Predicate Logic Practice with symbolizations Today s Lecture 3/30/10 Announcements Tests back today Homework: --Ex 9.1 pgs. 431-432 Part C (1-25) Predicate Logic Consider the argument: All
Lecture 3 Arguments Jim Pryor What is an Argument? Jim Pryor Vocabulary Describing Arguments
Lecture 3 Arguments Jim Pryor What is an Argument? Jim Pryor Vocabulary Describing Arguments 1 Agenda 1. What is an Argument? 2. Evaluating Arguments 3. Validity 4. Soundness 5. Persuasive Arguments 6.
Module 5. Knowledge Representation and Logic (Propositional Logic) Version 2 CSE IIT, Kharagpur
Module 5 Knowledge Representation and Logic (Propositional Logic) Lesson 12 Propositional Logic inference rules 5.5 Rules of Inference Here are some examples of sound rules of inference. Each can be shown
PRACTICE EXAM The state of Israel was in a state of mourning today because of the assassination of Yztzak Rabin.
PRACTICE EXAM 1 I. Decide which of the following are arguments. For those that are, identify the premises and conclusions in them by CIRCLING them and labeling them with a P for the premises or a C for
Dr. Carlo Alvaro Reasoning and Argumentation Distribution & Opposition DISTRIBUTION
DISTRIBUTION Categorical propositions are statements that describe classes (groups) of objects designate by the subject and the predicate terms. A class is a group of things that have something in common
SYLLOGISTIC LOGIC CATEGORICAL PROPOSITIONS
Prof. C. Byrne Dept. of Philosophy SYLLOGISTIC LOGIC Syllogistic logic is the original form in which formal logic was developed; hence it is sometimes also referred to as Aristotelian logic after Aristotle,
Logic Book Part 1! by Skylar Ruloff!
Logic Book Part 1 by Skylar Ruloff Contents Introduction 3 I Validity and Soundness 4 II Argument Forms 10 III Counterexamples and Categorical Statements 15 IV Strength and Cogency 21 2 Introduction This
Deductive Forms: Elementary Logic By R.A. Neidorf READ ONLINE
Deductive Forms: Elementary Logic By R.A. Neidorf READ ONLINE If you are searching for a book Deductive Forms: Elementary Logic by R.A. Neidorf in pdf format, in that case you come on to the correct website.
What Is On The Final. Review. What Is Not On The Final. What Might Be On The Final
What Is On he inal Review Everything that has important! written next to it on the slides Everything that I said was important ECE457 Applied Artificial Intelligence all 27 ecture #14 ECE457 Applied Artificial
Pastor-teacher Don Hargrove Faith Bible Church September 8, 2011
Pastor-teacher Don Hargrove Faith Bible Church http://www.fbcweb.org/doctrines.html September 8, 2011 Building Mental Muscle & Growing the Mind through Logic Exercises: Lesson 4a The Three Acts of the
A BRIEF INTRODUCTION TO LOGIC FOR METAPHYSICIANS
A BRIEF INTRODUCTION TO LOGIC FOR METAPHYSICIANS 0. Logic, Probability, and Formal Structure Logic is often divided into two distinct areas, inductive logic and deductive logic. Inductive logic is concerned
1 Clarion Logic Notes Chapter 4
1 Clarion Logic Notes Chapter 4 Summary Notes These are summary notes so that you can really listen in class and not spend the entire time copying notes. These notes will not substitute for reading the
What is the Nature of Logic? Judy Pelham Philosophy, York University, Canada July 16, 2013 Pan-Hellenic Logic Symposium Athens, Greece
What is the Nature of Logic? Judy Pelham Philosophy, York University, Canada July 16, 2013 Pan-Hellenic Logic Symposium Athens, Greece Outline of this Talk 1. What is the nature of logic? Some history
Selections from Aristotle s Prior Analytics 41a21 41b5
Lesson Seventeen The Conditional Syllogism Selections from Aristotle s Prior Analytics 41a21 41b5 It is clear then that the ostensive syllogisms are effected by means of the aforesaid figures; these considerations
HOW TO ANALYZE AN ARGUMENT
What does it mean to provide an argument for a statement? To provide an argument for a statement is an activity we carry out both in our everyday lives and within the sciences. We provide arguments for
PHILOSOPHY 102 INTRODUCTION TO LOGIC PRACTICE EXAM 1. W# Section (10 or 11) 4. T F The statements that compose a disjunction are called conjuncts.
PHILOSOPHY 102 INTRODUCTION TO LOGIC PRACTICE EXAM 1 W# Section (10 or 11) 1. True or False (5 points) Directions: Circle the letter next to the best answer. 1. T F All true statements are valid. 2. T
In this section you will learn three basic aspects of logic. When you are done, you will understand the following:
Basic Principles of Deductive Logic Part One: In this section you will learn three basic aspects of logic. When you are done, you will understand the following: Mental Act Simple Apprehension Judgment
Logic for Robotics: Defeasible Reasoning and Non-monotonicity
Logic for Robotics: Defeasible Reasoning and Non-monotonicity The Plan I. Explain and argue for the role of nonmonotonic logic in robotics and II. Briefly introduce some non-monotonic logics III. Fun,
HANDBOOK (New or substantially modified material appears in boxes.)
1 HANDBOOK (New or substantially modified material appears in boxes.) I. ARGUMENT RECOGNITION Important Concepts An argument is a unit of reasoning that attempts to prove that a certain idea is true by
Artificial Intelligence Prof. P. Dasgupta Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur
Artificial Intelligence Prof. P. Dasgupta Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Lecture- 10 Inference in First Order Logic I had introduced first order
Artificial Intelligence: Valid Arguments and Proof Systems. Prof. Deepak Khemani. Department of Computer Science and Engineering
Artificial Intelligence: Valid Arguments and Proof Systems Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras Module 02 Lecture - 03 So in the last
PHI Introduction Lecture 4. An Overview of the Two Branches of Logic
PHI 103 - Introduction Lecture 4 An Overview of the wo Branches of Logic he wo Branches of Logic Argument - at least two statements where one provides logical support for the other. I. Deduction - a conclusion
4.1 A problem with semantic demonstrations of validity
4. Proofs 4.1 A problem with semantic demonstrations of validity Given that we can test an argument for validity, it might seem that we have a fully developed system to study arguments. However, there
PHILOSOPHY IM 25 SYLLABUS IM SYLLABUS (2019)
PHILOSOPHY IM 25 SYLLABUS IM SYLLABUS (2019) IM SYLLABUS (2019): Philosophy Philosophy IM 25 Syllabus (Available in September) 1 Paper (3 hrs) 1. Introduction Since the time of the ancient Greeks, philosophy
6. Truth and Possible Worlds
6. Truth and Possible Worlds We have defined logical entailment, consistency, and the connectives,,, all in terms of belief. In view of the close connection between belief and truth, described in the first
Tutorial A03: Patterns of Valid Arguments By: Jonathan Chan
A03.1 Introduction Tutorial A03: Patterns of Valid Arguments By: With valid arguments, it is impossible to have a false conclusion if the premises are all true. Obviously valid arguments play a very important
SOME RADICAL CONSEQUENCES OF GEACH'S LOGICAL THEORIES
SOME RADICAL CONSEQUENCES OF GEACH'S LOGICAL THEORIES By james CAIN ETER Geach's views of relative identity, together with his Paccount of proper names and quantifiers, 1 while presenting what I believe
Essential Logic Ronald C. Pine
Essential Logic Ronald C. Pine Chapter 11: Other Logical Tools Syllogisms and Quantification Introduction A persistent theme of this book has been the interpretation of logic as a set of practical tools.
Philosophy 1100: Introduction to Ethics. Critical Thinking Lecture 1. Background Material for the Exercise on Validity
Philosophy 1100: Introduction to Ethics Critical Thinking Lecture 1 Background Material for the Exercise on Validity Reasons, Arguments, and the Concept of Validity 1. The Concept of Validity Consider
1/19/2011. Concept. Analysis
Analysis Breaking down an idea, concept, theory, etc. into its most basic parts in order to get a better understanding of its structure. This is necessary to evaluate the merits of the claim properly (is
Chapter 3: Basic Propositional Logic. Based on Harry Gensler s book For CS2209A/B By Dr. Charles Ling;
Chapter 3: Basic Propositional Logic Based on Harry Gensler s book For CS2209A/B By Dr. Charles Ling; cling@csd.uwo.ca The Ultimate Goals Accepting premises (as true), is the conclusion (always) true?
1.5 Deductive and Inductive Arguments
M01_COPI1396_13_SE_C01.QXD 10/10/07 9:48 PM Page 26 26 CHAPTER 1 Basic Logical Concepts 19. All ethnic movements are two-edged swords. Beginning benignly, and sometimes necessary to repair injured collective
LOGICAL THINKING CHAPTER DEDUCTIVE THINKING: THE SYLLOGISM. If we reason it is not because we like to, but because we must.
ISBN: 0-536-29907-2 CHAPTER 9 LOGICAL THINKING If we reason it is not because we like to, but because we must. WILL DURANT, THE MANSIONS OF PHILOSOPHY Thinking logically and identifying reasoning fallacies
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Exam Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Draw a Venn diagram for the given sets. In words, explain why you drew one set as a subset of
Baronett, Logic (4th ed.) Chapter Guide
Chapter 6: Categorical Syllogisms Baronett, Logic (4th ed.) Chapter Guide A. Standard-form Categorical Syllogisms A categorical syllogism is an argument containing three categorical propositions: two premises
Chapter 8 - Sentential Truth Tables and Argument Forms
Logic: A Brief Introduction Ronald L. Hall Stetson University Chapter 8 - Sentential ruth ables and Argument orms 8.1 Introduction he truth-value of a given truth-functional compound proposition depends
EXERCISES: (from
EXERCISES: (from http://people.umass.edu/klement/100/logic-worksheet.html) A. 2. Jane has a cat 3. Therefore, Jane has a pet B. 2. Jane has a pet 3. Therefore, Jane has a cat C. 2. It is not the case that
HANDBOOK. IV. Argument Construction Determine the Ultimate Conclusion Construct the Chain of Reasoning Communicate the Argument 13
1 HANDBOOK TABLE OF CONTENTS I. Argument Recognition 2 II. Argument Analysis 3 1. Identify Important Ideas 3 2. Identify Argumentative Role of These Ideas 4 3. Identify Inferences 5 4. Reconstruct the
Formal Logic. Mind your Ps and Qs!
Formal Logic Mind your Ps and Qs! Argument vs. Explanation Arguments and explanations often have a similar structure. They both have what we might (vaguely) call a basis and a result. They might both
GENERAL NOTES ON THIS CLASS
PRACTICAL LOGIC Bryan Rennie GENERAL NOTES ON THE CLASS EXPLANATION OF GRADES AND POINTS, ETC. SAMPLE QUIZZES SCHEDULE OF CLASSES THE SIX RULES OF SYLLOGISMS (and corresponding fallacies) SYMBOLS USED
Deduction. Of all the modes of reasoning, deductive arguments have the strongest relationship between the premises
Deduction Deductive arguments, deduction, deductive logic all means the same thing. They are different ways of referring to the same style of reasoning Deduction is just one mode of reasoning, but it is
Workbook Unit 17: Negated Categorical Propositions
Workbook Unit 17: Negated Categorical Propositions Overview 1 1. Reminder 2 2. Negated Categorical Propositions 2 2.1. Negation of Proposition A: Not all Ss are P 3 2.2. Negation of Proposition E: It is
Reasoning SYLLOGISM. follows.
Reasoning SYLLOGISM RULES FOR DERIVING CONCLUSIONS 1. The Conclusion does not contain the Middle Term (M). Premises : All spoons are plates. Some spoons are cups. Invalid Conclusion : All spoons are cups.
Informalizing Formal Logic
Informalizing Formal Logic Antonis Kakas Department of Computer Science, University of Cyprus, Cyprus antonis@ucy.ac.cy Abstract. This paper discusses how the basic notions of formal logic can be expressed
Faith indeed tells what the senses do not tell, but not the contrary of what they see. It is above them and not contrary to them.
19 Chapter 3 19 CHAPTER 3: Logic Faith indeed tells what the senses do not tell, but not the contrary of what they see. It is above them and not contrary to them. The last proceeding of reason is to recognize
Logic: Deductive and Inductive by Carveth Read M.A. CHAPTER VIII
CHAPTER VIII ORDER OF TERMS, EULER'S DIAGRAMS, LOGICAL EQUATIONS, EXISTENTIAL IMPORT OF PROPOSITIONS Section 1. Of the terms of a proposition which is the Subject and which the Predicate? In most of the
Early Russell on Philosophical Grammar
Early Russell on Philosophical Grammar G. J. Mattey Fall, 2005 / Philosophy 156 Philosophical Grammar The study of grammar, in my opinion, is capable of throwing far more light on philosophical questions
Thinking and Reasoning
Syllogistic Reasoning Thinking and Reasoning Syllogistic Reasoning Erol ÖZÇELİK The other key type of deductive reasoning is syllogistic reasoning, which is based on the use of syllogisms. Syllogisms are
A R G U M E N T S I N A C T I O N
ARGUMENTS IN ACTION Descriptions: creates a textual/verbal account of what something is, was, or could be (shape, size, colour, etc.) Used to give you or your audience a mental picture of the world around
Follow Will of the People. Your leftist h. b. ave often d1sgusted b h
Philosophy 101 (3/24/11) I ve posted solutions to HW #3 (study these!) HW #4 is due today Quiz #4 is next Thursday This will be re-do of the last quiz (on chs. 3&4) I ll give you the higher of your two
Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples
Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Section 1.1 Propositional Logic Page references correspond to locations of Extra Examples icons in the textbook. p.2, icon at
Criticizing Arguments
Kareem Khalifa Criticizing Arguments 1 Criticizing Arguments Kareem Khalifa Department of Philosophy Middlebury College Written August, 2012 Table of Contents Introduction... 1 Step 1: Initial Evaluation
(Some More) Vagueness
(Some More) Vagueness Otávio Bueno Department of Philosophy University of Miami Coral Gables, FL 33124 E-mail: otaviobueno@mac.com Three features of vague predicates: (a) borderline cases It is common
A Judgmental Formulation of Modal Logic
A Judgmental Formulation of Modal Logic Sungwoo Park Pohang University of Science and Technology South Korea Estonian Theory Days Jan 30, 2009 Outline Study of logic Model theory vs Proof theory Classical
Overview of Today s Lecture
Branden Fitelson Philosophy 12A Notes 1 Overview of Today s Lecture Music: Robin Trower, Daydream (King Biscuit Flower Hour concert, 1977) Administrative Stuff (lots of it) Course Website/Syllabus [i.e.,
Broad on Theological Arguments. I. The Ontological Argument
Broad on God Broad on Theological Arguments I. The Ontological Argument Sample Ontological Argument: Suppose that God is the most perfect or most excellent being. Consider two things: (1)An entity that
INTERMEDIATE LOGIC Glossary of key terms
1 GLOSSARY INTERMEDIATE LOGIC BY JAMES B. NANCE INTERMEDIATE LOGIC Glossary of key terms This glossary includes terms that are defined in the text in the lesson and on the page noted. It does not include
Geometry TEST Review Chapter 2 - Logic
Geometry TEST Review Chapter 2 - Logic Name Period Date Symbolic notation: 1. Define the following symbols. a b ~ c d e g a b c d a b c d 2. Consider the following legend: Let p = You love bananas. Let
1. Immediate inferences embodied in the square of opposition 2. Obversion 3. Conversion
CHAPTER 3: CATEGORICAL INFERENCES Inference is the process by which the truth of one proposition (the conclusion) is affirmed on the basis of the truth of one or more other propositions that serve as its
Assignment Assignment for Lesson 3.1
Assignment Assignment for Lesson.1 Name Date A Little Dash of Logic Two Methods of Logical Reasoning Joseph reads a journal article that states that yogurt with live cultures greatly helps digestion and
Example Arguments ID1050 Quantitative & Qualitative Reasoning
Example Arguments ID1050 Quantitative & Qualitative Reasoning First Steps to Analyzing an Argument In the following slides, some simple arguments will be given. The steps to begin analyzing each argument
HW3- Sets & Arguments (solutions) Due: Tuesday April 5, 2011
HW #3-SOLUTIONS Topics: Sets, categorical propositions, Venn diagrams, analyzing arguments, and critical thinking Please show your work and clearly indicate your answer. Although you are welcome to compare
Introduction to Logic
University of Notre Dame Spring, 2017 Arguments Philosophy has two main methods for trying to answer questions: analysis and arguments Logic is the the study of arguments An argument is a set of sentences,
2.1 Review. 2.2 Inference and justifications
Applied Logic Lecture 2: Evidence Semantics for Intuitionistic Propositional Logic Formal logic and evidence CS 4860 Fall 2012 Tuesday, August 28, 2012 2.1 Review The purpose of logic is to make reasoning
MCQ IN TRADITIONAL LOGIC. 1. Logic is the science of A) Thought. B) Beauty. C) Mind. D) Goodness
MCQ IN TRADITIONAL LOGIC FOR PRIVATE REGISTRATION TO BA PHILOSOPHY PROGRAMME 1. Logic is the science of-----------. A) Thought B) Beauty C) Mind D) Goodness 2. Aesthetics is the science of ------------.
PHIL2642 CRITICAL THINKING USYD NOTES PART 1: LECTURE NOTES
PHIL2642 CRITICAL THINKING USYD NOTES PART 1: LECTURE NOTES LECTURE CONTENTS LECTURE 1: CLAIMS, EXPLAINATIONS AND ARGUMENTS LECTURE 2: CONDITIONS AND DEDUCTION LECTURE 3: MORE DEDUCTION LECTURE 4: MEANING
Philosophy 57 Day 10
Branden Fitelson Philosophy 57 Lecture 1 Philosophy 57 Day 10 Quiz #2 Curve (approximate) 100 (A); 70 80 (B); 50 60 (C); 40 (D); < 40 (F) Quiz #3 is next Tuesday 03/04/03 (on chapter 4 not tnanslation)
Syllogism. Exam Importance Exam Importance. CAT Very Important IBPS/Bank PO Very Important. XAT Very Important BANK Clerk Very Important
1 About Disha publication One of the leading publishers in India, Disha Publication provides books and study materials for schools and various competitive exams being continuously held across the country.
Name: Course: CAP 4601 Semester: Summer 2013 Assignment: Assignment 06 Date: 08 JUL Complete the following written problems:
Name: Course: CAP 4601 Semester: Summer 2013 Assignment: Assignment 06 Date: 08 JUL 2013 Complete the following written problems: 1. Alpha-Beta Pruning (40 Points). Consider the following min-max tree.
Unit. Categorical Syllogism. What is a syllogism? Types of Syllogism
Unit 8 Categorical yllogism What is a syllogism? Inference or reasoning is the process of passing from one or more propositions to another with some justification. This inference when expressed in language
Artificial Intelligence Prof. P. Dasgupta Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur
Artificial Intelligence Prof. P. Dasgupta Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Lecture- 9 First Order Logic In the last class, we had seen we have studied
Artificial Intelligence I
Artificial Intelligence I Matthew Huntbach, Dept of Computer Science, Queen Mary and Westfield College, London, UK E 4NS. Email: mmh@dcs.qmw.ac.uk. Notes may be used with the permission of the author.
What would count as Ibn Sīnā (11th century Persia) having first order logic?
1 2 What would count as Ibn Sīnā (11th century Persia) having first order logic? Wilfrid Hodges Herons Brook, Sticklepath, Okehampton March 2012 http://wilfridhodges.co.uk Ibn Sina, 980 1037 3 4 Ibn Sīnā
Introduction to Logic
University of Notre Dame Fall, 2015 Arguments Philosophy is difficult. If questions are easy to decide, they usually don t end up in philosophy The easiest way to proceed on difficult questions is to formulate
In view of the fact that IN CLASS LOGIC EXERCISES
IN CLASS LOGIC EXERCISES Instructions: Determine whether the following are propositions. If some are not propositions, see if they can be rewritten as propositions. (1) I have a very refined sense of smell.
PHI 1500: Major Issues in Philosophy
PHI 1500: Major Issues in Philosophy Session 3 September 9 th, 2015 All About Arguments (Part II) 1 A common theme linking many fallacies is that they make unwarranted assumptions. An assumption is a claim
4.7 Constructing Categorical Propositions
4.7 Constructing Categorical Propositions We have spent the last couple of weeks studying categorical propositions. Unfortunately, in the real world, the statements that people make seldom have that form.