Simple Proofs in Propositional Logic
|
|
- Corey Phelps
- 7 years ago
- Views:
Transcription
1 Simple Proofs in Propositional Logic We do not need to use truth tables or the shorter truth table technique in order to asses the validity of arguments in propositional form. Instead, we can show the validity of an argument by deriving its conclusion from its premises using argument forms that are known to be valid. 69 Valid Argument Forms Each of the following argument forms can be shown to be valid (e.g., using truth tables). Once we have established that these argument forms are valid (and, possibly, committed them to memory), we can use them to derive the conclusion of an argument from its premises. If we can do this, we will have shown that the argument is valid. 70 1
2 Valid Argument Forms P ~(~P) ~(~P) P P v P P P; Q; P. Q P. Q P P. Q Q P > Q; P Q or P > Q; ~ Q ~P P > Q ~Q > ~P or ~Q > ~P P > Q Double negation Tautology Conjunction Simplification Modus Ponens Modus Tollens Transposition 71 P > Q; Q > R P > R Hypothetical syllogism P v Q; ~P Q P v Q; ~Q P P P v Q P Q v P or or ~(P. Q) ~P v ~Q or ~P v ~Q ~(P. Q) P > Q ~P v Q ~P v Q P > Q P > Q; R > S; P v R Q v S P > Q; R > S; ~Q v ~S ~P v ~R ~(P. ~P) or Disjunctive Syllogism Addition De Morgan s Rule Implication Constructive dilemma Destructive dilemma Noncontradiction 72 2
3 Simple Proof: An Example Let s say we had an argument that, once formalized, looked like this: 1. (A v B) > ~ C 2. ~C > D 3. A Therefore, 4. D Once again: If we can derive the conclusion from the premises using only known to be valid argument forms, we will have shown that the argument is valid (A v B) > ~ C (premise) to be proven: D 2. ~C > D (premise) 3. A (premise) 4. A v B (from 3 by addition) 5. ~C (from 1 by modus ponens) 6. D (from 2 by modus ponens) Valid (D was the conclusion of the original argument, and we have derived it from the premises of that argument using only known-to-be-valid argument forms) 74 3
4 Another example 1. A. B 2. A > ~C 3. B > ~D Therefore, 4. ~C. ~D A. B (premise) to be proven: ~C. ~D 2. A > ~C (premise) 3. B > ~D (premise) 4. A (from 1, simplification) 5. B (from 1, simplification) 6. ~C (from 2, modus ponens) 7. ~D (from 3, modus ponens) 8. ~C. ~D (from 6, 7, conjunction) Valid 76 4
5 Example from Exam #2 1. C > D (premise) to be proven: E 2. D > E (premise) 3. C (premise) 4. D (from 1 and 3, modus ponens) 5. E (from 2 and 4, modus ponens) 77 Something to Notice About Simple Proof If can derive the conclusion from an argument using only valid argument forms, you have shown that that argument is valid. However, if you cannot derive the conclusion this way, you have not shown that the argument is invalid. I.e., the simple proof technique requires a certain amount of ingenuity and you may have failed to hit upon a workable proof strategy. 78 5
6 So, if you cannot construct a proof for an argument, you cannot say for certain that the argument is invalid: It may be invalid or you may have failed to find the right proof strategy. (By contrast the truth table method, you ll recall, always provides an answer.) 79 Conditional Proof In conditional proof, a statement is assumed for the sake of argument. In particular, we assume the antecedent of a conditional and then see if that conditional follows from the other premises using valid argument forms. Notice that a conditional proof is conditional in two senses: i) the proof of the conclusion is conditional upon the assumption and ii) the conclusion proven is itself a conditional statement. 80 6
7 Conditional Proof: Simple Example 1. A > B (premise) to be proven: A > C 2. B > C (premise) 3. A (assumption) 4. B (from 1 and 3, modus ponens) 5. C (from 2 and 4, modus ponens) 6. A > C (from 3-5, conditional proof) 81 Making assumptions in this way may seem arbitrary. Remember, however, that > asserts only that if the antecedent is true, then the consequent cannot be false. So, if from the stated premises and some assumption A we can derive X, then we can derive from the stated premises if A, then X P (assumption) P > Q Conditional Proof 82 7
8 So, another rule (another valid argument form): P (assumption) P > Q Conditional Proof Meaning: If Q can be derived from the assumption P through a series of valid intermediate steps, represented here by the ellipsis ( ), then we can validly conclude P > Q. 83 Conditional proof is actually a fairly common technique in everyday reasoning. Imagine, e.g., that we believe that 1) we can solve the problems of Third World nations and still maintain a reasonable standard of living by developing alternative forms of energy and that 2) there will be a greater chance of lasting peace if we solve problems of Third World nations 84 8
9 This can be formalized as follows: E = we develop alternative forms of energy S = we solve problems of Third World nations R = we enjoy a reasonable standard of living P = there is greater chance of lasting peace 1. E > (S. R) (premise) to be proven: E > P 2. S > P (premise) 3. E (assumption) 4. S. R (from 1 and 3, modus ponens) 5. S (from 4, simplification) 6. P (from 2 and 6, modus ponens) 7. E > P (3-6, conditional proof) 85 Indirect Proof: Reductio ad Absurdum Reductio ad absurdum (< Latin reduce to absurdity ) is a proof technique in which we prove a statement by showing how the premises can entail a contradiction. I.e., they entail a statement that cannot be true, a statement of the form P. ~P The technique: If we assume X and show by a series of valid intermediate steps that it leads to a statement of the form P. ~P, then we can validly conclude ~X. 86 9
10 Reductio ad Absurdum: An Example 1. C > ~R (premise) to be proven: ~C 2. C > A (premise) 3. A > R (premise) 4. C (assumption) 5. ~R (from 1 and 4, modus ponens) 6. A (from 2 and 4, modus ponens) 7. R (from 3 and 6, modus ponens) 8. R. ~R (from 8 and 5, conjunction) 9. C > (R. ~R) (4-8, conditional proof) 10. ~(R. ~R) (noncontradiction) 11. ~C (from 9 and 10, modus tollens) 87 Two Fallacies Related to Conditionals modus ponens: P > Q; P Q modus tollens: P > Q; ~Q ~P Both of these argument forms are valid. They can be proven to be valid using truth tables and, moreover, they appear intuitively to be valid. They are basic to human thinking. Both of these argument forms are commonly misused, however 88 10
11 Conditional Fallacies Affirming the consequent: P > Q; Q P * Consider: If you got the job, you must have impressed the interviewer. You impressed the interviewer. Therefore, you must have got the job (!?) Denying the antecedent: P > Q; ~P ~Q * Consider: If Derek is in Saskatoon, then he is in Saskatchewan. Derek is not in Saskatoon. Therefore, he is not in Saskatchewan (!?) 89 Propositional Logic and Cogency We have only touched on the basics of formal logic in this course: It can be pursued in much greater depth (e.g., U of R, PHIL 250, 350, 351, MATH 221, 301; U of S, PHIL 241, 243, 343, CMPT 260, 417, etc.) For present purposes we should bear in mind the connection between formal logic and the topics in natural language (informal) logic that we encountered in Govier chs
12 we noted then that an argument is cogent iff it satisfies all of the ARG conditions. If an argument is deductively valid, then it satisfies the R and (if it premises are all true) G conditions. But just because an argument is valid does not mean that the A condition is satisfied. A special case of formally valid arguments failing the A test occurs in connection with dilemma arguments 91 Either employment levels will go up or there will be a revolt. Employment levels are not going up. Therefore, there will be a revolt. E v R; ~E R (valid disjunctive syllogism) But it the first premise really acceptable as it stands? Does anyone have good reason to believe that these are the only two alternatives? If we can specify a third alternative (a counterexample to premise 1) we can be said to have escaped through the horns of the dilemma 92 12
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 informationdef: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.
Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true
More informationHandout #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 informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationHypothetical Syllogisms 1
Phil 2302 Intro to Logic Dr. Naugle Hypothetical Syllogisms 1 Compound syllogisms are composed of different kinds of sentences in their premises and conclusions (not just categorical propositions, statements
More informationDISCRETE 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 informationRules of Inference Friday, January 18, 2013 Chittu Tripathy Lecture 05
Rules of Inference Today s Menu Rules of Inference Quantifiers: Universal and Existential Nesting of Quantifiers Applications Old Example Re-Revisited Our Old Example: Suppose we have: All human beings
More information1.2 Forms and Validity
1.2 Forms and Validity Deductive Logic is the study of methods for determining whether or not an argument is valid. In this section we identify some famous valid argument forms. Argument Forms Consider
More informationBeyond 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 informationA 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 informationDevelopment of a computer system to support knowledge acquisition of basic logical forms using fairy tale "Alice in Wonderland"
Development of a computer system to support knowledge acquisition of basic logical forms using fairy tale "Alice in Wonderland" Antonija Mihaljević Španjić *, Alen Jakupović *, Matea Tomić * * Polytechnic
More informationDEDUCTIVE & 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 information3. 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 informationMathematical 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 informationDeductive reasoning is the kind of reasoning in which, roughly, the truth of the input
Forthcoming in The Encyclopedia of the Mind, edited by Hal Pashler, SAGE Publishing. Editorial Board: Tim Crane, Fernanda Ferreira, Marcel Kinsbourne, and Rich Zemel. Deductive Reasoning Joshua Schechter
More informationP1. All of the students will understand validity P2. You are one of the students -------------------- C. You will understand validity
Validity Philosophy 130 O Rourke I. The Data A. Here are examples of arguments that are valid: P1. If I am in my office, my lights are on P2. I am in my office C. My lights are on P1. He is either in class
More informationDeductive versus Inductive Reasoning
Deductive Arguments 1 Deductive versus Inductive Reasoning Govier has pointed out that there are four basic types of argument: deductive, inductive generalization, analogical, and conductive. For our purposes,
More informationInvalidity in Predicate Logic
Invalidity in Predicate Logic So far we ve got a method for establishing that a predicate logic argument is valid: do a derivation. But we ve got no method for establishing invalidity. In propositional
More informationLogic 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 informationCS510 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 informationMath 3000 Section 003 Intro to Abstract Math Homework 2
Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these
More informationCHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.
CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:
More informationLogic Appendix. Section 1 Truth Tables CONJUNCTION EXAMPLE 1
Logic Appendix T F F T Section 1 Truth Tables Recall that a statement is a group of words or symbols that can be classified collectively as true or false. The claim 5 7 12 is a true statement, whereas
More informationPredicate 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 informationCultural Relativism. 1. What is Cultural Relativism? 2. Is Cultural Relativism true? 3. What can we learn from Cultural Relativism?
1. What is Cultural Relativism? 2. Is Cultural Relativism true? 3. What can we learn from Cultural Relativism? What is it? Rough idea: There is no universal truth in ethics. There are only customary practices
More informationCosmological Arguments for the Existence of God S. Clarke
Cosmological Arguments for the Existence of God S. Clarke [Modified Fall 2009] 1. Large class of arguments. Sometimes they get very complex, as in Clarke s argument, but the basic idea is simple. Lets
More information6.080/6.089 GITCS Feb 12, 2008. Lecture 3
6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
More informationPredicate Logic. Example: All men are mortal. Socrates is a man. Socrates is mortal.
Predicate Logic Example: All men are mortal. Socrates is a man. Socrates is mortal. Note: We need logic laws that work for statements involving quantities like some and all. In English, the predicate is
More informationDeveloping Critical Thinking Skills with The Colbert Report
Developing Critical Thinking Skills with The Colbert Report Why Teach Critical Thinking with The Colbert Report? A. Young people receive the majority of their information through popular culture. B. Using
More informationExamination paper for MA0301 Elementær diskret matematikk
Department of Mathematical Sciences Examination paper for MA0301 Elementær diskret matematikk Academic contact during examination: Iris Marjan Smit a, Sverre Olaf Smalø b Phone: a 9285 0781, b 7359 1750
More informationThe Mathematics of GIS. Wolfgang Kainz
The Mathematics of GIS Wolfgang Kainz Wolfgang Kainz Department of Geography and Regional Research University of Vienna Universitätsstraße 7, A-00 Vienna, Austria E-Mail: wolfgang.kainz@univie.ac.at Version.
More informationDiscrete 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 informationPhilosophical argument
Michael Lacewing Philosophical argument At the heart of philosophy is philosophical argument. Arguments are different from assertions. Assertions are simply stated; arguments always involve giving reasons.
More informationp: I am elected q: I will lower the taxes
Implication Conditional Statement p q (p implies q) (if p then q) is the proposition that is false when p is true and q is false and true otherwise. Equivalent to not p or q Ex. If I am elected then I
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationLecture Notes in Discrete Mathematics. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Lecture Notes in Discrete Mathematics Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 Preface This book is designed for a one semester course in discrete mathematics for sophomore or junior
More informationCSL105: Discrete Mathematical Structures. Ragesh Jaiswal, CSE, IIT Delhi
Propositional Logic: logical operators Negation ( ) Conjunction ( ) Disjunction ( ). Exclusive or ( ) Conditional statement ( ) Bi-conditional statement ( ): Let p and q be propositions. The biconditional
More informationSolutions Q1, Q3, Q4.(a), Q5, Q6 to INTLOGS16 Test 1
Solutions Q1, Q3, Q4.(a), Q5, Q6 to INTLOGS16 Test 1 Prof S Bringsjord 0317161200NY Contents I Problems 1 II Solutions 3 Solution to Q1 3 Solutions to Q3 4 Solutions to Q4.(a) (i) 4 Solution to Q4.(a)........................................
More informationWOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology
First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Mid-Session Test Summer Session 008-00
More informationLecture 2: Moral Reasoning & Evaluating Ethical Theories
Lecture 2: Moral Reasoning & Evaluating Ethical Theories I. Introduction In this ethics course, we are going to avoid divine command theory and various appeals to authority and put our trust in critical
More informationWhat Is Circular Reasoning?
What Is Circular Reasoning? Logical fallacies are a type of error in reasoning, errors which may be recognized and corrected by observant thinkers. There are a large number of informal fallacies that are
More informationLogical Agents. Explorations in Artificial Intelligence. Knowledge-based Agents. Knowledge-base Agents. Outline. Knowledge bases
Logical Agents Explorations in Artificial Intelligence rof. Carla. Gomes gomes@cs.cornell.edu Agents that are able to: Form representations of the world Use a process to derive new representations of the
More informationOne 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 informationKant s deontological ethics
Michael Lacewing Kant s deontological ethics DEONTOLOGY Deontologists believe that morality is a matter of duty. We have moral duties to do things which it is right to do and moral duties not to do things
More informationPHILOSOPHY 101: CRITICAL THINKING
PHILOSOPHY 101: CRITICAL THINKING [days and times] [classroom] [semester] 20YY, [campus] [instructor s name] [office hours: days and times] [instructor s e-mail] COURSE OBJECTIVES AND OUTCOMES 1. Identify
More information8. Inductive Arguments
8. Inductive Arguments 1 Inductive Reasoning In general, inductive reasoning is reasoning in which we extrapolate from observed experience (e.g., past experience) to some conclusion (e.g., about present
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationWhat Is Induction and Why Study It?
1 What Is Induction and Why Study It? Evan Heit Why study induction, and indeed, why should there be a whole book devoted to the study of induction? The first reason is that inductive reasoning corresponds
More informationPropositional Logic. A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both.
irst Order Logic Propositional Logic A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both. Are the following sentences propositions? oronto
More informationCorrespondence 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 informationCHAPTER 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 informationFull details of the course syllabus, support and examination arrangements are provided below.
A Level Studies in conjunction with Oxford College Study House is pleased to be able to offer students the opportunity to study A levels through our partner Oxford College who will provide study materials
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 Now that we have a basic understanding
More information1/9. Locke 1: Critique of Innate Ideas
1/9 Locke 1: Critique of Innate Ideas This week we are going to begin looking at a new area by turning our attention to the work of John Locke, who is probably the most famous English philosopher of all
More information0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text)
September 2 Exercises: Problem 2 (p. 21) Efficiency: p. 28-29: 1, 4, 5, 6 0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text) We discuss here a notion of efficiency that is rooted in the individual preferences
More informationIntroduction 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 informationFive High Order Thinking Skills
Five High Order Introduction The high technology like computers and calculators has profoundly changed the world of mathematics education. It is not only what aspects of mathematics are essential for learning,
More informationReviewfrom Last Class
Reviewfrom Last Class The most used fallacy on Earth! Ad Hominem Several Types of Ad Hominem Fallacies 1. Personal Attack Ad Hominem 2. Inconsistency Ad Hominem 3. Circumstantial Ad Hominem 4. Poisoning
More informationLogic and Reasoning Practice Final Exam Spring 2015. Section Number
Logic and Reasoning Practice Final Exam Spring 2015 Name Section Number The final examination is worth 100 points. 1. (5 points) What is an argument? Explain what is meant when one says that logic is the
More informationON WHITCOMB S GROUNDING ARGUMENT FOR ATHEISM Joshua Rasmussen Andrew Cullison Daniel Howard-Snyder
ON WHITCOMB S GROUNDING ARGUMENT FOR ATHEISM Joshua Rasmussen Andrew Cullison Daniel Howard-Snyder Abstract: Dennis Whitcomb argues that there is no God on the grounds that (i) God is omniscient, yet (ii)
More informationThe History of Logic. Aristotle (384 322 BC) invented logic.
The History of Logic Aristotle (384 322 BC) invented logic. Predecessors: Fred Flintstone, geometry, sophists, pre-socratic philosophers, Socrates & Plato. Syllogistic logic, laws of non-contradiction
More information196 Chapter 7. Logical Agents
7 LOGICAL AGENTS In which we design agents that can form representations of the world, use a process of inference to derive new representations about the world, and use these new representations to deduce
More informationTrust but Verify: Authorization for Web Services. The University of Vermont
Trust but Verify: Authorization for Web Services Christian Skalka X. Sean Wang The University of Vermont Trust but Verify (TbV) Reliable, practical authorization for web service invocation. Securing complex
More informationSlippery Slopes and Vagueness
Slippery Slopes and Vagueness Slippery slope reasoning, typically taken as a fallacy. But what goes wrong? Is it always bad reasoning? How should we respond to a slippery slope argument and/or guard against
More informationRead this syllabus very carefully. If there are any reasons why you cannot comply with what I am requiring, then talk with me about this at once.
LOGIC AND CRITICAL THINKING PHIL 2020 Maymester Term, 2010 Daily, 9:30-12:15 Peabody Hall, room 105 Text: LOGIC AND RATIONAL THOUGHT by Frank R. Harrison, III Professor: Frank R. Harrison, III Office:
More informationWe would like to state the following system of natural deduction rules preserving falsity:
A Natural Deduction System Preserving Falsity 1 Wagner de Campos Sanz Dept. of Philosophy/UFG/Brazil sanz@fchf.ufg.br Abstract This paper presents a natural deduction system preserving falsity. This new
More informationQuine on truth by convention
Quine on truth by convention March 8, 2005 1 Linguistic explanations of necessity and the a priori.............. 1 2 Relative and absolute truth by definition.................... 2 3 Is logic true by convention?...........................
More informationINTRODUCTION TO LOGIC
INTRODUCTION TO LOGIC A. Basic Concepts 1. Logic is the science of the correctness or incorrectness of reasoning, or the study of the evaluation of arguments. 2. A statement is a declarative sentence,
More informationChapter 5: Fallacies. 23 February 2015
Chapter 5: Fallacies 23 February 2015 Plan for today Talk a bit more about arguments notice that the function of arguments explains why there are lots of bad arguments Turn to the concept of fallacy and
More informationsimplicity hides complexity
flow of control backtracking reasoning in logic and in Prolog 1 simplicity hides complexity simple and/or connections of goals conceal very complex control patterns Prolog programs are not easily represented
More informationInductive Reasoning Page 1 of 7. Inductive Reasoning
Inductive Reasoning Page 1 of 7 Inductive Reasoning We learned that valid deductive thinking begins with at least one universal premise and leads to a conclusion that is believed to be contained in the
More informationTheorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive
Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime
More informationDeductive reasoning is the application of a general statement to a specific instance.
Section1.1: Deductive versus Inductive Reasoning Logic is the science of correct reasoning. Websters New World College Dictionary defines reasoning as the drawing of inferences or conclusions from known
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering
More informationChapter 1. Use the following to answer questions 1-5: In the questions below determine whether the proposition is TRUE or FALSE
Use the following to answer questions 1-5: Chapter 1 In the questions below determine whether the proposition is TRUE or FALSE 1. 1 + 1 = 3 if and only if 2 + 2 = 3. 2. If it is raining, then it is raining.
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationCritical Thinking. I m not sure what they expect when they ask us to critically evaluate
1 Critical Thinking Typical Comments I m not sure what they expect when they ask us to critically evaluate The word critical sounds so negative, as though you have to undermine everything The word analysis
More informationNP-Completeness and Cook s Theorem
NP-Completeness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:
More informationSchedule. Logic (master program) Literature & Online Material. gic. Time and Place. Literature. Exercises & Exam. Online Material
OLC mputational gic Schedule Time and Place Thursday, 8:15 9:45, HS E Logic (master program) Georg Moser Institute of Computer Science @ UIBK week 1 October 2 week 8 November 20 week 2 October 9 week 9
More informationIs Justified True Belief Knowledge?
Is Justified True Belief Knowledge? EDMUND GETTIER Edmund Gettier is Professor Emeritus at the University of Massachusetts, Amherst. This short piece, published in 1963, seemed to many decisively to refute
More informationCRITICAL THINKING REASONS FOR BELIEF AND DOUBT (VAUGHN CH. 4)
CRITICAL THINKING REASONS FOR BELIEF AND DOUBT (VAUGHN CH. 4) LECTURE PROFESSOR JULIE YOO Claims Without Arguments When Claims Conflict Conflicting Claims Conflict With Your Background Information Experts
More informationDoes rationality consist in responding correctly to reasons? John Broome Journal of Moral Philosophy, 4 (2007), pp. 349 74.
Does rationality consist in responding correctly to reasons? John Broome Journal of Moral Philosophy, 4 (2007), pp. 349 74. 1. Rationality and responding to reasons Some philosophers think that rationality
More informationor conventional implicature [1]. If the implication is only pragmatic, explicating logical truth, and, thus, also consequence and inconsistency.
44 ANALYSIS explicating logical truth, and, thus, also consequence and inconsistency. Let C1 and C2 be distinct moral codes formulated in English. Let C1 contain a norm N and C2 its negation. The moral
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More informationGödel s Ontological Proof of the Existence of God
Prof. Dr. Elke Brendel Institut für Philosophie Lehrstuhl für Logik und Grundlagenforschung g Rheinische Friedrich-Wilhelms-Universität Bonn ebrendel@uni-bonn.de Gödel s Ontological Proof of the Existence
More informationIncreasing the risk of injury and proof of causation on the balance of probabilities. Sandy Steel
Increasing the risk of injury and proof of causation on the balance of probabilities Sandy Steel A risk is a probability of a negative outcome. 1 The concept of risk plays several distinct roles in relation
More informationBoolean Design of Patterns
123 Boolean Design of Patterns Basic weave structures interlacement patterns can be described in many ways, but they all come down to representing the crossings of warp and weft threads. One or the other
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationA. Arguments are made up of statements, which can be either true or false. Which of the following are statements?
Critical Thinking University of St Andrews March 2007 Bullet point material is not on the students copies. Feel free to use the material as you see fit, depending on timing, ability, enthusiasm etc. Good
More informationArguments and Methodology INTRODUCTION
chapter 1 Arguments and Methodology INTRODUCTION We should accept philosophical views in general, and moral views in particular, on the basis of the arguments offered in their support. It is therefore
More informationThe Problem of Evil not If God exists, she'd be OOG. If an OOG being exists, there would be no evil. God exists.
24.00: Problems of Philosophy Prof. Sally Haslanger September 14, 2005 The Problem of Evil Last time we considered the ontological argument for the existence of God. If the argument is cogent, then we
More information2. Argument Structure & Standardization
2. Argument Structure & Standardization 1 Some Review So, we have been looking at arguments: What is and is not an argument. The main parts of an argument. How to identify one when you see it. In the exercises
More informationThe Refutation of Relativism
The Refutation of Relativism There are many different versions of relativism: ethical relativism conceptual relativism, and epistemic relativism are three. In this paper, I will be concerned with only
More informationFinal Draft. Forthcoming in Synthese. The final publication is available at www.springerlink.com. DOI 10.1007/s11229-013-0248-6
Final Draft. Forthcoming in Synthese. The final publication is available at www.springerlink.com. DOI 10.1007/s11229-013-0248-6 Interpreting Enthymematic Arguments Using Belief Revision Georg Brun, Hans
More informationPhilosophy 104. Chapter 8.1 Notes
Philosophy 104 Chapter 8.1 Notes Inductive reasoning - The process of deriving general principles from particular facts or instances. - "induction." The American Heritage Dictionary of the English Language,
More informationExhibit memory of previously-learned materials by recalling facts, terms, basic concepts, and answers. Key Words
The Six Levels of Questioning Level 1 Knowledge Exhibit memory of previously-learned materials by recalling facts, terms, basic concepts, and answers. who what why when where which omit choose find how
More informationThe Structure of L2 Classroom Interaction
3 The Structure of L2 Classroom Interaction One of the most important features of all classroom discourse is that it follows a fairly typical and predictable structure, comprising three parts: a teacher
More informationCritical Analysis So what does that REALLY mean?
Critical Analysis So what does that REALLY mean? 1 The words critically analyse can cause panic in students when they first turn over their examination paper or are handed their assignment questions. Why?
More informationChapter 1 Verificationism Then and Now
Chapter 1 Verificationism Then and Now Per Martin-Löf The term veri fi cationism is used in two different ways: the fi rst is in relation to the veri fi cation principle of meaning, which we usually and
More informationHarvard College Program in General Education Faculty of Arts and Sciences Harvard University. A Guide to Writing in Ethical Reasoning 15
Harvard College Program in General Education Faculty of Arts and Sciences Harvard University A Guide to Writing in Ethical Reasoning 15 A Guide to Writing in Ethical Reasoning 15 Professor Jay M. Harris
More information