7.4 Abbreviated Truth Tables
|
|
- Ralph Ethelbert McLaughlin
- 7 years ago
- Views:
Transcription
1 7.4 Abbreviated Truth Tables The full truth table method of Section 7.3 is extremely cumbersome. For example, an argument with only four statement letters requires a truth table with 2 4 = 32 rows. One with five requires a truth table with 2 5 = 64 rows. Obviously, truth tables of these sizes are simply impractical to construct. Abbreviated truth tables provide a much more efficient method for determining validity. The method The key insight behind the method If we can construct just one row of a truth table for an argument that makes the premises true and the conclusion false, then we will have shown the argument to be invalid. If we fail at such an attempt, we will have shown the argument to be valid. Recall the argument from the lecture for 7.3: Abortion is permissible only if fetuses are not innocent human beings or it is not always wrong to kill innocent human beings. But it is always wrong to kill innocent human beings. So abortion is not permissible. (A: Abortion is permissible; B: Fetuses are innocent human beings; W: It is always wrong to kill innocent human beings.)
2 We symbolized this sentence as follows: [A ( B W)]. The method applied to an invalid argument 1. Write down the symbolized argument thus (note that we ve dispensed with periods; they just get in the way): 2. Hypothesize that the premises are true and the conclusion false: T T F 3. Calculate the immediate consequences of this hypothesis: Copy the truth value assigned to W to its other occurrence: T T T F Calculate truth values of the compound statements whenever you know the truth values of their component parts. Thus, we can calculate that W is false in virtue of our assumption that W is true: 2
3 T F T T F Calculate truth values of the component parts of compound statements whenever you know the truth values of the compound statements (and copy any truth values assigned to statement letters to all other occurrences). Thus, we can calculate that A must be true given that A is false: T T F T T F T Furthermore, since (i) we ve hypothesized that [A ( B W)] is true and (ii) we have calculated that the antecedent A of [A ( B W)] must be true (given our initial hypothesis about A), it now follows that ( B W) must be true as well, lest we falsify our hypothesis that [A ( B W)] is true. (Recall that a conditional is false if its antecedent is true and its consequent is false.) Hence: T T T F T T F T But now that we have deduced that the disjunction ( B W) must be true (given our initial hypotheses), we know that, because its right disjunct W is false, its left disjunct B must be true for a disjunction is true if and only if at least one of its disjuncts is. Thus: T T T T F T T F T 3
4 And this, of course, enables us now to deduce that B is false, which completes the row: T T T F T F T T F T So we have identified a row that makes the premises of our argument true and the conclusion false, so we have thereby demonstrated that the argument is invalid when A is true, B is false, and W is true. We complete the abbreviated truth table by recording this invalidating truth value assignment into the table: A B W T F T T T T F T F T T F T Much shorter than the full truth table! To remind you: A B W T T T F F F F T F T T F T F T T F F T F T T T T F T F T F F T T T T F F F T T T F F F T T F T F T F T T F T F F T T T T F T T F F F T T T T F T Note that row 3 is exactly the row that we just constructed using the abbreviated truth table method. 4
5 The method applied to a valid argument What happens if the argument in question is valid? We demonstrate with a further example. We ll cut right to the chase with a symbolized argument without worrying about the English argument it symbolizes. We begin with the usual hypothesis that the premises are true and the conclusion false: T T T F Since the conclusion (W J) is false, its component statement (W J) must be true: T T T F T From the truth table schema for, the only way for (W J) to be true is if both W and J are true. So we record this information below the conclusion, and copy it into the rest of the table: T T T T T T T F T T T So far so good. Now, from the fact that Z is true, we infer that Z is false, and record this information into the table: T T T T F T T F T F F T T T 5
6 This then enables us to fill in the truth values for the two disjuncts in the second premise in accordance with the truth table schema for conditions: T T T T F F T T F F T F F T T T But now there is a problem: our hypothesis was that the three premises were all true. But the second premise is a disjunction with two false disjuncts. Hence, it is itself false. But this conflicts with our initial hypothesis. Hence, we have shown that it is not possible after all for the premises of the argument to be true and the conclusion false. We indicate the contradiction that arises when we assume otherwise by putting T/F underneath the main logical operator of the problematic premise (all of them, if there are more than one): T T T T F F T/F T F F T F F T T T When there are several ways the conclusion can be false If the conclusion can be made false in more than one way, then every way must be tried until either an invalidating assignment is found or all the ways of making the conclusion false are exhausted. Again, we start with the usual hypothesis: T T T F But note that, by the truth table schema for, there are three ways to make the conclusion false. So we simply pick one of the three to start with, viz., the one where both B and D are false: 6
7 T T T F F F Copying these to the rest of the row and calculating truth values we get: T F T T F T F F F Because of the truth table schema for and our hypothesis that (C D) is true, it follows that C has to be false as well: T F T F T F T F F F Copying this value to (A C) we get T F T F T F T F F F F But now we see that there will be no way to make (A C) true, since one of its conjuncts is false. Consequently, this particular way of making the conclusion false fails to yield an invalidating assignment, and we so indicate: T F T F T F T/F F F F F 7
8 Note: The Web Tutor will require you to complete the row. To do so, just assign an arbitrary value to A (it won t matter because (A C) will be false either way just because C is); F is easiest in this case: Now we need to try the two remaining ways of making the conclusion false; we will (arbitrarily) try the one where B is true and D is false first: T T T T F F But, when calculated out, this possibility fares no better than the first: F T T F T T T/F F T T T T F F So we must try the final case, where B is false and D is true: F T T F T T T/F F T T T T F F T T T F F T 8
9 But, when calculated out, this possibility too fails to yield an invalidating assignment: F T T F T T T/F F T T T T F F T F T T F T T T F T/F T F F T Only now that we have exhausted the possible ways of making the conclusion false are we permitted to say that the argument is valid; for on all the possible ways of making the conclusion false we found we were unable to make the premises true. If, however, any of these ways of making the conclusion false had yielded a row with true premises, the argument would have been shown to be invalid. To emphasize: to show invalidity, all it takes is one row, one truth value assignment to the statement letters, on which the premises are true and the conclusion false. 9
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationLecture 9 Maher on Inductive Probability
Lecture 9 Maher on Inductive Probability Patrick Maher Scientific Thought II Spring 2010 Two concepts of probability Example You know that a coin is either two-headed or two-tailed but you have no information
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 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 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 information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More informationLesson 9 Hypothesis Testing
Lesson 9 Hypothesis Testing Outline Logic for Hypothesis Testing Critical Value Alpha (α) -level.05 -level.01 One-Tail versus Two-Tail Tests -critical values for both alpha levels Logic for Hypothesis
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 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 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 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 informationBase Conversion written by Cathy Saxton
Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,
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 informationDEVELOPING HYPOTHESIS AND
Shalini Prasad Ajith Rao Eeshoo Rehani DEVELOPING 500 METHODS SEPTEMBER 18 TH 2001 DEVELOPING HYPOTHESIS AND Introduction Processes involved before formulating the hypotheses. Definition Nature of Hypothesis
More informationsome ideas on essays and essay writing
Disability and Dyslexia Service: Study Skills for Students some ideas on essays and essay writing why this document might be helpful for students: Before beginning work on an essay, it is vital to know
More informationPredicate Logic. For example, consider the following argument:
Predicate Logic The analysis of compound statements covers key aspects of human reasoning but does not capture many important, and common, instances of reasoning that are also logically valid. For example,
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 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 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 informationIntroduction to Hypothesis Testing OPRE 6301
Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about
More informationMcKinsey Problem Solving Test Top Tips
McKinsey Problem Solving Test Top Tips 1 McKinsey Problem Solving Test You re probably reading this because you ve been invited to take the McKinsey Problem Solving Test. Don t stress out as part of the
More informationColored Hats and Logic Puzzles
Colored Hats and Logic Puzzles Alex Zorn January 21, 2013 1 Introduction In this talk we ll discuss a collection of logic puzzles/games in which a number of people are given colored hats, and they try
More informationFull and Complete Binary Trees
Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full
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 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 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 informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
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 information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationSolving simultaneous equations using the inverse matrix
Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
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 informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationNote taking skills - from lectures and readings
Introduction Department of Lifelong Learning: Study Skills Series Note taking skills - from lectures and readings When you are at university, the sheer amount of information that is delivered to you can
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 informationOutline. Written Communication Conveying Scientific Information Effectively. Objective of (Scientific) Writing
Written Communication Conveying Scientific Information Effectively Marie Davidian davidian@stat.ncsu.edu http://www.stat.ncsu.edu/ davidian. Outline Objectives of (scientific) writing Important issues
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
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 information24. PARAPHRASING COMPLEX STATEMENTS
Chapter 4: Translations in Sentential Logic 125 24. PARAPHRASING COMPLEX STATEMENTS As noted earlier, compound statements may be built up from statements which are themselves compound statements. There
More informationModule 15 Exercise 3 How to use varied and correct sentence structures
Section 1A: Comprehension and Insight skills based on short stories Module 15 Exercise 3 How to use varied and correct sentence structures Before you begin What you need: Related text: Powder by Tobias
More informationAP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics
Ms. Foglia Date AP: LAB 8: THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,
More informationNo Solution Equations Let s look at the following equation: 2 +3=2 +7
5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are
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 informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
More informationObjections to Friedman s Shareholder/Stockholder Theory
Objections to Friedman s Shareholder/Stockholder Theory 1. Legal Morally Permissible: Almeder offers several criticisms of Friedman s claim that the only obligation of businesses is to increase profit
More informationNPV Versus IRR. W.L. Silber -1000 0 0 +300 +600 +900. We know that if the cost of capital is 18 percent we reject the project because the NPV
NPV Versus IRR W.L. Silber I. Our favorite project A has the following cash flows: -1 + +6 +9 1 2 We know that if the cost of capital is 18 percent we reject the project because the net present value is
More informationTypes of Error in Surveys
2 Types of Error in Surveys Surveys are designed to produce statistics about a target population. The process by which this is done rests on inferring the characteristics of the target population from
More informationMathematical 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 informationA Short Course in Logic Example 8
A Short ourse in Logic xample 8 I) Recognizing Arguments III) valuating Arguments II) Analyzing Arguments valuating Arguments with More than one Line of Reasoning valuating If then Premises Independent
More informationSection 4.1 Rules of Exponents
Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells
More informationBuilding Qualtrics Surveys for EFS & ALC Course Evaluations: Step by Step Instructions
Building Qualtrics Surveys for EFS & ALC Course Evaluations: Step by Step Instructions Jennifer DeSantis August 28, 2013 A relatively quick guide with detailed explanations of each step. It s recommended
More informationThesis Statement & Essay Organization Mini-Lesson (Philosophy)
Thesis Statement & Essay Organization Mini-Lesson (Philosophy) Lesson Objective Students will learn several strategies for organizing short, persuasive essays, preferably after they have started pre-writing.
More informationPHI 201, Introductory Logic p. 1/16
PHI 201, Introductory Logic p. 1/16 In order to make an argument, you have to make a claim (the conclusion) and you have to give some evidence for the claim (the premises). Bush tried to justify the war
More informationA Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution
A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September
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 informationHypothesis testing. c 2014, Jeffrey S. Simonoff 1
Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there
More informationChapter 9. Systems of Linear Equations
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
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 informationPlanning and Writing Essays
Planning and Writing Essays Many of your coursework assignments will take the form of an essay. This leaflet will give you an overview of the basic stages of planning and writing an academic essay but
More informationScience and Scientific Reasoning. Critical Thinking
Science and Scientific Reasoning Critical Thinking Some Common Myths About Science Science: What it is and what it is not Science and Technology Science is not the same as technology The goal of science
More informationFinding the last cell in an Excel range Using built in Excel functions to locate the last cell containing data in a range of cells.
Finding the last cell in an Excel range Using built in Excel functions to locate the last cell containing data in a range of cells. There are all sorts of times in Excel when you will need to find the
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 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 informationLecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
More informationStudy questions Give a short answer to the following questions:
Chapter 9 The Morality of Abortion 9.1 Homework Readings DW 15-17 Study questions Give a short answer to the following questions: 1. What are the two conflicting values in the abortion debate? 2. Explain
More informationDraft Copy: Do Not Cite Without Author s Permission
WHAT S WRONG WITH THE FUTURE OF VALUE ARGUMENT (1/8/2015) A. WHAT THE FUTURE OF VALUE ARGUMENT IS According to the future of value argument, what makes it wrong to kill those postnatal human beings we
More informationIndependent samples t-test. Dr. Tom Pierce Radford University
Independent samples t-test Dr. Tom Pierce Radford University The logic behind drawing causal conclusions from experiments The sampling distribution of the difference between means The standard error of
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 informationFormal Languages and Automata Theory - Regular Expressions and Finite Automata -
Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March
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 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 informationIntroduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
More informationThe Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
More informationEXTENDED LEARNING MODULE A
EXTENDED LEARNING MODULE A DESIGNING DATABASES AND ENTITY- RELATIONSHIP DIAGRAMMING Student Learning Outcomes 1. Identify how databases and spreadsheets are both similar and different. 2. List and describe
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 informationLast time we had arrived at the following provisional interpretation of Aquinas second way:
Aquinas Third Way Last time we had arrived at the following provisional interpretation of Aquinas second way: 1. 2. 3. 4. At least one thing has an efficient cause. Every causal chain must either be circular,
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More informationNUMBER SYSTEMS APPENDIX D. You will learn about the following in this appendix:
APPENDIX D NUMBER SYSTEMS You will learn about the following in this appendix: The four important number systems in computing binary, octal, decimal, and hexadecimal. A number system converter program
More informationChapter 14: Boolean Expressions Bradley Kjell (Revised 10/08/08)
Chapter 14: Boolean Expressions Bradley Kjell (Revised 10/08/08) The if statements of the previous chapters ask simple questions such as count
More informationChapter 4: The Logic of Boolean Connectives
Chapter 4: The Logic of Boolean Connectives 4.1 Tautologies and logical truth Logical truth We already have the notion of logical consequence. A sentence is a logical consequence of a set of sentences
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More information