Draft of August 01, 2021 Matheus Silva
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
An Overview of Logic, Proofs, Set Theory, and Functions Contents
An Overview of Logic, Proofs, Set Theory, and Functions aBa Mbirika and Shanise Walker Contents 1 Numerical Sets and Other Preliminary Symbols3 2 Statements and Truth Tables5 3 Implications 9 4 Predicates and Quantifiers 13 5 Writing Formal Proofs 22 6 Mathematical Induction 29 7 Quick Review of Set Theory & Set Theory Proofs 33 8 Functions, Bijections, Compositions, Etc. 38 9 Solutions to all exercises 42 Index 51 Preface: This handout is meant primarily for those students who are already familiar with most of the subject matter contained within (that is, those who have taken a proofs class before). Its purpose is to provide a foundation as a proofs refresher when taking classes like Real Analysis I or II, Abstract Algebra I or II, Number Theory, Discrete Mathematics, Linear Algebra, etc. LICENSE Creative Commons License (CC BY-NC-SA): This text, including the art and illustrations, are available under the Creative Commons license (CC BY-NC- SA), allowing anyone to reuse, revise, remix and redistribute the text. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/4.0/ Section 1 NUMERICAL SETS AND OTHER PRELIMINARY SYMBOLS Page 3 1 Numerical Sets and Other Preliminary Symbols The following are numerical sets that you should familiarize yourself with: natural numbers 1 N = f1; 2; 3;:::g integers Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g a rational numbers Q = b j a; b 2 Z and b 6= 0 real numbers R = frational and irrational numbersg p complex numbers C = fa + bi j a; b 2 R and i = −1g p Gaussian integers Z[i] = fa + bi j a; b 2 Z and i = −1g 2πi Eisenstein integers Z[ρ] = fa + bρ j a; b 2 Z and ρ = e 3 g even integers 2Z = f2k j k 2 Zg odd integers 2Z + 1 = f2k + 1 j k 2 Zg arithmetic progression aZ + b = fak + b j k 2 Zg where a; b fixed CULTURAL QUESTION 1: Why are the integers denoted Z? ANSWER: The German word for \numbers" is Zahlen. -
Emergent Properties of Terrorist Networks, Percolation and Social Narrative
Emergent Properties of Terrorist Networks, Percolation and Social Narrative Maurice Passman Adaptive Risk Technology, Ltd. London, UK [email protected] Philip V. Fellman American Military University Charles Town, WV [email protected] Abstract In this paper, we have initiated an attempt to develop and understand the driving mechanisms that underlies fourth-generation warfare (4GW). We have undertaken this from a perspective of endeavoring to understand the drivers of these events (i.e. the 'Physics') from a Complexity perspective by using a threshold-type percolation model. We propose to integrate this strategic level model with tactical level Big Data, behavioral, statistical projections via a ‘fractal’ operational level model and to construct a hierarchical framework that allows dynamic prediction. Our initial study concentrates on this strategic level, i.e. a percolation model. Our main conclusion from this initial study is that extremist terrorist events are not solely driven by the size of a supporting population within a socio-geographical location but rather a combination of ideological factors that also depends upon the involvement of the host population. This involvement, through the social, political and psychological fabric of society, not only contributes to the active participation of terrorists within society but also directly contributes to and increases the likelihood of the occurrence of terrorist events. Our calculations demonstrate the links between Islamic extremist terrorist events, the ideologies within the Muslim and non-Muslim population that facilitates these terrorist events (such as Anti-Zionism) and anti- Semitic occurrences of violence against the Jewish population. In a future paper, we hope to extend the work undertaken to construct a predictive model and extend our calculations to other forms of terrorism such as Right Wing fundamentalist terrorist events within the USA. -
Argument Forms and Fallacies
6.6 Common Argument Forms and Fallacies 1. Common Valid Argument Forms: In the previous section (6.4), we learned how to determine whether or not an argument is valid using truth tables. There are certain forms of valid and invalid argument that are extremely common. If we memorize some of these common argument forms, it will save us time because we will be able to immediately recognize whether or not certain arguments are valid or invalid without having to draw out a truth table. Let’s begin: 1. Disjunctive Syllogism: The following argument is valid: “The coin is either in my right hand or my left hand. It’s not in my right hand. So, it must be in my left hand.” Let “R”=”The coin is in my right hand” and let “L”=”The coin is in my left hand”. The argument in symbolic form is this: R ˅ L ~R __________________________________________________ L Any argument with the form just stated is valid. This form of argument is called a disjunctive syllogism. Basically, the argument gives you two options and says that, since one option is FALSE, the other option must be TRUE. 2. Pure Hypothetical Syllogism: The following argument is valid: “If you hit the ball in on this turn, you’ll get a hole in one; and if you get a hole in one you’ll win the game. So, if you hit the ball in on this turn, you’ll win the game.” Let “B”=”You hit the ball in on this turn”, “H”=”You get a hole in one”, and “W”=”you win the game”. -
Logical Reasoning
updated: 11/29/11 Logical Reasoning Bradley H. Dowden Philosophy Department California State University Sacramento Sacramento, CA 95819 USA ii Preface Copyright © 2011 by Bradley H. Dowden This book Logical Reasoning by Bradley H. Dowden is licensed under a Creative Commons Attribution- NonCommercial-NoDerivs 3.0 Unported License. That is, you are free to share, copy, distribute, store, and transmit all or any part of the work under the following conditions: (1) Attribution You must attribute the work in the manner specified by the author, namely by citing his name, the book title, and the relevant page numbers (but not in any way that suggests that the book Logical Reasoning or its author endorse you or your use of the work). (2) Noncommercial You may not use this work for commercial purposes (for example, by inserting passages into a book that is sold to students). (3) No Derivative Works You may not alter, transform, or build upon this work. An earlier version of the book was published by Wadsworth Publishing Company, Belmont, California USA in 1993 with ISBN number 0-534-17688-7. When Wadsworth decided no longer to print the book, they returned their publishing rights to the original author, Bradley Dowden. If you would like to suggest changes to the text, the author would appreciate your writing to him at [email protected]. iii Praise Comments on the 1993 edition, published by Wadsworth Publishing Company: "There is a great deal of coherence. The chapters build on one another. The organization is sound and the author does a superior job of presenting the structure of arguments. -
Summary of the Full License, Which Is Available On-Line at to Harmony
Proofs and Concepts the fundamentals of abstract mathematics by Dave Witte Morris and Joy Morris University of Lethbridge incorporating material by P. D. Magnus University at Albany, State University of New York Preliminary Version 0.78 of May 2009 This book is offered under the Creative Commons license. (Attribution-NonCommercial-ShareAlike 2.0) The presentation of logic in this textbook is adapted from forallx An Introduction to Formal Logic P. D. Magnus University at Albany, State University of New York The most recent version of forallx is available on-line at http://www.fecundity.com/logic We thank Professor Magnus for making forallx freely available, and for authorizing derivative works such as this one. He was not involved in the preparation of this manuscript, so he is not responsible for any errors or other shortcomings. Please send comments and corrections to: [email protected] or [email protected] c 2006–2009 by Dave Witte Morris and Joy Morris. Some rights reserved. Portions c 2005–2006 by P. D. Magnus. Some rights reserved. Brief excerpts are quoted (with attribution) from copyrighted works of various authors. You are free to copy this book, to distribute it, to display it, and to make derivative works, under the following conditions: (1) Attribution. You must give the original author credit. (2) Noncommercial. You may not use this work for commercial purposes. (3) Share Alike. If you alter, transform, or build upon this work, you may distribute the resulting work only under a license identical to this one. — For any reuse or distribution, you must make clear to others the license terms of this work. -
35. Logic: Common Fallacies Steve Miller Kennesaw State University, [email protected]
Kennesaw State University DigitalCommons@Kennesaw State University Sexy Technical Communications Open Educational Resources 3-1-2016 35. Logic: Common Fallacies Steve Miller Kennesaw State University, [email protected] Cherie Miller Kennesaw State University, [email protected] Follow this and additional works at: http://digitalcommons.kennesaw.edu/oertechcomm Part of the Technical and Professional Writing Commons Recommended Citation Miller, Steve and Miller, Cherie, "35. Logic: Common Fallacies" (2016). Sexy Technical Communications. 35. http://digitalcommons.kennesaw.edu/oertechcomm/35 This Article is brought to you for free and open access by the Open Educational Resources at DigitalCommons@Kennesaw State University. It has been accepted for inclusion in Sexy Technical Communications by an authorized administrator of DigitalCommons@Kennesaw State University. For more information, please contact [email protected]. Logic: Common Fallacies Steve and Cherie Miller Sexy Technical Communication Home Logic and Logical Fallacies Taken with kind permission from the book Why Brilliant People Believe Nonsense by J. Steve Miller and Cherie K. Miller Brilliant People Believe Nonsense [because]... They Fall for Common Fallacies The dull mind, once arriving at an inference that flatters the desire, is rarely able to retain the impression that the notion from which the inference started was purely problematic. ― George Eliot, in Silas Marner In the last chapter we discussed passages where bright individuals with PhDs violated common fallacies. Even the brightest among us fall for them. As a result, we should be ever vigilant to keep our critical guard up, looking for fallacious reasoning in lectures, reading, viewing, and especially in our own writing. None of us are immune to falling for fallacies. -
DISPOSITION CONCEPTS and EXTENSIONAL LOGIC ------ARTHUR PAP------Not Be "Contrary-To-Fact") : If I Pull the Trigger, the Gun Will Fire
DISPOSITION CONCEPTS AND EXTENSIONAL LOGIC ------ARTHUR PAP------ not be "contrary-to-fact") : if I pull the trigger, the gun will fire. It would be sad if belief in such an implication were warranted only by knowledge of the truth of antecedent and consequent separately, for in that case it would be impossible for man to acquire the power of Disposition Concepts and Extensional Logic even limited control over the course of events by acquiring warranted beliefs about causal connections. Notice further that frequently pre sumed knowledge of a causal implication is a means to knowledge of the truth, or at least probability, of the antecedent; if this is an acid then it will turn blue litmus paper red; the reaction occurred; thus the One of the striking differences between natural languages, both con hypothesis is confirmed. Knowledge of the consequences of suppositions versational and scientific, and the extensional languages constructed by is independent of knowledge of the truth-values of the suppositions, no logicians is that most conditional statements, i.e., statements of the form matter whether the consequences be logical or causal. "if p, then q", of a natural language are not truth-functional. A state The difference between material implication and natural implica tion has been widely discussed. The logician's use of "if p, then q" ment compounded out of simpler statements is truth-functional if its in the truth-functional sense of "not both p and not-q", symbolized by truth-value is uniquely determined by the truth-values of the component "p :J q", is fully justified by the objective of constructing an adequate statements. -
Better Thinking and Reasoning Chapter 11 Test Answer
Name ________________________________ Better Thinking and Reasoning Chapter 11 Test Page 1 Answer each question. 1. Define formal fallacy. an error in the form of reasoning (caught by using a truth table) 2. Define informal fallacy. an error in the relationship of the premises (caught by examining context) 3. When is a valid argument unsound? when it has a false premise 4. What should you always do after you have completed your essay? rewrite it and check it for logical fallacies Classify the informal fallacy in each deductive argument as accident, begging the question, composition, division, or equivocation. 5. Anyone who thrusts a knife into another person should be arrested on the grounds of attempted murder. Since surgeons regularly put knives into their victims, they should be arrested for attempted murder. accident 6. Every atom in this chair is invisible. Therefore, the chair is invisible. composition 7. Please be quiet. You are making far too much noise. begging the question 8. A watermelon is a fruit. Therefore, a small watermelon is a small fruit. equivocation 9. The average American family has 2.4 children. The Smiths are an average American family. Therefore the Smiths have 2.4 children. division Classify each deductive argument as displaying a false dichotomy or having one of these three formal fallacies: affirming the consequent, denying the antecedent, or improper negation. 10. If your pet is a whale or a dolphin, then your pet is a mammal. Judy is not a mammal. Therefore, Judy is not a whale or not a dolphin. improper negation 11. If Probst won the election, then we would have a Democrat for mayor. -
The Role of Reasoning in Constructing a Persuasive Argument
The Role of Reasoning in Constructing a Persuasive Argument <http://www.orsinger.com/PDFFiles/constructing-a-persuasive-argument.pdf> [The pdf version of this document is web-enabled with linking endnotes] Richard R. Orsinger [email protected] http://www.orsinger.com McCurley, Orsinger, McCurley, Nelson & Downing, L.L.P. San Antonio Office: 1717 Tower Life Building San Antonio, Texas 78205 (210) 225-5567 http://www.orsinger.com and Dallas Office: 5950 Sherry Lane, Suite 800 Dallas, Texas 75225 (214) 273-2400 http://www.momnd.com State Bar of Texas 37th ANNUAL ADVANCED FAMILY LAW COURSE August 1-4, 2011 San Antonio CHAPTER 11 © 2011 Richard R. Orsinger All Rights Reserved The Role of Reasoning in Constructing a Persuasive Argument Chapter 11 Table of Contents I. THE IMPORTANCE OF PERSUASION.. 1 II. PERSUASION IN ARGUMENTATION.. 1 III. BACKGROUND.. 2 IV. USER’S GUIDE FOR THIS ARTICLE.. 2 V. ARISTOTLE’S THREE COMPONENTS OF A PERSUASIVE SPEECH.. 3 A. ETHOS.. 3 B. PATHOS.. 4 C. LOGOS.. 4 1. Syllogism.. 4 2. Implication.. 4 3. Enthymeme.. 4 (a) Advantages and Disadvantages of Commonplaces... 5 (b) Selection of Commonplaces.. 5 VI. ARGUMENT MODELS (OVERVIEW)... 5 A. LOGIC-BASED ARGUMENTS. 5 1. Deductive Logic.. 5 2. Inductive Logic.. 6 3. Reasoning by Analogy.. 7 B. DEFEASIBLE ARGUMENTS... 7 C. THE TOULMIN ARGUMENTATION MODEL... 7 D. FALLACIOUS ARGUMENTS.. 8 E. ARGUMENTATION SCHEMES.. 8 VII. LOGICAL REASONING (DETAILED ANALYSIS).. 8 A. DEDUCTIVE REASONING.. 8 1. The Categorical Syllogism... 8 a. Graphically Depicting the Simple Categorical Syllogism... 9 b. A Legal Dispute as a Simple Syllogism.. 9 c. -
Logical Reasoning, World Knowledge, and Mental Imagery: Interconnections in Cognitive Processes
Memory & Cognition 1986, 14 (4), 299-307 Logical reasoning, world knowledge, and mental imagery: Interconnections in cognitive processes CATHERINE A. CLEMENT and RACHEL JOFFE FALMAGNE Clark University, Worcester, Massachusetts To better understand the role of problem content in verbal reasoning, the effect of two aspects of problem representation on conditional reasoning was examined. Specifically, this study focused on the effect of availability of knowledge schemata and mental imagery on recognition of indeter minacy. Four groups of20 adults solved syllogisms that varied in imagery value and in tendency to access knowledge schemata (assessed by ratings of the relatedness of antecedent and conse quent clauses of premises). When problems both were high in imagery value and had related clauses, performance was significantly better on indeterminate syllogisms. Access to schemata may permit elaborative processing and the generation of counterexamples to invalid inferences; imagery may support representation of problems and generation of elaborative information in memory. It has been well documented that meaningful thematic The experiments reported in this article focus on con material can influence the reasoning process in deductive ditional reasoning in standardconditionalreasoningtasks, reasoning tasks, most notably by improving subjects' that is, evaluating implicationsof statementsformed with differentiation between logically necessary and logically the conditional connective "ifp then q." There are four indeterminate inferences (e.g., Kuhn, 1977; Wason & basic syllogistic forms associated with the connective; two Johnson-Laird, 1972). However, the particular proper yield determinate solutions and two are indeterminate.1 ties of this material that are effectiveand the reasons why To distinguishdeterminatefrom indeterminateproblems, they are effective have not been characterized in specific subjectsmust realizethat the conditionalrelation between terms. -
Fallacies.Pdf
2 Fallacies Fallacies Ex. God must exist, because the Bible says so, and we can trust the Bible because God inspired it. False analogy An argument extending an analogy beyond its limits. Logicians have categorized many types of fallacies that Ex. Saint Patrick likened the Trinity to the three leafs of commonly occur in arguments; knowing their names can help a clover. Clovers are mortal, so the Trinity must also be you in intelligently critiquing arguments. Here, I’ve selected mortal. ten common fallacies. False dilemma An argument for something by arguing against The page http://www.datanation.com/fallacies/ an opposing possibility, when in fact another alternative discusses other types of fallacies. is possible. Ad hominem argument Somebody counterargues by appeal- Ex. If you don’t own a car, you’ll never be able to go ing to characteristics of the person making the argu- anywhere. ment. Hasty generalization An argument appealing to a small num- Ex. Why should we listen to your arguments against ber of particular cases. abortion? You’re just following the Catholic Church’s Ex. Catholic colleges always have excellent football pro- teaching. grams: Look at SJU and Notre Dame. Ad populum argument An argument based on the opinion Post hoc reasoning An argument that since something hap- of a large number of people. pened after taking an action, the action must have caused Ex. Same-sex couples should not be allowed to marry it. because the majority of Americans believe it should con- Ex. When the speed limit was reduced to 55 mph, traffic tinue to be illegal. -
On the Pragmatic Approach to Counterpossibles
Philosophia https://doi.org/10.1007/s11406-018-9979-4 On the Pragmatic Approach to Counterpossibles Maciej Sendłak1 Received: 5 December 2017 /Revised: 6 February 2018 /Accepted: 3 May 2018 # The Author(s) 2018 Abstract Nina Emery and Christopher Hill proposed a pragmatic approach toward the debate about counterpossibles—i.e., counterfactuals with impossible anteced- ents. The core of this approach is to move the burden of the problem from the notion of truth value into the notion of assertion. This is meant to explain our pre- theoretical intuitions about counterpossibles while claiming that each and every counterpossible is vacuously true. The aim of this paper is to indicate a problematic aspect of this view. Keywords Counterpossibles.Counterfactuals.Pragmatics.Assertion.Possibleworlds. Impossible Worlds The subject of this paper is counterpossibles; i.e., subjunctive conditionals of the form ‘If it were the case that A, then it would be the case that C’ (‘A>C’), where ‘A’ expresses impossibility (necessary falseness).1 The problem of counterpossibles is the problem that arises in the face of worlds semantics for counterfactuals, combined with the assumption that there are no impossible worlds. The consequence of these is that every counterfactual with an impossible antecedent is vacuously true. This, however, is difficult to square with pre-theoretical intuitions that some counterpossibles are false. Consequently, there is a question of whether it is justified to introduce a modification of 1Earlier versions of this material were presented in Szczecin (University of Szczecin) at the seminar of “Cognition & Communication Research Group” and in Munich (Ludwig Maximilian University of Munich) at “The ninth European Congress of Analytic Philosophy.” I am grateful to the participants of these meetings for their helpful comments and discussions, especially to Arkadiusz Chrudzimski and Joanna Odrowąż- Sypniewska.