1 Culture, Dialectics, and Reasoning About
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Chapter 5: Methods of Proof for Boolean Logic
Chapter 5: Methods of Proof for Boolean Logic § 5.1 Valid inference steps Conjunction elimination Sometimes called simplification. From a conjunction, infer any of the conjuncts. • From P ∧ Q, infer P (or infer Q). Conjunction introduction Sometimes called conjunction. From a pair of sentences, infer their conjunction. • From P and Q, infer P ∧ Q. § 5.2 Proof by cases This is another valid inference step (it will form the rule of disjunction elimination in our formal deductive system and in Fitch), but it is also a powerful proof strategy. In a proof by cases, one begins with a disjunction (as a premise, or as an intermediate conclusion already proved). One then shows that a certain consequence may be deduced from each of the disjuncts taken separately. One concludes that that same sentence is a consequence of the entire disjunction. • From P ∨ Q, and from the fact that S follows from P and S also follows from Q, infer S. The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from it. You repeat this process for each disjunct. So it doesn’t matter which disjunct is true—you get the same conclusion in any case. Hence you may infer that it follows from the entire disjunction. In practice, this method of proof requires the use of “subproofs”—we will take these up in the next chapter when we look at formal proofs. -
In Defence of Constructive Empiricism: Metaphysics Versus Science
To appear in Journal for General Philosophy of Science (2004) In Defence of Constructive Empiricism: Metaphysics versus Science F.A. Muller Institute for the History and Philosophy of Science and Mathematics Utrecht University, P.O. Box 80.000 3508 TA Utrecht, The Netherlands E-mail: [email protected] August 2003 Summary Over the past years, in books and journals (this journal included), N. Maxwell launched a ferocious attack on B.C. van Fraassen's view of science called Con- structive Empiricism (CE). This attack has been totally ignored. Must we con- clude from this silence that no defence is possible against the attack and that a fortiori Maxwell has buried CE once and for all, or is the attack too obviously flawed as not to merit exposure? We believe that neither is the case and hope that a careful dissection of Maxwell's reasoning will make this clear. This dis- section includes an analysis of Maxwell's `aberrance-argument' (omnipresent in his many writings) for the contentious claim that science implicitly and per- manently accepts a substantial, metaphysical thesis about the universe. This claim generally has been ignored too, for more than a quarter of a century. Our con- clusions will be that, first, Maxwell's attacks on CE can be beaten off; secondly, his `aberrance-arguments' do not establish what Maxwell believes they estab- lish; but, thirdly, we can draw a number of valuable lessons from these attacks about the nature of science and of the libertarian nature of CE. Table of Contents on other side −! Contents 1 Exordium: What is Maxwell's Argument? 1 2 Does Science Implicitly Accept Metaphysics? 3 2.1 Aberrant Theories . -
Buddhism and Holistic Versus Analytic Thought
CULTURE, RELIGION AND COGNITION: BUDDHISM AND HOLISTIC VERSUS ANALYTIC THOUGHT by Alain Samson A thesis submitted for the degree of Doctor of Philosophy (PhD) 2007 University of London The London School of Economics and Political Science Institute of Social Psychology UMI Number: U615882 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U615882 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Declaration I certify that the thesis I have presented for examination for the MPhil/PhD degree of the London School of Economics and Political Science is solely my own work other than where I have clearly indicated that it is the work of others (in which case the extent of any work carried out jointly by me and any other person is clearly identified in it). The copyright of this thesis rests with the author. Quotation from it is permitted, provided that full acknowledgement is made. This thesis may not be reproduced without the prior written consent of the author. I warrant that this authorization does not, to the best of my belief, infringe the rights of 2 British ity<;(. -
Deduction (I) Tautologies, Contradictions And
D (I) T, & L L October , Tautologies, contradictions and contingencies Consider the truth table of the following formula: p (p ∨ p) () If you look at the final column, you will notice that the truth value of the whole formula depends on the way a truth value is assigned to p: the whole formula is true if p is true and false if p is false. Contrast the truth table of (p ∨ p) in () with the truth table of (p ∨ ¬p) below: p ¬p (p ∨ ¬p) () If you look at the final column, you will notice that the truth value of the whole formula does not depend on the way a truth value is assigned to p. The formula is always true because of the meaning of the connectives. Finally, consider the truth table table of (p ∧ ¬p): p ¬p (p ∧ ¬p) () This time the formula is always false no matter what truth value p has. Tautology A statement is called a tautology if the final column in its truth table contains only ’s. Contradiction A statement is called a contradiction if the final column in its truth table contains only ’s. Contingency A statement is called a contingency or contingent if the final column in its truth table contains both ’s and ’s. Let’s consider some examples from the book. Can you figure out which of the following sentences are tautologies, which are contradictions and which contingencies? Hint: the answer is the same for all the formulas with a single row. () a. (p ∨ ¬p), (p → p), (p → (q → p)), ¬(p ∧ ¬p) b. -
Leibniz's Ontological Proof of the Existence of God and the Problem Of
Leibniz’s Ontological Proof of the Existence of God and the Problem of »Impossible Objects« Wolfgang Lenzen (Osnabrück) Abstract The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived – without any further assumptions about the external world – from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from contradiction. A complete proof will rather have to consist of two parts. First, a proof of premise (1) God is possible. Second, a demonstration of the “remarkable proposition” (2) If God is possible, then God exists. The present contribution investigates an interesting paper in which Leibniz tries to prove proposition (2). It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ (denoting the concept of existence) as follows: (3) g =df ιxE(x). Principle (2) which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows: (4) ◊E(ιxE(x)) → E(ιxE(x)). At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to (4), the possibility of the necessary being has to be understood in the sense of something which possibly exists. -
Contrastive Empiricism
Elliott Sober Contrastive Empiricism I Despite what Hegel may have said, syntheses have not been very successful in philosophical theorizing. Typically, what happens when you combine a thesis and an antithesis is that you get a mishmash, or maybe just a contradiction. For example, in the philosophy of mathematics, formalism says that mathematical truths are true in virtue of the way we manipulate symbols. Mathematical Platonism, on the other hand, holds that mathematical statements are made true by abstract objects that exist outside of space and time. What would a synthesis of these positions look like? Marks on paper are one thing, Platonic forms an other. Compromise may be a good idea in politics, but it looks like a bad one in philosophy. With some trepidation, I propose in this paper to go against this sound advice. Realism and empiricism have always been contradictory tendencies in the philos ophy of science. The view I will sketch is a synthesis, which I call Contrastive Empiricism. Realism and empiricism are incompatible, so a synthesis that merely conjoined them would be a contradiction. Rather, I propose to isolate important elements in each and show that they combine harmoniously. I will leave behind what I regard as confusions and excesses. The result, I hope, will be neither con tradiction nor mishmash. II Empiricism is fundamentally a thesis about experience. It has two parts. First, there is the idea that experience is necessary. Second, there is the thesis that ex perience suffices. Necessary and sufficient for what? Usually this blank is filled in with something like: knowledge of the world outside the mind. -
Indigenous and Cultural Psychology
Indigenous and Cultural Psychology Understanding People in Context International and Cultural Psychology Series Series Editor: Anthony Marsella, University of Hawaii, Honolulu, Hawaii ASIAN AMERICAN MENTAL HEALTH Assessment Theories and Methods Edited by Karen S. Kurasaki, Sumie Okazaki, and Stanley Sue THE FIVE-FACTOR MODEL OF PERSONALITY ACROSS CULTURES Edited by Robert R. McCrae and Juri Allik FORCED MIGRATION AND MENTAL HEALTH Rethinking the Care of Refugees and Displaced Persons Edited by David Ingleby HANDBOOK OF MULTICULTURAL PERSPECTIVES ON STRESS AND COPING Edited by Paul T.P. Wong and Lilian C.J. Wong INDIGENOUS AND CULTURAL PSYCHOLOGY Understanding People in Context Edited by Uichol Kim, Kuo-Shu Yang, and Kwang-Kuo Hwang LEARNING IN CULTURAL CONTEXT Family, Peers, and School Edited by Ashley Maynard and Mary Martini POVERTY AND PSYCHOLOGY From Global Perspective to Local Practice Edited by Stuart C. Carr and Tod S. Sloan PSYCHOLOGY AND BUDDHISM From Individual to Global Community Edited by Kathleen H. Dockett, G. Rita Dudley-Grant, and C. Peter Bankart SOCIAL CHANGE AND PSYCHOSOCIAL ADAPTATION IN THE PACIFIC ISLANDS Cultures in Transition Edited by Anthony J. Marsella, Ayda Aukahi Austin, and Bruce Grant TRAUMA INTERVENTIONS IN WAR AND PEACE Prevention, Practice, and Policy Edited by Bonnie L. Green, Matthew J. Friedman, Joop T.V.M. de Jong, Susan D. Solomon, Terence M. Keane, John A. Fairbank, Brigid Donelan, and Ellen Frey-Wouters A Continuation Order Plan is available for this series. A continuation order will bring deliv- ery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher. -
Logic, Proofs
CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. 1.1.1. Connectives, Truth Tables. Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation p “not p” Conjunction p¬ q “p and q” Disjunction p ∧ q “p or q (or both)” Exclusive Or p ∨ q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p → q “p if and only if q” ↔ The truth value of a compound proposition depends only on the value of its components. Writing F for “false” and T for “true”, we can summarize the meaning of the connectives in the following way: 6 1.1. PROPOSITIONS 7 p q p p q p q p q p q p q T T ¬F T∧ T∨ ⊕F →T ↔T T F F F T T F F F T T F T T T F F F T F F F T T Note that represents a non-exclusive or, i.e., p q is true when any of p, q is true∨ and also when both are true. -
Logic, Sets, and Proofs David A
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical statements with capital letters A; B. Logical statements be combined to form new logical statements as follows: Name Notation Conjunction A and B Disjunction A or B Negation not A :A Implication A implies B if A, then B A ) B Equivalence A if and only if B A , B Here are some examples of conjunction, disjunction and negation: x > 1 and x < 3: This is true when x is in the open interval (1; 3). x > 1 or x < 3: This is true for all real numbers x. :(x > 1): This is the same as x ≤ 1. Here are two logical statements that are true: x > 4 ) x > 2. x2 = 1 , (x = 1 or x = −1). Note that \x = 1 or x = −1" is usually written x = ±1. Converses, Contrapositives, and Tautologies. We begin with converses and contrapositives: • The converse of \A implies B" is \B implies A". • The contrapositive of \A implies B" is \:B implies :A" Thus the statement \x > 4 ) x > 2" has: • Converse: x > 2 ) x > 4. • Contrapositive: x ≤ 2 ) x ≤ 4. 1 Some logical statements are guaranteed to always be true. These are tautologies. Here are two tautologies that involve converses and contrapositives: • (A if and only if B) , ((A implies B) and (B implies A)). In other words, A and B are equivalent exactly when both A ) B and its converse are true. -
Atomistic Empiricism Or Holistic Empiricism?*
ATOMISTIC EMPIRICISM OR HOLISTIC EMPIRICISM?∗ I. Atomistic Empiricism Traditional empiricism was atomistic: it assumed that scientific knowledge may be tested through direct comparison of the “atoms” of knowledge, i.e. particular sentences, with the “atoms” of experience, that is separate results of observation and experiments, or through the confrontation of small units (if only sensible) of knowledge with small units (“elementary” units) of experience. “Atomistic” tendencies in empiricism can be best observed in positivism. Especially the so-called doctrine of logical atomism in which the notions of atomic sentence and molecular sentence come to the fore, proclaimed mainly by Russell and Wittgenstein, represents decisively the viewpoint of atomistic empiricism. One of the fundamental dogmas of neopositivism is the thesis on the possibility of empirical confirmation or refutation of every statement, taken separately, by means of particular elements of experience.1 In principle, the thesis is also shared by a critic of some of other dogmas of logical empiricism – K.R. Popper. II. The Concept of Holistic Empiricism It appears that the systemic character of knowledge makes it impossible to juxtapose separate theoretical statements with experimental data; what is more, theoretical hypotheses (laws) are not only and cannot be on the whole separately tested but mostly are not (and cannot be) separated ∗ Translation of “Empiryzm atomistyczny czy empiryzm holistyczny?” Studia Filozoficzne, 12, 1975, 149-160. 1 W. Mejbaum writes that empiricism of neopositivists is based on a conviction about the divisibility of experimental material into particular building blocks: impressions in the subjectivist version, and facts in the objectivist version. We subordinate sentences to those events, and a set of those sentences may be considered to be a starting point of a programme of logical reconstruction of knowledge. -
Thinking Patterns Or Errors of Chinese Singaporeans
Thinking Patterns or Errors of Chinese Singaporeans D R . FOO, KOONG HEAN JAMES COOK UNIVERSITY, SINGAPORE CAMPUS 21 OCTOBER 2008 Sources of Paper Feedback on CBT and Chinese clients from MHPs—presented at the 1st Asian CBT Conference Personal experience in CBT with clients Survey with class in Positive Psychology Sources of Paper Basing on Automatic (Negative)Thoughts by Aaron T. Beck and Common False Beliefs by Adele B. Lynn Combining and Modifying the said Thoughts and Beliefs Proposed new definitions and labels Singapore in Brief an Asian state a complex metropolis embracing a mix of Eastern and Westernized values, attitudes and lifestyles Chinese Singaporeans make up over 75% of the 4.5 million population of Singapore Aaron T. Beck’s List of (-ve) Automatic Thoughts Label Explanation All–or-nothing (Also called black-and-white, polarized, or dichotomous thinking): You view a situation in only thinking two categories instead of on a continuum. Example: “If I’m not a total success, I’m a failure.” Catastrophizing (Also called fortune telling): You predict the future negatively without considering other, more likely outcomes. Example: “I’ll be so upset, I won’t be able to function at all.” Disqualifying or You unreasonably tell yourself that positive experiences, deeds, or qualities do not count. discounting the Example: “I did that project well, but that doesn’t mean I’m competent; I just got lucky.” positive Emotional You think something must be true because you “feel” (actually believe) it so strongly, ignoring reasoning or discounting evidence to the contrary. Example: “I know I do a lot of things okay at work, but I still feel like I’m a failure.” Labelling You put a fixed, global label on yourself or others without considering that the evidence might more reasonably lead to a less disastrous conclusion. -
On the Indemonstrability of the Principle of Contradiction
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 6-23-2003 On the Indemonstrability of the Principle of Contradiction Elisabeta Sarca University of South Florida Follow this and additional works at: https://scholarcommons.usf.edu/etd Part of the American Studies Commons Scholar Commons Citation Sarca, Elisabeta, "On the Indemonstrability of the Principle of Contradiction" (2003). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/1465 This Thesis is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. On the Indemonstrability of the Principle of Contradiction by Elisabeta Sarca A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Department of Philosophy College of Arts and Sciences University of South Florida Major Professor: John P. Anton, Ph.D. Kwasi Wiredu, Ph.D. Willis Truitt, Ph.D. Eric Winsberg, Ph.D. Date of Approval: June 23, 2003 Keywords: logic, metaphysics, aristotle, mill, russell-whitehead © Copyright 2003, Elisabeta Sarca Acknowledgements I would like to express my gratitude to Dr. John Anton for his invaluable advice and guidance throughout the preparation of this thesis and to Dr. Kwasi Wiredu, Dr. Willis Truitt and Dr. Eric Winsberg for their help and encouragement and for their participation in the defense of my thesis. I want to take this opportunity to thank the Department of Philosophy and the Graduate School for granting me the University Fellowship for graduate studies at University of South Florida.