9: Consistency and Inconsistency
- Page ID
- 22012
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Your goal is to maintain the consistency of your beliefs as you add new beliefs. This chapter is devoted to exploring how to achieve this goal. All of us want to remove any inconsistent beliefs we have, because if we don’t then we are accepting something impossible. We aren’t like the red queen in Alice in Wonderland who said she could believe six impossible things before breakfast.
- 9.1: Recognizing Inconsistency and Contradiction
- This page explores the concepts of inconsistency and logical consistency in statements. It explains how groups of statements can be inconsistent if they cannot be true together and distinguishes between logical and factual inconsistencies. The discussion includes examples and emphasizes the role of context and language interpretation in assessing consistency, particularly highlighting ambiguities like the term "some.
- 9.2: Identifying Self-Contradictions and Oxymorons
- This page examines self-contradiction, focusing on logical inconsistencies in statements and distinguishing them from falsehoods. It explores oxymorons and discusses their effects on credibility, using examples from politics and film dialogues to illustrate these concepts. The importance of clear and consistent communication is a central theme, reinforced by various examples, including comments on capital punishment and humorous phrases.
- 9.3: Inconsistency with Presuppositions
- This page explores presupposition in language, using humor to illustrate how statements rely on unacknowledged assumptions. It highlights how comedians exploit these contradictions, showcasing examples such as Gandhi's comment on Western civilization and a political scandal. The page concludes with a story of Harpy, a boy whose claim of divine instruction to steal exemplifies the absurdity of conflicting reasoning with societal norms.
- 9.4: Refuting General Statements by Finding Counterexamples
- This page explores generalizations, differentiating between universal and statistical types, emphasizing the role of counterexamples and context. It details scientific testing, highlighting how hypotheses are verified through observation and experimentation, with refutation being key to scientific advancement. Additionally, it includes exercises on general claims and scientific methods, using the historical discovery of Uranus and Neptune as a case study.
- 9.5: Resolving Inconsistencies
- This page emphasizes the necessity of clarity in communication and moral principles to avoid misunderstandings, illustrated by a Peanuts comic. It addresses ethical dilemmas from conflicting morals and proposes revisions to resolve these inconsistencies. Additionally, it stresses the careful integration of new information to avoid contradictions, asserting that belief is not absolute and highlighting the importance of evaluating all evidence when drawing conclusions.
- 9.6: Review of Major Points
- This page examines inconsistency and its role in assessing improbability, highlighting contradictions in statements, self-contradictory phrases, and speaker cues. It explains the use of counterexamples to challenge general claims and advises discarding unsupported information to restore consistency. The importance of gathering more information is stressed, alongside the need to recognize counterevidence and hold balanced beliefs.
- 9.7: Glossary
- This page explores logical concepts such as contradictions (oxymorons), contrary statements (both can be false but not true), and counterexamples that disprove general statements. It also covers generalizations about groups, hypotheses as claims, logically inconsistent statements, inherent presuppositions, successful refutations, and self-contradictions in statements.
- 9.8: Exercises
- This page delves into inconsistency in statements through examples, emphasizing the principle of charity in interpretations and the importance of identifying contradictions in beliefs and actions. It encourages critical thinking by evaluating logical relationships among claims and highlights that at least one statement must be false in inconsistent scenarios.