10: Deductive Reasoning
- Page ID
- 22019
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In any argument, the arguer intends the reasons to adequately support the conclusion, to imply it. For example, if you want people to conclude that your product is the best buy for them, you ought to give them some good reasons. What makes the reasons good enough is that they imply that your product is the best buy for them. This chapter explores how the notion of implication lies at the heart of logical reasoning. There are two kinds of implication that can be involved—deductive or inductive. This chapter focuses on deductive arguments, and the main goal of a deductive argument is to satisfy the standard of being deductively valid. We will define “deductive validity” very soon.
- 10.1: Implying with Certainty vs. with Probability
- This page covers deductive and inductive arguments, explaining how conclusions relate to premises. It distinguishes between deductively valid arguments, which guarantee true conclusions from true premises, and inductively strong arguments, which suggest probable conclusions. The text also explores the concept of certainty in logical relationships, illustrating how one statement can entail another and emphasizing the need for clarity and specificity in logical reasoning to avoid ambiguity.
- 10.2: Distinguishing Deduction from Induction
- This page outlines the evaluation of arguments, focusing on validity, soundness, and inductive strength. It distinguishes between deductive and inductive reasoning, explaining that a strong inductive argument differs from a deductively sound one. Common misunderstandings of terms are addressed, and real-world examples illustrate the complexities of inductive reasoning.
- 10.3: Review of Major Points
- This page explores the concept of implication, categorizing it into certain and probable types associated with inconsistency and improbability. It presents three evaluation standards for arguments—deductive validity, deductive soundness, and inductive strength—highlighting that inductive strength is a matter of degree. The term "proof" is defined as ambiguous, differentiating between mathematical proofs as deductively sound and scientific proofs as strong inductive arguments.
- 10.4: Glossary
- This page outlines the distinctions between deductive and inductive arguments, emphasizing that deductive arguments seek validity and soundness, where soundness ensures true premises. It clarifies the concepts of certainty versus probability in implications and highlights that inductive arguments strive for high probability in their conclusions.
- 10.5: Exercises
- This page contains exercises on reasoning and implications, stressing the careful evaluation of arguments without excessive application of the principle of charity. It explores deductive validity, soundness, and the nuances among certainty, probability, and ambiguity. Key scenarios like President Kennedy's assassination and bacterial reproduction serve as thought experiments to enhance analytical skills.