13.6: Evaluating Arguments
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Evaluating Arguments
Last, let us turn to the most important topic this chapter will take up: argument evaluation. To evaluate an argument is to decide whether it is good or bad. We have an intuitive ability to evaluate arguments—we can usually distinguish good arguments from bad ones just by looking at them. This native ability is not, however, infallible; in fact, there are certain contexts and kinds of cases where we tend to make mistakes. Thus it is important to ask: what is the difference between a good argument and a bad one? Put otherwise, we need to know what makes a good argument, good, and a bad argument, bad. Let us start with an example.
Ex. 9:
Premise: The largest city in the U.S. is located in Nebraska.
Premise: New York City is the largest city in the U.S.
Conclusion: So, New York City is located in Nebraska.
Clearly this is a bad argument. The problem is that one of its premises is false—we know that the largest city in the U.S. is not in the state of Nebraska. This example shows one way in which an argument can be bad: when it has false premises. Let us say the following:
An argument is factually correct when (and only when) all of its premises are true. It is factually incorrect otherwise.
Thus, the argument in Ex. 9 is factually incorrect because not all of its premises are true. In addition, whether an argument is factually correct or not is solely a matter of whether the premises are true—an argument with all true premises but a false conclusion is still factually correct. Factual correctness is not, however, the only feature of an argument relevant to its evaluation. Consider the following case.
Ex. 10:
Premise: Selena passed her driver’s license exam.
Conclusion: So, Selena will pass her calculus exam.
Let us say that it is true that Selena passed her driver’s license exam. Even so, clearly this is a bad argument. This is a poor argument because the premise does not support the truth of the conclusion. That is, the premise, though true, does not give us good or sufficient reason to believe the conclusion is true. We will refer to this feature of arguments as logical strength and say:
An argument is logically strong when (and only when) the premises—if true—provide strong support for the truth of the conclusion. An argument is logically weak otherwise.
It is crucial to see that these features of arguments are independent of one another. An argument may be factually incorrect, but logically strong (see, e.g. Ex. 9), factually correct, but logically weak (Ex. 10), both correct and strong, or both incorrect and weak. In light of these distinctions we can distinguish good arguments from bad ones in the following way. Let us say that:
An argument is good, henceforth sound, when (and only when) it is both factually correct and logically strong. An argument is unsound otherwise.
Ideally, all of our reasoning would proceed by means of sound arguments. However, we are always working with limited information, and this means that we all sometimes endorse unsound arguments. Nevertheless, we can take steps to limit these kinds of errors.