5.2: Standard Arguments
- Page ID
- 83420
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)5.2.1: Modus Ponens
standard argument with form
\(\begin{aligned}
&p \rightarrow q \\
&q \rightarrow r \\
&\hline p \rightarrow r
\end{aligned}\)
| \(p \rightarrow q\) |
| \(p\) |
| \(q\) |
Verify the validity of the modus ponens standard argument.
Solution
Verify the validity by ensuring that each row in the truth table with premises all true also has the conclusion true.
|
(pr) |
(c) |
(pr) |
|
|
\(p\) |
\(q\) |
\(p \rightarrow q\) |
|
|
\(T\) |
\(T\) |
\(T\) |
\(\checkmark\) argument is valid |
|
\(T\) |
\(F\) |
\(F\) |
|
|
\(F\) |
\(T\) |
\(\ast\) |
|
|
\(F\) |
\(F\) |
\(\ast\) |
The argument in Example 5.1.2 has modus ponens form. So it is valid, even though the first premise and the conclusion are not actually true.
5.2.2 Modus tollens
standard argument with form
\(\begin{aligned} &p \rightarrow q \\ &\neg q \\ & \hline \neg p \end{aligned}\)
Verify the validity of the modus tollens standard argument.
Solution
Verify the validity by ensuring that each row in the truth table with premises all true also has the conclusion true.
|
(pr) |
(pr) |
(c) |
|||
|
\(p\) |
\(q\) |
\(p \rightarrow q\) |
\(\neg q\) |
\(\neg p\) |
|
|
\(T\) |
\(T\) |
\(T\) |
\(F\) |
\(\ast\) |
|
|
\(T\) |
\(F\) |
\(F\) |
\(\ast\) |
\(\ast\) |
|
|
\(F\) |
\(T\) |
\(T\) |
\(F\) |
\(\ast\) |
|
|
\(F\) |
\(F\) |
\(T\) |
\(T\) |
\(T\) |
\(\checkmark\) argument is valid |
The argument in Example 5.1.1 has modus tollens form.
5.2.3 Law of Syllogism
standard argument with form
\(\begin{aligned} &p \rightarrow q\\ &q \rightarrow r \\ &\hline p \rightarrow r \end{aligned}\)
We already verified that the Law of Syllogism is valid in Worked Example 5.1.4.
The Law of Syllogism may be extended to chains of conditionals of arbitrary (finite) length.
standard argument with form
\(\begin{aligned} &p_1 \rightarrow p_2\\ &p_2 \rightarrow p_3 \\&\vdots \phantom{\rightarrow p_n} \\ &p_{n-1} \rightarrow p_n \\ & \hline p_1 \rightarrow p_n \end{aligned}\)
We will verify that the extended Law of Syllogism is a valid argument using mathematical induction in Section 7.2.
\(\begin{aligned} &\text{If I don't study hard this term, I won't master the course material.} \\ &\text{If I don't master the course material, I will fail the course.} \\ &\text{If I fail the course, I will have to take it again next year.} \\ &\text{If I take it again next year, I will have to study harder.} \\ &\hline \text{Therefore, if I don't study hard this term, I will have to study harder next year.} \end{aligned}\)