5.2: Argumentos estándar
- Page ID
- 118382
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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
argumento estándar con forma
\ (\ comenzar {alineado}
&p\ fila derecha q\\
&q\ fila derecha r\\
&\ hline p\ fila derecha r
\ final {alineado}\)
| \(p \rightarrow q\) |
| \(p\) |
| \(q\) |
Verificar la validez del argumento estándar modus ponens.
Solución
Verificar la validez asegurando que cada fila en la tabla de verdad con premisas todas verdaderas también tenga la conclusión verdadera.
|
(pr) |
c) |
(pr) |
|
|
\(p\) |
\(q\) |
\(p \rightarrow q\) |
|
|
\(T\) |
\(T\) |
\(T\) |
\(\checkmark\)argumento es válido |
|
\(T\) |
\(F\) |
\(F\) |
|
|
\(F\) |
\(T\) |
\(\ast\) |
|
|
\(F\) |
\(F\) |
\(\ast\) |
El argumento en el Ejemplo 5.1.2 tiene forma de modus ponens. Por lo que es válido, aunque la primera premisa y la conclusión no sean realmente ciertas.
5.2.2 Modus tollens
argumento estándar con forma
\(\begin{aligned} &p \rightarrow q \\ &\neg q \\ & \hline \neg p \end{aligned}\)
Verificar la validez del argumento estándar modus tollens.
Solución
Verificar la validez asegurando que cada fila en la tabla de verdad con premisas todas verdaderas también tenga la conclusión verdadera.
|
(pr) |
(pr) |
c) |
|||
|
\(p\) |
\(q\) |
\(p \rightarrow q\) |
\(\neg q\) |
\(\neg p\) |
|
|
\(T\) |
\(T\) |
\(T\) |
\(F\) |
\(\ast\) |
|
|
\(T\) |
\(F\) |
\(F\) |
\(\ast\) |
\(\ast\) |
|
|
\(F\) |
\(T\) |
\(T\) |
\(F\) |
\(\ast\) |
|
|
\(F\) |
\(F\) |
\(T\) |
\(T\) |
\(T\) |
\(\checkmark\)argumento es válido |
El argumento en el Ejemplo 5.1.1 tiene la forma modus tollens.
5.2.3 Ley del silogismo
argumento estándar con forma
\(\begin{aligned} &p \rightarrow q\\ &q \rightarrow r \\ &\hline p \rightarrow r \end{aligned}\)
Ya verificamos que la Ley del Silogismo es válida en el Ejemplo Trabajado 5.1.4.
La Ley del Silogismo puede extenderse a cadenas de condicionales de longitud arbitraria (finita).
argumento estándar con forma
\(\begin{aligned} &p_1 \rightarrow p_2\\ &p_2 \rightarrow p_3 \\&\vdots \phantom{\rightarrow p_n} \\ &p_{n-1} \rightarrow p_n \\ & \hline p_1 \rightarrow p_n \end{aligned}\)
Verificaremos que la Ley extendida del silogismo es un argumento válido utilizando la inducción matemática en la Sección 7.2.
\(\begin{aligned} &\text{If I don't study hard this term, I won't master the course material.} \\ &\text{If I don't master the course material, I will fail the course.} \\ &\text{If I fail the course, I will have to take it again next year.} \\ &\text{If I take it again next year, I will have to study harder.} \\ &\hline \text{Therefore, if I don't study hard this term, I will have to study harder next year.} \end{aligned}\)