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'modus ponens'
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| Title of object: |
modus ponens |
| Canonical Name: |
ModusPonens |
| Type: |
Definition |
| Created on: |
2007-03-18 13:52:06 |
| Modified on: |
2007-03-19 15:22:12 |
| Classification: |
msc:03B35, msc:03B05, msc:03B22 |
| Synonyms: |
modus ponens=rule of detachment
modus ponens=detachment |
Preamble:
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
\usepackage{xypic}
\usepackage{pst-plot}
\usepackage{psfrag}
% define commands here
\newtheorem{prop}{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{ex}{Example}
\newcommand{\real}{\mathbb{R}} |
Content:
\emph{Modus ponens} is a rule of inference that is commonly found in many logics where the binary logical connective $\to$ called implication is defined. Informally, it states that
\begin{quote}
\begin{center}
if $a$ and $a\to b$ are formulas, then $b$ is a formula.
\end{center}
\end{quote}
Modus ponens is also called the \emph{rule of detachment}: the formula $b$ can be ``detached'' from the formula $a\to b$ provided that $a$ is also a formula.
An example of this rule is the following: From the premises ``It is raining.''
and ``If it rains, then my laundry will se soaked.'', we may draw the conclusion
``My laundry will be soaked.''.
Two common ways of mathematically denoting modus ponens are the following:
$$\frac{a,a\to b}{b}\qquad\mbox{ or }\qquad (a\mbox{ and }a\to b)\Rightarrow b.$$
One formal way of looking at modus ponens is to define it as a partial function $\Rightarrow: F\times F\to F$, where $F$ is a set of formulas in a language $L$ where a binary operation $\to$ is defined, such that
\begin{enumerate}
\item
$\Rightarrow(x,y)$ is defined whenever $x,y\in F$ and $y=x\to z$ for some $z\in L$, and
\item
when this is the case, $z\in F$ and $\Rightarrow(x,y):=z$;
\item
$\Rightarrow$ is not defined otherwise.
\end{enumerate} |
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