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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: |
2008-06-23 12:15:29 |
| Classification: |
msc:03B35, msc:03B05, msc:03B22 |
| Synonyms: |
modus ponens=rule of detachment
modus ponens=detachment
modus ponens=modus ponendo ponens |
Preamble:
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\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:
\textbf{Modus ponens} is a rule of inference that is commonly found in many logics where the binary logical connective $\rightarrow$ or the binary logical relation $\Rightarrow$ called logical implication are defined. Informally, it states that
\begin{center}
If $a$ and $a \Rightarrow b$ are theorems, then $b$ is a theorem.
\end{center}
Modus ponens is also called the \emph{rule of detachment}: the theorem $b$ can be ``detached'' from the theorem $a \Rightarrow b$ provided that $a$ is also a theorem.
An example of this rule is the following: From the premisses ``It is raining'',
and ``If it rains, then my laundry will be soaked'', we may draw the conclusion
``My laundry will be soaked''.
Two common ways of mathematically denoting modus ponens are the following:
$$\frac{a, a \Rightarrow b}{b} \qquad \mbox{ or } \qquad a, a \Rightarrow b \vdash b.$$
One formal way of looking at modus ponens is to define it as a partial function $\vdash : F \times F \to F,$ where $F$ is a set of formulas in a language $L$ where a binary operation $\Rightarrow$ is defined, such that
\begin{enumerate}
\item
$\vdash(x, y)$ is defined whenever $x, y \in F$ and $y \equiv (x \Rightarrow z)$ for some $z \in L$, and
\item
when this is the case, $z \in F$ and $\vdash(x, y) := z$;
\item
$\vdash$ is not defined otherwise.
\end{enumerate} |
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