\Rightarrow vs. \implies, and "does not imply" symbol

Question

Is it better to use \Rightarrow or \implies to symbolize logical implications? Why?

Also, if I write \not \Rightarrow, I get a symbol that means "does not imply." But if I type \not \implies, it doesn't look right. How do I get the corresponding "does not imply" symbol?

Answer 1

To answer your first question, you should use \implies, not \Rightarrow. \Rightarrow is far too small to give a readable result and is not spaced properly. Knuth specially defined \iff to be used for equivalence and \implies is the same but for implication (from the amsmath package). An implication is not a relation like > and, therefore, needs to be spaced according to how it is used. Two thick spaces (which are about an en-space) precede and follow an implication because it's more important than a relation. (Basic rule of math spacing: the more important an operator, the wider the space around it). Compare the readability of the following formulas:

The first line uses \Rightarrow and is the least legible because the main part of the formula (the implication) is difficult to identify. The second line uses \Longrightarrow and is better, but there's not enough space to set the arrow apart from the surrounding symbols (the two inequalities are spaced as much as the arrow). Finally, in the third line (which uses \implies), the additional space highlights the arrow from the rest and so improves the readability of the formula.

Concerning your second question, the simplest way to negate nearly any symbol whatever its length is to use \centernot from the centernot package:

\documentclass{article}

\usepackage{amsmath}
\usepackage{centernot}

\begin{document}

$A \centernot\implies B$

\end{document}

Of course, wrapping \centernot\implies inside a \notimplies macro like Werner did is a good idea.

Answer 2

I'll answer the second part of your question, since the first seems more subjective.

You can use

\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\newcommand{\notimplies}{%
  \mathrel{{\ooalign{\hidewidth$\not\phantom{=}$\hidewidth\cr$\implies$}}}}

to represent "does not imply". This provides \notimplies that sets a relational symbol with \not overlaid \implies (technically, the other way around):

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\newcommand{\notimplies}{%
  \mathrel{{\ooalign{\hidewidth$\not\phantom{=}$\hidewidth\cr$\implies$}}}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\Y}{\mathcal{Y}}
\begin{document}
\[
  \X\Rightarrow\Y \quad \X\not\Rightarrow\Y \qquad
  \X\implies\Y \quad \X\notimplies\Y
\]
\end{document}

Of course, from this point the spacing can also be modified, if needed.

For a quick lesson on \ooalign, see \subseteq + \circ as a single symbol (“open subset”).

Answer 3

It's not perfect, but I use

\newcommand{\notimplies}{\;\not\!\!\!\implies}

which looks like this:

Answer 4

The conventions in other parts of mathematics may differ, but in logic texts, implication is virtually never written with long arrows. Implication is most commonly denoted by \rightarrow (= \to) or \supset, occasionally \Rightarrow. Long arrows (\longrightarrow and \Longrightarrow) are used for sequent arrows.

Answer 5

It's my answer of your second question:

$A \rlap{\(\quad\not\)}\implies B$

or

$$
A \rlap{$\quad\not$}\implies B
$$

whick looks like this:

Perhaps, it's not the best solution, but it's short and doesn't need external packages.