Forcing a figure not to skip onto the next page

Question

When I include the following figure it doesn't go directly below the text after the semicolon but skips onto the next page (it does however appear below the semicolon). The image is the size I want it to be. I just need to know how to force it to appear on the same page without changing the size. How can I force the figure not only to appear below the semicolon in the following example, but force it not to skip onto the next page?

\documentclass[12pt,a4paper]{article} 
\usepackage{float}  
\usepackage{graphicx}

\begin{document} 
  The sequent calculus proof of this uses contraction and is therefore not derivable in linear logic (unless special modalities are used):
\begin{figure}[H]
\includegraphics[width=10cm,scale=18]{prooftreeseqmg2.pdf}
\centering
\end{figure}

\end{document}

Answer 1

Your image is a displayed formula (that could by directly produced with LaTeX, I believe), so treat it as a displayed formula:

\documentclass[12pt,a4paper]{article} 
\usepackage{graphicx}

\begin{document} 
The sequent calculus proof of this uses contraction and is therefore
not derivable in linear logic (unless special modalities are used):
\[
\includegraphics[width=10cm]{prooftreeseqmg2.pdf}
\]

\end{document}

There's no rule that says \includegraphics should be in a figure environment. To the contrary, it can go everywhere: as far as TeX is concerned, the image is just like a big box.

Note that specifying scale=18 and width=10cm is redundant: just use the width; it's probably better to say something like

\includegraphics[width=.7\columnwidth]{prooftreeseqmg2.pdf}

(adjust the factor to your liking).

If the PDF image has white margins, use trim:

\includegraphics[
  clip,
  trim=1cm 8cm 1cm 1cm,
  width=.7\columnwidth,
]{prooftreeseqmg2.pdf}

experimenting what dimensions are the good ones. The trimming lengths are in the order “left–bottom–right—top”.

Using the image you posted, here's the example:

\documentclass[12pt,a4paper]{article}
\usepackage{graphicx}

\begin{document}
The sequent calculus proof of this uses contraction and is therefore
not derivable in linear logic (unless special modalities are used):
\[
\fbox{\includegraphics[
  clip,
  trim=6cm 41cm 3cm 6cm,
  width=10cm]{prooftreeseqmg2}}
\]
Some text after the display.
\end{document}

Remove the \fbox that I just added for debugging.

Answer 2

To illustrate, what we (Ulrike Fischer, LaRiFaRi. Egreg and me) said in comment:

This I obtained width slightly modified your code (I don't use float package, for image I use example-image:

\documentclass[12pt,a4paper]{article}
\usepackage{graphicx,float}
%\usepackage{float}% it is not necessary, 
                   % avoid placement [H] if it is possible

   \begin{document}
The sequent calculus proof of this uses contraction and is therefore not derivable in linear logic (unless special modalities are used):
\begin{figure}[h]
\centering%usually is on begining of environment where ti should act
\includegraphics[width=10cm,scale=18]{example-image}
\end{figure}
And one more figure, with `width=10cm, without `scale` option:
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{example-image}% instead example image I 
                                           % include snip of figure 
                                           %as you updated in your question
\end{figure}

But use only scale: it can not be fit into page ...
\begin{figure}
\centering
\includegraphics[scale=18]{example-image}
\end{figure}
    \end{document}

Test of this code will show, what happens, when the figure is to large ...

Edit: Just to show, that comments of OP is not valid, I cut it from full page with snipping tool image, which she/he provide in upgraded question:

and include it into my MWE. I can be clearly seen, that this image is in the same page. If this is not the case at OP, then only what I can conclude, that original image is bigger as it is or in the real case is something what is not shown in my or egreg answer.