Here is a solution using the\stackinset command from the stackengine package:
\documentclass{article}
\usepackage{stackengine}
\newcommand{\xor}{\mathrel{\stackinset{c}{}{c}{-0.15ex}{
}{
}}}
\begin{document}
\[ A \xor B\]%
\end{document}

math mode - How to make the exclusive or in LaTex - TeX - LaTeX Stack Exchange
pdftex - How to write sign "|" (or) in LaTeX? - TeX - LaTeX Stack Exchange
symbols - Logical "and" character in TeX (⋀) - TeX - LaTeX Stack Exchange
Symbol for "but not both"
Here is a solution using the\stackinset command from the stackengine package:
\documentclass{article}
\usepackage{stackengine}
\newcommand{\xor}{\mathrel{\stackinset{c}{}{c}{-0.15ex}{
}{
}}}
\begin{document}
\[ A \xor B\]%
\end{document}

You can roll your own:
\documentclass{article}
\usepackage{tikz}
\newcommand*\circledlor{\tikz[baseline=(char.base)]{\node[shape=circle,draw,inner sep=1pt] (char) {\lor};}}
\begin{document}
\(\circledlor\)
\end{document}

If you're trying to typeset "the conditional probability of event E given event H", you should write it -- in math mode -- as
P(E \mid H)
or, better yet,
\Pr(E \mid H)
The macro \Pr is a "math operator" and its output (the letter pair "Pr") is typeset using upright Latin letters. The macro \mid inserts a vertical bar and surrounds it with whitespace, as is appropriate for a relational operator symbol.
In TeX and LaTeX, there is an absolutely fundamental distinction between text mode and math mode. Do familiarize yourself with the basics of LaTeX coding. A good starting point is the document A (Not So) Short Introduction to LaTeX, which is available in about two dozen languages.
A full MWE (minimum working example):

\documentclass{article}
\begin{document}
or, better yet,
.
\end{document}
You can use either | or \mid. Using \mid is better as this adds the appropriate amount of spacing around the operator.
More importantly, you should enclose all of these in to put them in math-mode. You will get an error if you use \ell or mid outside of math-mode.
Compare:

The full code:
\documentclass{article}
\usepackage{}
\begin{document}
text: P(E
H) or P(E|H) or P(E
H)
math-mode:
or
or
\end{document}
$((a\implies b) \land (c\implies a)) \implies (c \implies b)$
\land and \wedge are synonymous.
In case you don't like the default \wedge symbol, you can try the one provided by the mathabx package:

EDIT: Oh the shame, not even an example? Based on egreg's solution:
\documentclass{article}
\usepackage{amsmath}
\usepackage{mathabx}
\begin{document}
% from egreg's solution
\end{document}
Note that the use of mathabx is just an aesthetic choice. Since you are writing logic, egreg's approach is the way to go, since it's more semantic.
The question basically amounts to A or B but not both and I'm wondering if there is some other symbol you need besides the normal or symbol?
This is what I want to do. I did it with $\sum$ and $\int$, but they should be done with different methods. How can I do it differently?