Logical "and" character in TeX (⋀)

$((a\implies b) \land (c\implies a)) \implies (c \implies b)$

\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
$((a \implies b) \land (c \implies a))  (c \implies  b)$
\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.

use $\wedge$ or $\bigwedge$