Is there a symbol for a repeated logical and operation? For example, given a predicate $P$ and a set of variables: $\{x_1,x_2,x_3,x_4,...\}$, is there a single symbol similar to $\sum_{k=1}^n$ or $\bigcap_{k=1}^n$ to denote the following?
$$P(x_1)\land{P}(x_2)\land{P}(x_3)\land{P}(x_4)\land\dots$$

It's called bounded quantifier. The most common notation for what you wish to do is
$$\bigwedge_{k=1}^n P(x_k)$$
produced by \bigwedge_{k=1}^{n} P(x_k), but I've also seen
$$\forall k \in \{1,\ldots, n\}.\ P(x_k).$$
For $\lor$ there is $\bigvee_{k=1}^{n}$ produced by \bigvee_{k=1}^{n}.
I hope this helps $\ddot\smile$
