Is there notation for the logical and/or of lots of items? You can add lots of things together using summation notation $$\sum_{n=1}^5 n = 15$$
Is there a similar operator for logical "and" and "or"?
I considered $\exists$/$\forall$, but that doesn't allow me to easily specify how to determine the answer in some cases.
 A: Interestingly, you can quite easily use a statement like $$(\forall i\in I)(\exists a_i\in A_i)$$ to get repeated "there exist" statements and similarly you could use $$(\forall i\in I)(\forall a_i \in A_i)$$ to get repeated "for all" statements. You would just need to define the set $I$ and the sets $A_i$. Note that each $a_i$ is not necessarily unique, and in the second statement, $a_i$ is unique only when $|A_i|=1$.
A: Use \bigvee and \bigwedge ,
$$  \bigvee_{i=1}^{100} X_i  $$
and
$$  \bigwedge_{i \in I} X_i  \text{,}  $$
respectively.
See also What is the meaning of  $\bigvee$ (bigvee) operator
Be aware that these are also used for meets and joins (lattice theory).  Depending on your context, it could be a good idea to explicitly introduce this notation.
A: You mentioned in a comment that this came up in an algorithmic context. In programming terms, the general expression for this is reduce(operation, iterable) (of course, this implies that you're dealing with an associative binary operator; reduce(mean, iterable) isn't going to get the mean of the iterable), e.g. reduce(or, (f(_) for _ in range(k))). There's also any(f(_) for _ in range(k)). Those are the Python syntax, but if you're doing psuedo code, this should be clear even to CS people who aren't familiar with Python. In mathematical logic terms, it would be $\exists n:f(n)$.
