Count the number of fish caught in total if we are given the number of fishermen who caught at least $n$ fish. 
Some fishermen caught some fish. No one caught more than 20 fish. $a_i$ is the number of fishermen who caught at least $i$ fish. How many fish were caught?

So my guess is that the number of fish caught has to be $$ a_1+a_2+\ldots + a_{20}$$
I reason that if a fisher man caught $n$ fish then he will appear in the tally $n$ times. 
Is this logic correct?

Yes, you are right. Here is a different (more concrete, I guess) way of seeing it:
Put out barrels numbered 1 through 20 and tell each fisherman to put their first fish in barrel 1, their second in barrel 2, and so on. Each fish gets put in a barrel, and the number of fish is therefore necessarily equal to the sum of fish in each barrel. However, we also have that $a_i$ is the number of fish in barrel $i$, since any fisherman who got at least $i$ fish will put one fish into barrel $i$. Thus we get your result.
