Full Idea
Compactness is a virtue - it makes the consequence relation more manageable; but it is also a limitation - it limits the expressive power of the logic.
Gist of Idea
Compactness makes consequence manageable, but restricts expressive power
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Reference
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.44
A Reaction
The major limitation is that wholly infinite proofs are not permitted, as in Idea 10977.
Related Idea
Idea 10977 Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite) [Read]