Full Idea
We must recognise that first-order classical logic is inadequate to describe all valid consequences, that is, all cases in which it is impossible for the premisses to be true and the conclusion false.
Gist of Idea
Not all validity is captured in first-order logic
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Reference
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.54
A Reaction
This is despite the fact that first-order logic is 'complete', in the sense that its own truths are all provable.