Full Idea
Being 'compact' means that if we have an inconsistency or an entailment which holds just because of the truth-functors and quantifiers involved, then it is always due to a finite number of the propositions in question.
Gist of Idea
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
Source
David Bostock (Intermediate Logic [1997], 4.8)
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.183
A Reaction
Bostock says this is surprising, given the examples 'a is not a parent of a parent of b...' etc, where an infinity seems to establish 'a is not an ancestor of b'. The point, though, is that this truth doesn't just depend on truth-functors and quantifiers.