Full Idea
Tarski discovered how to give a compositional semantics for predicate calculus, defining truth in terms of satisfaction, and showing how satisfaction for a complicated formula depends on satisfaction of the simple subformulas.
Gist of Idea
Tarski built a compositional semantics for predicate logic, from dependent satisfactions
Source
report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Vann McGee - Logical Consequence 4
Book Reference
'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.37
A Reaction
The problem was that the subformulas may contain free variables, and thus not be sentences with truth values. 'Satisfaction' can handle this, where 'truth' cannot (I think).