Full Idea
The 'satisfaction' relation may be thought of as a function from models, assignments, and formulas to the truth values {true,false}.
Gist of Idea
'Satisfaction' is a function from models, assignments, and formulas to {true,false}
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
Book Reference
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.5
A Reaction
This at least makes clear that satisfaction is not the same as truth. Now you have to understand how Tarski can define truth in terms of satisfaction.