  The main focus of this paper is on bisimulation-invariant MSO, and more
particularly on giving a novel model-theoretic approach to it. In model theory,
a model companion of a theory is a first-order description of the class of
models in which all potentially solvable systems of equations and non-equations
have solutions. We show that bisimulation-invariant MSO on trees gives the
model companion for a new temporal logic, "fair CTL", an enrichment of CTL with
local fairness constraints. To achieve this, we give a completeness proof for
the logic fair CTL which combines tableaux and Stone duality, and a fair CTL
encoding of the automata for the modal {\mu}-calculus. Moreover, we also show
that MSO on binary trees is the model companion of binary deterministic fair
CTL.
