  We prove that the satisfaction relation $\mathcal{N}\models\varphi[\vec a]$
of first-order logic is not absolute between models of set theory having the
structure $\mathcal{N}$ and the formulas $\varphi$ all in common. Two models of
set theory can have the same natural numbers, for example, and the same
standard model of arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$,
yet disagree on their theories of arithmetic truth; two models of set theory
can have the same natural numbers and the same arithmetic truths, yet disagree
on their truths-about-truth, at any desired level of the iterated
truth-predicate hierarchy; two models of set theory can have the same natural
numbers and the same reals, yet disagree on projective truth; two models of set
theory can have the same $\langle H_{\omega_2},{\in}\rangle$ or the same
rank-initial segment $\langle V_\delta,{\in}\rangle$, yet disagree on which
assertions are true in these structures.
  On the basis of these mathematical results, we argue that a philosophical
commitment to the determinateness of the theory of truth for a structure cannot
be seen as a consequence solely of the determinateness of the structure in
which that truth resides. The determinate nature of arithmetic truth, for
example, is not a consequence of the determinate nature of the arithmetic
structure $\mathbb{N}=\{0,1,2,\ldots\}$ itself, but rather, we argue, is an
additional higher-order commitment requiring its own analysis and
justification.
