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.
Categories: math.LO