SATISFACTION IS NOT ABSOLUTE

doi:10.1017/S1755020325100907

SATISFACTION IS NOT ABSOLUTE

2025-09-01

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DOI：10.1017/S1755020325100907

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We prove that the satisfaction relation N |= phi[a(->)] of first-order logic is not absolute between models of set theory having the structure N and the formulas phi all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic (N, +, , 0, 1, <), 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 (H-omega 2, is an element of) or the same rank-initial segment (V-delta, is an element of), 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 N = { 0, 1, 2, ... } itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.

Keywords:

truth

satisfaction class

set theory

SATISFACTION IS NOT ABSOLUTE

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