Kripke’s Intrinsic Truths in Boolean Systems

Studies in Logic, Vol. 18, No. 6 (2025): 19–42                             PII: 1674-3202(2025)-06-0019-24

Ming Hsiung, Zhaolong Yuan

Abstract. In Kripke’s theory of truth, the largest intrinsic fixed point—like the least fixed point—is of special theoretical interest among all fixed points. However, for intrinsic yet ungrounded sentences (i.e., those belonging to the largest intrinsic fixed points but not to the least fixed point), only sporadic examples have been provided so far, and a universal criterion for deciding such sentences remains unknown. This paper aims to establish a general criterion for determining intrinsic truth in Boolean systems of self-referential sentences under Kleene’s strong valuation scheme. To achieve this, we first present a known result about the definability of three-valued functions within Kleene’s strong logic. Then, by reducing the problem of determining the fixed points to a calculation problem in propositional logic, we demonstrate a truth-functional characteristic for the intrinsic truths in Boolean systems. We thus find an effective method for constructing intrinsic truths in a first-order language for Peano arithmetic. We also discuss the applicability of our findings to Kleene’s weak valuation scheme.