Abstract Gödel’s Incompleteness Theorems suggest that no single formal system can capture the entirety of one’s mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those implicit assumptions. This notion of implicit commitment motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn’t been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, we carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system $S$, it can be derived that a uniform reflection principle for $S$—stating that all numerical instances of theorems of $S$ are true—must be contained in $S$’s implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language.

Abstract We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal {M}$ , or an axiomatization S thereof, we find a modal logic M such that a modal sentence $\varphi $ is a theorem of M if and only if the sentence $\varphi ^*$ obtained by translating the modal operator with the truth predicate is true in $\mathcal {M}$ or a theorem of S under all such translations. To this end, we introduce a novel version of possible worlds semantics featuring both classical and nonclassical worlds and establish the completeness of a family of noncongruent modal logics whose internal logic is nonclassical with respect to this semantics.

Iterated reflection principles have been employed extensively to unfold epistemic commitments that are incurred by accepting a mathematical theory. Recently this has been applied to theories of truth. The idea is to start with a collection of Tarski-biconditionals and arrive by iterated reflection at strong compositional truth theories. In the context of classical logic, it is incoherent to adopt an initial truth theory in which |$A$| and ‘|$A$| is truen’ are inter-derivable. In this article, we show how in the context of a weaker logic, which we call Basic De Morgan Logic, we can coherently start with such a fully disquotational truth theory and arrive at a strong compositional truth theory by applying a natural uniform reflection principle a finite number of times.