  G\"odel'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 \emph{implicit} assumptions. This notion of \emph{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, \L elyk and Nicolai 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.
