In this paper, we explore the idea that sets depend on, or are grounded in, their members.  It is said that a set depends on each of its members, and not vice versa.  Members do not depend on the sets that they belong to.  We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members.  We extend the modal truth conditions to include impossible worlds and give a more satisfactory account of  the dependence of a set on its members. Focusing on the case of singletons, we articulate a logical framework in which to evaluate set-theoretic dependence claims, using a normal first-order modal logic.  We show that on this framework the dependence of a singleton on its single members follows from logic alone. However, the converse does not hold.

Anti-exceptionalism about logic takes logical theories to be continuous with scientific theories. Treating logical theories and scientific theories on a par allows one to provide a justification for logic that is compatible with standard criteria for theory choice in the sciences. But if the anti-exceptionalist position understands logical theories in this way, then it becomes open to more general questions about what logical theories are and when logical theories should be treated as equivalent. Though these are familiar questions that arise in philosophical discussions of scientific theories, they are rarely directed toward logical theories. It is the latter question of theoretical equivalence in logic that this paper addresses. To answer this question, while maintaining the spirit of anti-exceptionalism, one should appeal to standard criteria for theoretical equivalence recognized in the philosophy of science. This paper compares two accepted criteria of equivalence for scientific theories-one syntactic and one semanticin the context of logical anti-exceptionalism, and argues that the syntactic approach leads to undesirable consequences. The anti-exceptionalist should therefore take a semantic approach when evaluating whether logical theories, understood as scientific theories, are equivalent. This paper argues for a particular semantic approach, in terms of categorical equivalence, to determine whether logical theories are equivalent.

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Linnebo and Pettigrew (Philos Q 64:267-283, 2014) have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They recognize that this version of structuralism is vulnerable to the well-known problem of non-rigid structures. This paper offers a solution to the problem for this version of structuralism. The solution involves expanding the languages used to describe mathematical structures. We then argue that this solution is philosophically acceptable to those who endorse mathematical structuralism based on Fregean abstraction principles.