 The model-theoretic definition is one of the most successful mathematical explications of a philosophical concept to date. It promises to capture both the necessity of logical consequence—by looking at truth over all models, and the formality of logical consequence—by varying the interpretations of the nonlogical vocabulary across models: an argument is valid no matter what the nonlogical vocabulary means. Yet, models are just sets, which are merely mathematical objects. How do they account for the range of possibilities, or circumstances required? John Etchemendy (1990) offers two perspectives for understanding models. On the representational approach, each model is taken to represent a possible world. If an argument preserves truth over models, we are then guaranteed that it preserves truth over possible worlds, and if we accept the identification of necessity with truth in all possible worlds, we have the necessary truth preservation of logical consequence. The problem with this approach is that it identifies logical consequence with metaphysical consequence, and it gives no account of the formality of logical consequence. On the representational approach, there is no basis for a distinction between the logical and the nonlogical vocabulary, and there is no explanation of why the interpretations of the nonlogical vocabulary are maximally varied. The second perspective on models is afforded by the interpretational approach, by which each model assigns extensions to the nonlogical vocabulary from the actual world: what varies between models is not the world depicted but the meaning of the terms. Here, the worry is that necessity isn’t captured. For instance, on the usual division of the vocabulary into logical and nonlogical, identity is considered a logical term, and can be used to form statements about the cardinality of the domain (e.g., ‘‘there are at least two things’’) which are true under every reinterpretation, but perhaps are not necessarily true. On this approach, there is no basis for considering models with domains other than the universe of what actually exists, and specifically, there is no explanation of model theory’s use of domains of different sizes. Each approach, as described here, is flawed with respect to our analysis of logical consequence as necessary and formal. The interpretational approach, by looking only at the actual world fails to account for necessity, and the representational approach fails to account for formality (for details, see Etchemendy 1990, Sher 1996, and Shapiro 1998, and for refinements see Etchemendy 2008). A possible response to Etchemendy would be to blend the representational and the interpretational perspectives, viewing each model as representing a possible world under a re-interpretation of the nonlogical vocabulary (Shapiro 1998, see also Sher 1996 and Hanson 1997 for alternative responses).