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Tarski's definition of logical consequence can take different forms when implemented in second order languages, depending on what counts as a model. In the canonical, or standard, version, a model is just an ordinary structure and the (monadic) second-order variables are meant to range over all subsets of its domain. We discuss the dependence of canonical second-order consequence on set theory and raise doubts on the assumption that canonical consequence is a definite relation.doi:10.1387/theoria.421 doaj:0fd8f87adda94489bca490f8727ddb5d fatcat:efbtngjivjhobkvioh7llx76ya