First-order logic is currently by far the most important formal logic, even up to the extent of having the status of the one and only canonical logic. This state of affairs has, however, been criticized by, e.g., Hintikka. According to Hintikka, first-order logic is missing a crucial operator that enables one to declare independence patterns between variables. Independence declarations do occur in the practice of mathematics, but there exists a whole range of other operators equally (or more) common in the everyday language of mathematics that are not available in first-order logic. We investigate a novel logic based on two such operators. We show that the resulting language exactly captures the expressive power of Turing machines, thereby being maximally expressive within the realm of systematic computability. The extra operators, or logical constants, of the logic, are natural from the point of view of natural language. However, the self-referential flavor of the operators lead to some intriguing puzzles concerning compositionality.