The standard emphasis in formal logical systems is ‘bottom up’. We need to design a fully specified vocabulary and set of construction rules, and then produce complete constructions of formulas, their evaluation, and inferential behavior. This feature makes for explicitness and rigor, but it also leads to system imprisonment. The notions that we define are relative to formal systems. This is one of the reasons why outsiders have so much difficulty grasping logical results: there is usually some parameter relativizing the statement to some formal system, whether first-order logic or some other system.
Obviously this problem propagates through a number of philosophical discussions in which symbolic systems figure, and as van Benthem notes, the only solution for it is to pay close attention to how these systems and natural practice interact. I am very firmly in agreement with both Catarina Dutilh Novaes and Johan van Benthem here. Part of the solution, I think, is always to look to see how things look if you use a different system; but there is more needed.