Is English consequence compact?

By mimicking the standard definition for a formal language, we define what it is for a natural language to be compact. We set out a valid English argument none of whose finite subarguments is valid. We consider one by one objections to the argument's logical validity and then dismiss them. The conclusion is that English—and any other language with the capacity to express the argument—is not compact. This rules out a large class of logics as the correct foundational one, for example any sound and complete logic, and in particular first-order logic. The correct foundational logic is not compact.