In my posts to date on Tarski, I have noted that he argued that natural languages were inconsistent. This claim needs some explanation as it does not simply mean that inconsistent statements can be stated in natural languages, or that people can hold inconsistent beliefs.
Consistency in this context is a defined property of logical or mathematical systems. Such a system is consistent if there is no statement in the system that is both demonstrably true and demonstrably not true. In claiming that natural languages are not consistent, Tarski assumes that natural languages have (at least partially) a logical structure that allows statements to be demonstrated to be true or not true.
This assumption is highly plausible. For example, if you understand the meaning of the connectives “If … then …” in English, then you almost certainly have to accept the Modus Ponens rule of inference. You can go through similar exercises with connectives like “and” and “or” and pretty soon you get to a set of inference rules that would determine a full system of logic.
If English has an inbuilt logic, then it makes sense that natural languages like English must be (logically) consistent or inconsistent.