When we defined the interpretation of propositions, we insisted that a proposition has a truth value that is either true or false. We also noted that propositions build only a part of all the strings which are syntactically correct among other strings which can be formulated in a given formal language $L$. But how about all the other strings, for which we cannot assign a truth value? Important types of such statements are paradoxes.
Let a formal language $L$ be given, in which the valuation function $[]_I$ of $PL0$ (law of excluded middle) holds. A paradox is a string $s\in L$, for which the interpretation $[[s]]_I=undefined,$ i.e. for which it is not possible to assign a truth value, and which apparently contradicts itself.
Motivations: 2 3