Informally, any suppressed premise or background framework of thought necessary to make an argument valid, or a position tenable. More formally, a presupposition has been defined as a proposition whose truth is necessary for either the truth or the falsity of another statement. Thus if p presupposes q, q must be true for p to be either true or false. In the theory of knowledge of Collingwood, any propositions capable of truth or falsity stand on a bed of ‘absolute presuppositions’ which are not properly capable of truth or falsity, since a system of thought will contain no way of approaching such a question (a similar idea was later voiced by Wittgenstein in his work On Certainty).
It was influentially suggested by Strawson, in opposition to Russell's theory of definite descriptions, that ‘There exists a King of France’ is a presupposition of ‘The King of France is bald’, the latter being neither true nor false, if there is no King of France. It is, however, a little unclear whether the idea is that no statement at all is made in such a case, or whether a statement is made, but fails of being either true or false. The former option preserves classical logic, since we can still say that every statement is either true or false, but the latter does not, since in classical logic the law of bivalence holds, and ensures that nothing at all is presupposed for any proposition to be true or false. The introduction of presupposition therefore means that either a third truth value is found, ‘intermediate’ between truth and falsity, or that classical logic is preserved, but it is impossible to tell whether a particular sentence expresses a proposition that is a candidate for truth and falsity, without knowing more than the formation rules of the language. Each suggestion carries costs, and there is some consensus that at least where definite descriptions are involved, examples like the one given are equally well handled by regarding the overall sentence as false when the existence claim fails, and explaining the data that Strawson relied upon as the effects of implicature.